Electrical Network Load Monitoring by Viivi Jokk

Electrical Network Load Monitoring Mati Meldorf

Tallinn 2010

Cover design: Viivi Jokk ÂŠ Mati Meldorf, 2010 ISBN 978-9949-23-031-0

Foreword For electrical network operation control and planning, it is not only necessary to forecast the network load, but also to analyse and simulate the load under different conditions (low or high outdoor temperature, probable load deviations, different trend scenarios, etc.). Loads are treated with a mathematical model, which considers regular changes of the load, randomness, temperature dependency, controllability, as well as voltage and frequency sensitivity. In the book's introductory chapter, the mathematical model of the load is compared to traditional forecast models. The formation of the load model and its application are also observed. It is demonstrated that load models are principally different from forecast models. While the forecast models only consider data (time series), the load models describe the physical behaviour of the load. In the second chapter, the main relations of the mathematical load model are presented. The practical implementations of those relations are then observed in chapter three. The fourth chapter is dedicated to the problems of programming, where the load model is presented as an object-based program. Programming is also discussed in some other sections (chap. 1.5, 5.4, etc.); readers who are not interested in implementing the monitoring methodology in computer programs may wish to skip those parts of the book. However, algorithmic presentation of the load model not only examines with programming issues but also supplements the description of the mathematical load model. The fifth chapter presents possible practical applications of the load model for resolving various problems. According to the principles presented, the mathematical model describes the load but does not directly give us the necessary practical values. These load characteristics can be found on the basis of mathematical load model. The structure of the mathematical load model is the same for all loads. The model parameters must be estimated in order to make the model correspond with the actual load. Problems concerning the estimation process (e.g., different quantity and quality of the load data) are observed in chapter six.

In the seventh chapter, the application of load models to compute the operation of the electrical network is observed. The traditional approach is possible, if the necessary load characteristics (e.g., load forecast) are found, based on the mathematical load model and used as initial data when computing the network state. It is also possible to integrate the load models into the calculating procedures of operation of the network. The material is then summarized in the final chapter. The principles of mathematical modelling of the load, and the possibilities of monitoring the electric network operation are observed. The examples presented correspond to the Nordic (Estonian, Finnish) conditions; therefore, some circumstances, such as temperature dependency, may be significantly different in the case of southern countries. Nevertheless, the observable load model remains universal. Differences of load character reflect on the parameters of the mathematical model, which are estimated in the practical application of the model. This book is intended for engineers and other specialists who forecast loads, analyse electrical network operations, operate in the electricity market, etc. Engineers developing application programs for mathematical modelling of the load may use this material as introductory. The author is grateful first to the Faculty of Power Engineering at Tallinn University of Technology, under whose support he has had the opportunity to study electrical load problems for over forty years. The principles described in this book have been practiced in energy systems of Eesti Energia (Estonia), Lenenergo (Russia), and Imatran Voima (Finland). The experience and feedback from their personnel have played a significant role on the material presented. Closer scientific collaboration has been carried out with Technical Research Centre of Finland (VTT), with Professor Matti Lehtonen, and with Dr. Anssi Sepp채l채. The author has received theoretical counselling and help from Toomas T채ht, Professor of Mathematics. The author would like to express his gratitude to all of them. The author is deeply grateful to Mr. R. Douglas Hanisch for thoroughly proofreading the manuscript. Finally, I am very grateful to my wife Tiiu for her encouragement and support over many years.

Contents 1 Introduction ........................................................................................ 7 1.1 Load Models and Forecast Models............................................... 7 1.2 Principles of Load Modelling ..................................................... 10 1.3 Estimation of the Load Model.................................................... 12 1.4 Load Monitoring......................................................................... 14 1.5 The Component Presentation of the Load Model....................... 17 2 Mathematical Model of Load .......................................................... 21 2.1 General Form of the Load Model ............................................... 21 2.2 Mathematical Expectation and Standard Deviation of Load...... 23 2.3 Temperature Dependency of Load ............................................. 31 2.4 Stochastic Component of Load .................................................. 36 3 Realization of the Mathematical Model.......................................... 45 3.1 Hierarchy of Model Parameters.................................................. 45 3.2 The Changes of Load Level ....................................................... 47 3.2.1 Trend of Load .................................................................. 47 3.2.2 Load Cases and Scenarios ............................................... 49 3.3 Temperature Dependency and Stochasticity of Load................. 51 3.4 Special Days and Periods ........................................................... 53 4 Object-Mode Presentation of the Mathematical Model ................ 59 4.1 Load Net Object ......................................................................... 59 4.2 Load Object and Temperature Object ........................................ 61 4.3 Model Object .............................................................................. 64 4.3.1 Object Structure............................................................... 64 4.3.2 Regular Changes.............................................................. 66 4.3.3 Temperature Dependency................................................ 68 4.3.4 Stochasticity..................................................................... 70 4.3.5 Special Day Calendar ...................................................... 71 5 Load Monitoring............................................................................... 73 5.1 Load Characteristics ................................................................... 73 5.1.1 Primary Load Characteristics .......................................... 73 5.1.2 Derived Load Characteristics .......................................... 76 5.1.3 Simplified Load Models .................................................. 77 5.2 Load Forecasting and Simulation ............................................... 79 5.3 Load Analysis............................................................................. 83 5.4 Computation of Load Characteristics .......................................... 87

6 Model Estimation..............................................................................91 6.1 Estimation Principles ..................................................................91 6.1.1 Model Co-ordinates .........................................................92 6.1.2 Temperature Dependency Model.....................................94 6.1.3 Other Model Components................................................98 6.2 Load Research ..........................................................................100 6.3 Model Editing ...........................................................................105 6.4 Estimation Object .....................................................................111 7 Electrical Network Operation Monitoring ...................................113 7.1 Transmission Network..............................................................114 7.2 Observable Distribution Network.............................................118 7.2.1 State Estimation .............................................................119 7.2.2 Model Editing ................................................................122 7.2.3 Operation Monitoring ....................................................124 7.3 Unobservable Distribution Network.........................................126 7.4 Modelling of the Distribution Network Operation ...................128 7.5 Distribution Network Object ....................................................132 8 Summary..........................................................................................133 8.1 Load Monitoring.......................................................................133 8.2 Monitoring of the Electrical Network Operation......................138 8.3 The Component Presentation of the Monitoring System..........141 Register ................................................................................................143

Introduction

1 Introduction The nature of a power system's operation depends on its load. Load changes regularly, depends on weather conditions, and has a stochastic nature. As in a power system, the generation and consumption of power must be in balance at every moment, the amount and time of electricity generated, transmitted and distributed is determined by load (Fig. 1.1). While the requirement of the active power balance is valid for the entire power system, it is possible to optimise generation of individual units. The reactive power balance must also be ensured locally (i.e., in certain parts of the network and in larger nodes of the grid). The power system load is controllable to some extent, for example, through electrical tariffs or using the load dependency of operational parameters (voltage and frequency). Sometimes load is controlled directly by operating personnel as well. Nevertheless, possibilities of load control remain secondary, as adjustable features of the power system operation. The behaviour of the load and the regularities of the load change are the primary features of the power system operation.

Figure 1.1 Principal scheme of power system

1.1

Load Models and Forecasting Models

In power system planning and operation, load is treated as a power system's total load, electrical network busload, or load of the individual consumer. Load is needed for short- and long-term forecasts, analyses, and simulations. Practically necessary form of load data depends on the application. Often the mathematical expectation is needed as a long-term forecast or for some other purposes. Depending on the task, the mathematical expectation corresponds to electrical power or current at a certain moment, or the mean value of a period (hour, day, week, or year). Often the conditional mathematical expectation is needed as a short-term 7

Introduction forecast. The temperature dependency of load belongs mostly to shortterm forecast. In the case of long-term forecast and load analyses, the temperature dependency may be simulated. The stochastic nature of the load is described by load standard deviation and distribution function. Depending on the application, other values, (i.e., load characteristics for describing load in different viewpoint) may also be used. In the conventional approach, the necessary load characteristics (mainly short-term forecast) are found using a formal approach based directly on load data, which is given as time-series. A number of methods, based on regression analyses, time-series models, neural networks, etc., are used. Load forecast methods are taken according to the character of initial data (amount of data) and on the needed result (e.g., forecast lead time). The focus of forecast models is on the application of formal mathematical methods under specific circumstances. The physical nature of the load is considered only superficially. Although forecast models may give practical results in certain conditions, load characteristics may be determined much more accurately and diversely by composing a mathematical model of the load. By developing the model, the physical nature of the load is examined and described quantitatively. In the model, namely load is described â&#x20AC;&#x201C; not the existing load data (time-series) as in the case of the conventional approach. Load is considered as an object that is being modelled. Active and reactive power or current is only load data, i.e., different phenomena of load. Despite its mathematical methods, the forecast models may be considered as trivial. Indeed, short-period data (often a week or two), on which forecasting methods are based, do not express all load regularities. Load should be monitored for at least a few years to determine its features. Also, statistically, short-term data is insufficient, considering load's stochastic nature. Both the load level and shape of the load curve may be exceptional for a period preceding forecasting. Contrasting situations may occur when considering the influence of the weather on load. As outdoor temperature may deviate from normal temperature for longer periods (weeks), significant model errors may occur when the weather changes. Although forecasting models are being improved (by considering special days, temperature dependency factors, etc.), these are only the first steps towards a model based on the physical nature of load.

Introduction The load model's substantial advantages over forecast models are especially clear when considering possible applications of both models. While forecast models are meant only for short-term load forecasting, the load model allows for much more. Besides forecasting, it is possible to analyse and simulate load. For example, causes of load deviation may be investigated, load at normal temperature or during exceptional weather conditions may be found, etc. Also, the forecast anticipation time is not limited. The long-term forecast accuracy is ensured by the fact that the model describes load to the full extent, considering seasonal changes of load level and load curve shape, trend etc. During long-term forecast, economical and technical conditions of load forming may be considered. Also, for a longer time period, it is possible to simulate load by varying, for example, weather conditions and load trend scenarios. At times, a lot of empty hopes have been applied to the formal mathematical methods for solution of practical problems. For the last few decades, it has been done under the name of artificial intelligence, where the methods for load forecast, using neural networks and knowledge systems, belong. Although artificial intelligence offers many interesting and useful information for base studies, its practical applications are not worth mentioning. Overall, it is an attempt to solve complex problems using inadequate means. The nature of the phenomenon is not observed. It can be said in this case that the scientific method, which demands empirical research and development of the theories based on survey and experiments, is ignored. In science history, the scientific method is associated with Galileo Galilee, to whom we owe numerous advancements in science and in technology. The earlier approach, associated with Aristotle, offers us the necessary philosophical means for research, but which is ineffective practically. Hence, trying to solve practical problems only with formal mathematical methods means a return to principles of Aristotle and forgetting the Galilean principles. Hence, the load and forecasting models have little in common. Preferably, the load model could be compared to pattern curves of load, which are used in some applications. When the temperature dependency formula is added to the pattern curve, then it is possible to speak of the simplified load model (primitive model). When applying the load model, both primitive and trivial models can occur in exceptional situations. While estimating the model parameters â&#x2C6;&#x2019; and when there is not enough necessary data for observed load â&#x2C6;&#x2019; then the load model with similar 9

Introduction nature should be used, and the result would be similar to using a pattern curve (primitive model). When the nature of the load is so irregular that its regularities do not become apparent by the estimation of load model, then we can speak about the trivial model. However, the load model is not another magic method that automatically solves all problems but rather a tool for specialists. The experience and know-how of the operation and planning engineers is of primary importance. The load model, with appropriate application programs, enables specialists to implement the engineering visions effectively.

1.2

Principles of Load Modelling

Derived from the idea of mathematical modelling of the load: The structure of the model does not depend on the amount of data available. The load is modelled, not the data. The structure of the model does not depend on the possible applications. It is not important whether the model is used for long- or short-time forecasts or for forecasting at all. The complexity of the model is derived from handling of the load. The structure of the model depends on how the load is defined and what is the main purpose to be considered when the load changes. When forming the mathematical model, the goal will be investigation of inherent regularities of load and quantitative description of these regularities. During modelling, no attention is paid to possible features (e.g., sampling frequency) or the amount of load data. Indeed, the nature of load does not depend on how it is measured. The necessary characteristics of the load (short- or long-term forecast, etc.) also remain secondary. Needed load characteristics are defined based on applications and are found by appropriate programs, not even with program modules related to the mathematical model of load. In the model, the following changes of load are considered: Regular changes, which are periodical changes in the course of a day, a week, and a year, trend and load changes on special days. Temperature dependency, the importance of which is rather high in the case of electrical heating and cooling. In the model, temperature dependency inertia, non-linearity, and time changes are considered. Dependency on operational parameters, which occurs as voltage and frequency sensitivity.

Introduction Â&#x192;

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Randomness is especially noticeable in smaller distribution network loads. Such loads have rather high standard deviation in relation to mathematical expectation. In smaller loads, large deviations may occur from time to time that do not match with normal distribution. Controllability. Load is mostly controlled indirectly â&#x2C6;&#x2019; for example, through electrical tariffs. Direct control by dispatch personnel also occurs. Controllability may be considered if, for example, switching in the distribution network causes transmission grid busload change.

During modelling, load is considered as an object, which includes a name, connection point, characteristic composition of consumers, and other properties. Quantitatively, active and reactive power and current describe load. Load data may be regular (time-series) and also as particular values (e.g., yearly energy). The main component of the model is load mathematical expectation. Mathematical expectation describes regular load changes in standard conditions such as normal temperature as well as rated frequency and voltage. Mathematical expectation is principally non-stochastic. Load stochastic measure is standard deviation, which is also considered as time variable. Temperature dependency of the load may be high in the case of electrical heating and in other cases of weather-sensitive electricity consumption. Temperature dependency is specified as load deviation, which is caused by outdoor temperature deviation in relation to normal temperature. The normal temperature (mathematical expectation of temperature) is generally the average outdoor temperature of 30 years at any given time of the year. Load temperature dependency is characterized by a delay of about 24 hours. If the actual temperature corresponds to normal temperature, considering the delay, then temperature dependency is absent. Also, it is possible to consider other weather factors like sun radiation (cloudiness), wind speed, air humidity, etc. However, it should be emphasized that, practically speaking, only weather factors which are treated (incl. forecasted) quantitatively by meteorological services should be considered. Dependency on operational parameters is considered through sensitivity of load to voltage and frequency. Frequency dependency is typical to active load. This must be considered, for example, at power systems balancing to ensure necessary energy from the electricity market.

Introduction Voltage, which influences mostly reactive power, is important for treatment of electrical network busload. Mathematical presentation of the stochastic component considers autocorrelation of load deviation, which is necessary for obtaining short-term load forecasts. Peak component, which causes load large deviations, is also considered. Large deviations are especially characteristic of distribution network loads. This is one reason why load is not normally distributed. Regular components of the model (mathematical expectation, temperature dependency, etc.) generally describe load well enough. But still loads exist where changes are so irregular that they are impossible to describe accurately. In such cases, the load mathematical expectation and standard deviation are considered as constant, and the stochastic component will be decisive. We may consider it as a trivial model of load. Based on stochastic component model, it is feasible to analyse and forecast load, presented by trivial model but the accuracy will remain lower than in the case of the normal model. Such a description of the trivial model complies well with the previous discussion about load forecast models triviality.

1.3

Estimation of the Load Model

The structure of the mathematical model is the same for all loads. In order to describe particular loads, the model parameters must be estimated according to the load data. In the purpose of estimation, the regular load data (time-series) and other quantitative and qualitative data will be used. If the existing data is not enough to evaluate all parameters of the model, then type-models (i.e., a typical set of model parameters) may be used. In such cases, the suitable amount of load parameters are evaluated by load statistical data, and the rest of the parameters will comply with the type-model. The result of estimation will be a complete model for all loads independently from the amount of used data. If there is more data available, the result will be more accurate â&#x20AC;&#x201C; the model will describe the load better and enables one to determine more accurate load characteristics. The main estimation phase, initial estimation, will be done during the load research. If, for example, a certain electrical network is observed, the result will be type-models of the electrical network load. Only loads 12

Introduction that have enough data and change of which is regular (typical) are considered. Next a suitable type-model will be fixed to each individual load, and an additional number of parameters will be estimated, which will form the models for all loads. This phase of the estimation is called model editing. However, the estimation method described does not require application of type-models. If there is enough data, the unique model may be estimated for each load.

Figure 1.2 Estimation scheme of load models

The categories of load model estimation are shown in Fig. 1.2. Loads, of the electrical network considered, are classified into groups during load research; groups are then classified into classes, and classes into types. Load model group and class components will be estimated based on such load data, the character of which may be considered as typical and data for which is sufficient (several yearsâ&#x20AC;&#x2122; hourly readings). Parameters that do not belong to model components are called model factors and are appropriate to each load. Such type of estimation scheme is necessary for regularly (through SCADA system) non-measurable loads, data for which is insufficient. Also, in the case of availability of necessary data, the application of type-components is appropriate because it enables one to decrease the number of estimated parameters and therefore increase the reliability of the model. The structure of the model is composed so that its components, where most of the parameters belong, are relatively stable. Therefore, these are not necessary to update if the character of

Introduction loads change over time. Only model factors must be adjusted (edited) on the basis of new load data. From the point of view of practical application of the mathematical model, the estimation procedure is essential. The principle is that all model parameters of the load must be estimated. The simplified models, due to a lack of load data, are not observed. Different means of estimation are possible depending on the extent and amount of raw data, but also the necessary accuracy of the model. The purpose of estimation is that the model corresponds to the given load. However, the nature of the practically necessary load characteristics (short-term or long-term forecast, etc.) is not considered during the estimation.

1.4

Load Monitoring

Mathematical model describes load changes. The model does not directly give practically necessary values (e.g., load forecast). These values (i.e., load characteristics) may be achieved based on the model. For example, the following initial load characteristics may be considered: A[P] – load data E[P] – load mathematical expectation C[P] – expected deviation of load (conditional mathematical expectation of load deviation) I[T, P] – temperature dependency of load I[Z[T], P] – simulated temperature dependency of load. Based on these following characteristics may be derived: E[P] or E[P] + I[Z[T], P] – long-term load forecast E[P] + I[T, P] + C[P] – shot-term load forecast A[P] – I[T, P] – normalized load A[P] – I[T, P] + I[Z[T], P] – simulated load. Possible load characteristics are described in more detail in Chapter 5. Electrical network loads are rarely interesting alone. Load values are generally needed for solving network operational problems as initial data. Short- and long-term load forecasts are necessary, as well as load analysis and simulation results, at different given conditions. The main purposes of load treatment are: short-term forecasting of total load analyses and simulation of total load 14

Introduction Â&#x192; Â&#x192; Â&#x192;

short-term forecasting of electrical network busloads analysis and simulation of electrical network busloads treatment of electricity consumers load.

The best-known task considering load is short-term forecast of the total load of electrical power system (or region). The goal may be power system operation planning, including optimal load distribution between generating units or operation in electricity market. Although it is possible to carry out short-term forecast with forecasting models, the load model application provides more accurate and versatile results. The power system operation is also considered over longer time periods (e.g., a year), as it is necessary to plan fuel resources and establish electricity-trading contracts. As the forecast anticipation time is not limited in load model, then no computational problems occur. In longterm planning, special importance must be given to load simulation, as it is necessary to find load values at different weather conditions and consider different load increase scenarios (trends). In the case of longterm forecasts, specialists may also consider regional economic development plans and expected technical changes in the network. Busloads are the main initial data for calculating the electrical network operation. In a shorter time period (few days ahead) it is necessary to optimise network operation, ensure sufficient reliability and electricity quality. It is also possible to find necessary values of busloads with forecasting models, but, because of relatively significant randomness, the results are non-reliable. Therefore, approximation methods are used, where, for example, the total load forecasted would be distributed between buses according to previously defined coefficients, etc. Load models also allow for treatment of small loads. The truth is that randomness of the load cannot be avoided. Whereas the model reliably describes load mathematical expectation, standard deviation, temperature dependency, and other characteristics then finding the reliable confidence interval of forecast may compensate the randomness to a certain extent. In electrical network long-term planning, load values are needed at various conditions such as low or high outdoor temperature, possible load deviations, place and time of connecting new electricity consumers. Load cases and scenarios, which are caused by commutation in lower level network, switching of reactive power compensators, electricity

Introduction consumersâ&#x20AC;&#x2122; development alternatives, and other circumstances, are of considerable importance in treatment of electrical network. In the case of free electricity markets, all market participants must ensure their balance. For that reason, electricity consumption must be forecasted and electricity delivery contracts must be made. Balance settlement will be done later where energy unbalances are found, and consumers will pay higher tariffs for unbalances. In order to increase the efficiency of the electricity market, Nordic countries develop a system to get all electricity consumersâ&#x20AC;&#x2122; load data by remote measurement. Data about consumersâ&#x20AC;&#x2122; actual loads enable to make balance settlement fair. These data may also be used for load forecasts. For that purpose, given load model is appropriate. A large number of electricity consumers is not an obstacle. The mathematical model enables one to calculate all practically necessary load values for shorter and longer time periods. The result will be reliable information about load data, which enables companies to plan fuel resources and electrical network transmission capability for longer time periods and with greater economic efficiency, which results in the correct timing of investments (e.g., possibility to postpone the investment) and increases the reliability of the network operation (overload prevention, voltage stability enhancement, relay protection, correct operation at minimum and maximum load level, etc.). Accurate load forecasts in a shorter time period enable to increase the quality of electricity and achieve economic efficiency, which clearly emerges in the case of the electricity market. The application of load models requires additional contribution compared to forecasting models and other traditional load treatment methods. First of all, it is necessary to understand what the load models offer. During initial application of the model, it may be necessary to order load research from specialists. During everyday conditions, the formation of electrical network load is monitored, and, if necessary, load models are edited. Also, application programs, to which load models belong, are needed. However, all practical actions are done on the engineering level and do not require any additional costs. On the other hand, benefits achieved from load model applications may be significant.

Introduction

1.5

The Component Presentation of the Load Model

For practical application the mathematical model of the load, a computer program must be developed. Because of the variety of applications, the development of a program is difficult. Moreover, independent programs are generally not needed. The information concerning the load is used as initial data for computing the electrical network operation, electrical market transactions, and for other purposes. The handling of load should be carried out within an appropriate application program, not an independent program. The best solution in this situation is to use component technology of programming. Present day programming is based on different standard components, which are included in the programming environment, or are obtained additionally. Those components could be developed for different purposes, including for handling electrical network loads. Different programming languages can use the same components. Diverse component models are known. In this work, the ActiveX component technology is used. Since the structure of the mathematical model is the same for all loads, the program components are also the same. Hence, it is not important whether we deal with a large power system with loads up to many gigawatts, or an electrical network with busloads of a few kilowatts, active or reactive power, if values of the short-term or long-term forecast are needed, etc. â&#x20AC;&#x201C; in all cases, the program components are the same. Application programs that use the program components, their purposes, and user interfaces are different. The program components are based on objects. An object is a combination of code and data that can be treated as a unit. Objects support properties, methods, and events. Properties are data that describe an object. Procedures that can operate on the object are called its methods. Events are procedures, which are triggered when objects are changed. Objects are encapsulated; their code and their data are accessible through the properties and methods of objects. Object is defined by a class. An operating object is a copy, an instance of a class. The properties present particularly characteristic values of the object. The object methods also process large amounts of data, which, in common applications are in databases or in files. In current work, extensive data is saved into binary files, which are not accessible outside the 17

Introduction objects. They form buffers, where the objects save data and use it afterwards. The load monitoring system objects do not handle other data, which are not located in buffers. Similar objects form object classes, based on which the required instances are derived. The necessary number of object classes for handling the load is not large. The observable objects are, for example, instances of OLoad and OTemp classes, which represent the load and temperature. The load models belong to class Model and the model estimation tools to the class EModel. Additionally, the class ElmoNet, joining the specific loads (e.g., belonging to the same electrical network), temperature objects, and their models, is observed. The abovementioned object classes form the ActiveX-type code component ElmoExe.dll. For computing the distribution network state, object class Disco and corresponding code component ElmoDisco.dll are developed.

Figure 1.3 Component ElmoExe.dll application in dispatching system MicroSCADA

The objects of component ElmoExe.dll can be used for all kinds of necessary handling and processing actions of load data. General operations, associated with the user interface, are not included in the component. The possible implementation of component ElmoExe.dll in ABB dispatching system MicroSCADA is presented in Fig. 1.3. Here the load data, gathered from the electrical network through dispatching system, is saved using the component ElmoExe.dll into buffer files. The data is prepared and saved by program segment, which is developed with SCIL-

Introduction language, used in dispatching system MicroSCADA. Through that segment, the necessary information (e.g., load forecast, prepared by component ElmoExe.dll) is also passed into the dispatching system. All necessary access to the dispatching system database and user interface is the responsibility of the dispatching system. Component ElmoExe.dll does not deal with that. The data processing with component ElmoExe.dll is based on the mathematical model of the load. The parameters' initial estimation takes place during the load research. Hereafter, the model parameters are edited from time to time by a person (the load manager, using the load monitoring program ElmoSet), or automatically. The involvement of the load manager is necessary when the nature of the load changes â&#x20AC;&#x201C; loads adding or dropping, the network scheme changing, etc. The load manager can also investigate reasons why the load deviates from the model. Reasons could include accidents in the network, irregularities of the operation of consumers, or errors in the measuring system. For example, when in substation, the consumers are supplied through two transformers and when one transformer transducer is out of order, and then the substation load measurements are a half of the real value. This kind of a decrease does not give any signals in the dispatching system, and the situation could remain unnoticed for a long time. The load manager works only on an engineering basis. He observes the loads, especially those for which the program ElmoSet has discovered extraordinarily large deviations from the normal value, and, when needed, makes some adjustments to the load models using the program ElmoSet. The load manager does not need any specific knowledge of the structure of the load model. The load monitoring system could also be implemented with other applications. Handling the load data and computing the load characteristics is done by the component ElmoExe.dll. The preparation of the load data and performance of the user interface is the task of the application program. The program ElmoSet is necessary for load research and initial estimation of the load model. When the load is relatively stable (total load of network, etc.), then there is no need for the load manger. The necessary model editing (adaptation) can be done automatically.

Mathematical Model of Load

2 Mathematical Model of Load Intensity of electricity consumption in certain regions is called the load of the power system. The load is formed cumulatively from different electrical devices. The load is generally treated as a total load, which is the sum of lower-level individual loads and losses of the electrical network. Load can be treated as a certain object, which can be characterized with general data, load data, and a mathematical model of load. General data consists of the name of the load, connection point, marginal power, the type of the load, electricity consumers’ composition, etc. Load data describes all kinds of quantitative information about the load. Quantitative information includes, in particular, active and reactive power and current as well as bus voltage, outdoor temperature, and other values, which are used for load treatment. Load data may be both regular (timeseries) and non-regular, single data (e.g., yearly energy, minimum and maximum values, etc.). Load is often considered in a narrower sense than power or current. A mathematical model describes principal changes of load. The model considers regular changes, temperature dependency, and the stochastic nature of load. Regular changes are load trend; seasonal, weekly, and daily periods; and load level on special days (e.g., holidays). Voltage and frequency dependency are also taken into account. In this chapter, the main components of the mathematical model of load are described. The application of the model in different situations is described in chapter three.

2.1

General Form of Load Model

The mathematical model describing load changes (active power, reactive power, or current) consists of three basic components: P (t ) = E (t ) + Γ (t ) + Θ (t ) where E(t) is the mathematical expectation of the load Γ (t ) is the temperature-sensitive part of the load Θ (t ) is the stochastic component of the load.

Mathematical expectation E(t) describes regular changes of a load, such as general trend and seasonal, weekly, and daily periodicity. Mathemati21

Mathematical Model of Load

cal expectation is principally non-stochastic and corresponds to the normal temperature. The temperature-sensitive part of a load Γ (t ) describes load deviations, caused by deviations of outdoor temperature from the normal temperature. The normal temperature (mathematical expectation of the temperature) is the average outdoor temperature of the last 30 years on any given hour of the year. Besides other features, temperature dependency of loads is characterized by a delay of about 24 hours. If the actual outdoor temperature corresponds to the normal temperature (considering delay), the value of the temperature-sensitive part of the load is zero. It is appropriate to scale the temperature-sensitive part of the load by rate of the temperature dependency of load R(t) Γ (t ) = R(t )γ (t ) where R(t) is the rate of the temperature dependency of the load γ (t ) is the relative temperature dependency component. Here the rate R(t), that represents the temperature sensitivity of load (load increase when the temperature rises by 1 ºC), determines the level of temperature dependency and supports the consideration of its timely changes. Component γ (t ) describes other regularities of the temperature dependency, e.g. inertia etc. The stochastic component Θ (t ) describes stochastic deviations of load. The deviations of load are stochastically correlated. It is possible to observe the stochastic component of the load by expected deviation ζ (t ) , which describes the conditional mathematical expectation of the stochastic component and normally distributed non-correlated residual deviation (white noise) ξ (t ) . In addition, it is necessary to observe peak deviations of the load by the component π (t ) , which describes large positive or negative deviations that do not correspond to the normal distribution. When scaling the stochastic component by standard deviation of the load S(t), the result is

Θ (t ) = S (t )[ζ (t ) + ξ (t ) + π (t )]

Expected deviation of load ζ (t ) is needed, for example, for short-term load forecast. It is possible to find the short-term forecast, as load conditional mathematical expectation, by adding deviation and temperature dependency to the load mathematical expectation. The result is 22

Mathematical Model of Load

Eτ [P(t )] = E (t ) + R (t )γ τ (t ) + S (t )ζ τ (t ) where τ is lead time in the units (an hour or a part of it) of sampling steps. However, when analyzing the load and controlling the model adequacy, the one-step forecast E1 [P (t )] , which is named as an expected value (short term) of load, is required. Consequently, the mathematical model of a load is P(t ) = E (t ) + R(t )γ (t ) + S (t )[ζ (t ) + ξ (t ) + π (t )] According to this model, E(t) is the mathematical expectation of a load E[P(t )] = E (t ) on the conditions that both stochastic deviations and influence of temperature are missing. S(t) is, in its turn, standard deviation of load σ [P(t )] = S (t ) on the condition that influence of temperature is given and belongs to the mathematical expectation E[P(t )] = E (t ) + R (t )γ (t ) It must be pointed out that E(t) and S(t) are first of all components of the model. What real mathematical expectation and standard deviation will be like depends on given conditions.

2.2

Mathematical Expectation and Standard Deviation of Load

For development of load model it is rational to describe the mathematical expectation, standard deviation, and rate of temperature dependency as a function of three arguments: yearly (general) time t, daily time h, and type of day l as follows: 1 P (t , h, l ) = E (t , h, l ) + R (t , h, l )γ (t ) + S (t , h, l )[ζ (t ) + ξ (t ) + π (t )] Timely changes of mathematical expectation, standard deviation, and rate of load can be described by the following expressions: E (t , h, l ) = M T ( h )G El N(t )

Since the yearly time t, daily time h, and type of day l are the functions of general time t, we can use different indications to mark the functions, depending on the purpose. It is possible to use, e.g., both P(t) or P(t,h,l).

Mathematical Model of Load

S (t , h, l ) = M T ( h )G Sl N(t ) R (t , h, l ) = M T ( h )G Rl N(t ) where M(h ) and N(t ) are vector functions which include components corresponding to daily and annual load changes. G El , G Sl , and G Rl are matrices consisting of parameters depending on day type l . Respectively, ⎡ ν 0 (t ) ⎤ ⎡ μ0 (h) ⎤ ⎢ ν (t ) ⎥ ⎢ μ (h) ⎥ 1 ⎥ ⎥ , N(t ) = ⎢ 1 M( h ) = ⎢ ⎢ L ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ν NAC (t )⎦ ⎣ μ MDC ( h )⎦ and G El = g Elks , G Sl = g Slks , G Rl = g Rlks where k = 0...MDC and s = 0...NAC. Here the vector components, corresponding to index 0, are trivial μ 0 ( h ) ≡ 1 , ν 0 (t ) ≡ 1 The number of non-trivial components MDC and NAC is, for example, 4...5. Matrices G E (l ) , G S (l ) , and G R (l ) can be developed into series G El ≅ al 0 G 0 + al1G1 + al 2 G 2 + K + al , NSC G NSC G Sl ≅ bl 0 G 0 + bl1G1 + bl 2 G 2 + K + bl , NSC G NSC G Rl ≅ cl 0 G 0 + cl1G1 + cl 2 G 2 + K + cl , NSC G NSC

where G 0 = 1 . The result obtained is E (t , h, l ) = M T ( h )∑ (alr G r )N(t ) r

S (t , h, l ) = M ( h )∑ (blr G r )N(t ) T

R (t , h, l ) = M ( h )∑ (clr G r )N(t ) T

where r = 0…NSC. Actually, NSC = 10…12. Types of day l = 1…NTP correspond primarily to normal weekdays (l = 1…7). In addition, special days (holidays, pre-holidays, post-holidays, etc.), which l > 7, are observed. The number of special days depends on the calendar (the country) and required accuracy of model-

Mathematical Model of Load

ling. The total number of type of days NTP may be up to 50…60. In a simplified case, the special day is considered as a similar weekday (holiday – Sunday, pre-holiday – Friday etc). In that case, the number of different types of days is 7. Model parameters alr , blr , and clr can be normalized, based on calculation of mean value of mathematical expectation, standard deviation, and rate of load model 1 1 1 a = ∑ a lr g r 00 , b = ∑ blr g r 00 , c = ∑ clr g r 00 7 l ,r 7 l ,r 7 l ,r where g r 00 is the element of matrix G r with index 00. Here the summing up is done in the range of ordinary weekdays l = 1…7 (special days are not considered). In the model, elements of matrices are replaced as follows: a lr ⇒ a ⋅ alr , blr ⇒ b ⋅ blr , clr ⇒ c ⋅ clr Parameters cannot be normalized if the calculated mean value is too small (zero) or negative, which may occur in the case of reactive power and current. In that case, a = b = c = 1. The load level and shape may change. Fast changes may be considered in the model as a step change of the model factors starting at a certain time. Long-term load increase or decrease is presented as an additional component of the model – trend. Trend component is given as quadratic function

[

]

A(t ) = a 1 + α1 (t − t0 ) + α 2 (t − t0 ) 2 where α1 and α 2 are factors and t0 is the moment in time at which the computation of trend starts. Beside mathematical expectation, trend also belongs to the standard deviation and the rate of the temperature dependency as

[ ] C (t ) = c[1 + α (t − t ) + α (t − t ) ] B ( t ) = b 1 + α1 ( t − t 0 ) + α 2 ( t − t 0 ) 2 1

where factors α1 and α 2 as well as t0 are the same, as in the case of mathematical expectation. Load voltage and frequency sensitivity are described as quadratic functions U (u ) = 1 + μ1u + μ 2 u 2 , where u = UV / U N − 1 F ( f ) = 1 + ν 1 f + ν 2 f 2 , where f = FV / FN − 1

Mathematical Model of Load

Here μ1 , μ 2 ,ν 1 ,ν 2 are factors and UV , U N and F V , F N are observed as rated values of voltage and frequency, respectively. Voltage and frequency sensitivity are considered only in connection with mathematical expectation. Consequently E (t , h, l ) = A(t )U (u ) F ( f ) M T ( h )∑ (alr G r )N(t ) r

S (t , h, l ) = B (t )M ( h )∑ (blr G r )N(t ) T

R (t , h, l ) = C (t )M T ( h )∑ (clr G r )N(t ) r

Components μ i (h ) (i = 1…MDC) of vector function M (h ) are depicted by points (in table form). The number of points depends on the density of sampling data, which may be once an hour or more. As the time range is one day, the minimum number of points, therefore, is 24. The components, corresponding to annual changes ν j (t ) (j = 1…NAC), are approximated with the Fourier series

⎛ 2π ⎞⎤ ⎛ 2π ⎞ kt ⎟ + a ′jk′ cos⎜ kt ⎟⎥ T ⎝ T ⎠⎦ ⎝ ⎠ k =1 ⎣ Series order MB is usually 4 or 5. The number of parameters for one component of vector function N(t) is therefore 8 or 10. In Figs 2.1 and 2.2, some examples of the vector function components, which are also called co-ordinate functions of the model, are shown. MB

⎡

ν j (t ) = ∑ ⎢a ′jk sin⎜

Besides load, regular changes of temperature must also be modelled. The structure of the temperature model is principally the same as for the load model, but with some simplifications. As for temperature, there is no need to distinguish different day types; therefore, there is no need for developing parameter matrices into series. The temperature model consists of only one type of G E and G S matrix, which, together with coordinate functions, determine the mathematical expectation (normal temperature) and standard deviation of temperature T(t,h) ET (t , h ) = E[T (t , h )] = M TT ( h )G TE N T (t )

S T (t , h ) = σ [T (t , h )] = M TT ( h )G TS N T (t )

Mathematical Model of Load

Figure 2.1 Components of vector function

M(h )

Figure 2.2 Components of vector function N(t )

Thus, mathematical expectation describes the load's probable values at standard conditions. Possible load deviations are caused by weather (temperature) influence and stochastic reasons. Figures 2.3 and 2.4 describe the comparison of load mathematical expectation and actual load values. Here the load deviation is caused mainly by the temperature influence. 27

Mathematical Model of Load

Figure 2.3 Load actual value (1), mathematical expectation (2), and standard deviation (3), weekly values

Figure 2.4 Load actual value (1), mathematical expectation (2), and standard deviation (3), hourly values

If the temperature influence will be eliminated – i.e., considering load normalized values P (t ) − R(t )γ (t ) – the situation is different (Figs 2.5

Mathematical Model of Load

and 2.6). It should be mentioned that the estimation of mathematical expectation would be done solely on the base of load normalized values.

Figure 2.5 Normalized load (1) and mathematical expectation (2), weekly values

Figure 2.6 Normalized load (1) and mathematical expectation (2), hourly values

Mathematical Model of Load

In these examples, the load is 25â&#x20AC;Ś100 MW, and standard deviation is only a small percentage of mathematical expectation, as is characteristic of such load levels. Stochasticity of lower level loads is larger. Figures 2.7 and 2.8 describe an example of a household with electrical heating,

Figure 2.7 Household actual load (1), mathematical expectation (2), and standard deviation (3), weekly values

Figure 2.8 Household actual load (1), mathematical expectation (2), and standard deviation (3), hourly values

Mathematical Model of Load

where power is given in kWs. Here, too, the temperature dependency is significant, but the standard deviation reaches 50% of mathematical expectation, and even more in some instances. In Fig. 2.7, a cold winter is reflected. In some cases, the load changes are so irregular that it is not possible to find a definite description. In such cases, load mathematical expectation and standard deviation may be presented as constant E[P (t )] = a σ [P(t )] = b In the case of such a trivial model, temperature dependency is not considered, but we may consider the difference between load value and mathematical expectation as a stochastic component and find the expected deviation, residual deviation and peak deviation of the load. Also, trend may be considered in the case of trivial model. So it is possible to analyze and forecast loads with trivial model.

2.3

Temperature Dependency of Load

Electrical network load depends on the outdoor temperature and other meteorological factors such as sun radiation (cloudiness), wind speed, humidity, etc. To avoid unreasonable complexity of the model and possible estimation difficulties, we consider only the basic factor – outdoor temperature. It must be emphasised that only factors that are treated (including forecasts) by meteorological services in a quantitative way will be considered. Principally other weather factors can be considered through transformed value of temperature, i.e. through effective temperature. Of course, this value is also treated by meteorological services. Load temperature dependency is largest where electrical heating or air conditioning is used. For example, in Lapland (Finland), where electrical heating is of great importance, and outdoor temperature deviations are large, the load increase caused by the outdoor temperature may be up to 100% compared to the load at normal temperature. Typically, however, the temperature dependency of the load is smaller, especially in the case of industrial loads. The temperature dependency component is depicted as Γ (t ) = R(t )γ (t )

Mathematical Model of Load

Figure 2.9 Temperature sensitivity of the load, weekly values

Figure 2.10 Temperature sensitivity of the load, hourly values

The rate of temperature dependency R(t) matches with the temperature sensitivity of load – the load increase when the temperature rises by 1 ºC. The temperature sensitivity is changeable seasonally, daily, and by the type of days. An example of changes of the temperature sensitivity on weekly and hourly levels is presented in Figs 2.9 and 2.10, respectively. The rate, the unit of which is MW/ºC is, in this case, negative – the rise 32

Mathematical Model of Load

of the temperature causes a fall of load and vice versa. in. The highest effect of the temperature on load is in winter. The peak in daily temperature sensitivity at 23.00 is apparently caused by switching of heating devices at that time. In the first approach, the temperature effect on the load may be expressed as Γ (t ) = R (t ) ΔT (t ) where ΔT (t ) = T (t ) − E[T (t )] is the temperature deviation from normal temperature, and thus γ (t ) = ΔT (t ) . Actually, a more precise presentation of the relative temperature dependency component γ (t ) is necessary. It should include a more detailed description of the temperature dependency of the load, first of all, considering the delay of the temperature influence (inertia). To present the relative temperature dependency component, it is appropriate to apply the time series ARIMA model (Integrated Autoregressive – Moving Average Model), which is also called the Box-Jenkins model, as follows Ψ ( B) γt = T ΔTt Φ T ( B) Here ΔTt = Tt − E[Tt ] is the deviation of temperature from the mathematical expectation (normal temperature), and Φ T (B ) , ΨT (B ) are polynomials of the shift operator B ( Bx t = x t −1 ) . If operators Φ T (B ) and ΨT (B ) are presented in the form of Φ T ( B ) = 1 − ϕB , ΨT ( B ) = ψB m where ϕ , ψ , and m are parameters, then the temperature dependency model of the load corresponds to transfer function, shown in Fig. 2.11 with parameters

H 0 = m, H =

1−ϕ

, γ∞ =

1−ϕ

Figure 2.11 Transfer function of temperature dependency

Mathematical Model of Load

According to this transfer function, the temperature change influences load after H 0 hours, and H is time constant of temperature dependency. For example, if H 0 = 5 and H = 10 , then temperature change will affect the load fully after about H 0 + 2 H = 25 hours (i.e., on the next day). In addition to level and inertia, one must also consider changes of temperature dependency in time and non-linearity. Time response is taken into account by considering the variability of parameters of the temperature dependency model. It is essential to consider seasonal variability, because the influence of temperature in summer, for example, differs significantly from that of winter. If necessary, weekly and daily changes of temperature dependency may also be taken into account. A non-linearity phenomenon occurs when, within certain temperature values, the characteristics of temperature dependency changes. For example, in summer, when the temperature dependency is mostly missing in Northern countries, load increase can be recognized if the temperature falls below 14 °C or rises above 25 °C. In winter, the load increase speed may decrease if the temperature falls below –25 °C. These phenomena can be explained by use of additional cooling equipment in the summer and achieving the maximum output of heating equipment or finishing outdoor works in the winter. Similar phenomenon can also be recognized during other periods. These phenomena are in conjunction with temperature deviation from a certain value instead of mathematical expectation of temperature. In the model, non-linearity is considered by adding the so-called marginal components, which will be activated if the temperature falls below or reaches above the pre-set values T1 or T2 . According to that, the temperature dependency component will consist of three components γ t = γ 0t + γ 1t + γ 2t where Φ T 0 ( B )γ 0t = ΨT 0 ( B )(Tt − E[Tt ]) ⎧ΨT 1 ( B )(Tt − T1 ), Φ T 1 ( B )γ 1t = ⎨ 0, ⎩ ⎧ΨT 2 ( B )(Tt − T2 ), Φ T 2 ( B )γ 2 t = ⎨ 0, ⎩

if Tt < T1 if Tt ≥ T1

if Tt > T2 if Tt ≤ T2

The temperature dependency model as a whole will be

Mathematical Model of Load

[

(1 − ϕB )γ t = B m ψ 0 (Tt − E[Tt ]) + ψ 1 (Tt − T1 ) T <T + ψ 2 (Tt − T2 ) T >T t

]

Figure 2.12 Actual value (1), mathematical expectation (2), and expected value (3) of load, and actual value (4), and normal temperature (5), weekly values

Figure 2.13 Actual value (1), mathematical expectation (2), and expected value (3) of load, and actual value (4), and normal temperature (5), daily values

Mathematical Model of Load

Figures 2.12 and 2.13 are examples of load temperature dependency with weekly and daily values. In addition to actual values of temperature and load the mathematical expectations and expected value of load and normal temperature are presented. Here, the expected value of load (long term) E (t ) + R (t )γ (t ) is suggested as the value which is obtained by adding temperature dependency to the mathematical expectation. Depending on the accuracy of modelling needed, other load temperature dependency details may be considered. For example, load may not decrease in an ordinary fashion after a long, cold winter period (a week or more) when temperatures return to normal levels. Concerning the model, this means that the time constant H increases. A problem is how to represent load temperature dependency of accumulative electrical heating. In that case, temperature dependency is related not so much to the power of the heating system but to the duration of the turn-on time.

2.4

Stochastic Component of the Load

The randomness of the load is considered in a mathematical model by a stochastic component Θ (t ) = S (t )[ζ (t ) + ξ (t ) + π (t )] The level of the stochasticity is expressed by standard deviation of the load S(t), which is changeable over time. Examples of standard deviation at weekly and daily levels are presented in Fig. 2.14 and Fig. 2.15. From the examples, it appears that the value of standard deviation basically corresponds to the changes of the load (mathematical expectation). Standard deviation is higher in the winter and in the evenings, smaller in the summer and at nights. However, a closer examination shows that the changing regularities of the standard deviation do not always match the changes of the mathematical expectation. Load stochastic deviation can be described with the ARIMA model as Ψ( B ) ϑt = ξt Φ( B) where ϑt is the relative value of random deviation ϑ (t ) = [P (t ) − E (t ) − R (t )γ (t )]/ S (t ) in the time interval t, Φ (B ) , and Ψ (B ) are linear operators, and ξ t is residual deviation – white noise. Also, transfer function may be observed 36

Mathematical Model of Load

Figure 2.14 Standard deviation of the load, weekly values

Figure 2.15 Standard deviation of the load, hourly values

F( B ) =

Î¨( B ) ÎŚ( B)

So that 37

Mathematical Model of Load

ϑt = F( B )ξ t Actually, the operators Φ (B ) and Ψ (B ) are presented as Φ ( B ) = (1 − ϕ1 B − ... − ϕ MF B MF )(1 − ϕ M B M )(1 − ϕ N B N ) Ψ ( B ) = (1 − ψ 1 B − ... − ψ MP B MP )(1 − ψ M B M )(1 − ψ N B N ) Here, the first part of the operators considers the after-effect of load deviations, which precede the present time interval. The second and third part of the operators considers duration from one day backward and one week backward. The daily displacement factors MF and MP are actually within 1…2, and if the sampling frequency is once an hour, then M = 24 and N = 168. Thus, the model of the stochastic component consists of eight parameters ϕ 1 , ϕ 2 , ϕ 24 , ϕ 168 , ψ 1 , ψ 2 , ψ 24 , and ψ 168 . When describing the value of residual deviation ξ t the large deviations of load are omitted. The following criterion may be used ξ t < cSσ ξ where cs is the confidence factor (e.g., 2.7) and σ ξ standard deviation of residual deviation. Possible large deviations belong to load peak component π t . Actually, the load's stochastic component is treated recursively. For each time interval (an hour or part of it), value of the deviation ζ t is found by the Box-Jenkins model. If deviation ϑt − ζ t is suitable according to the previous criterion, then ϑt − ζ t = ξ t and peak component value π t = 0. If not, then the peak component value differs from zero and ϑt − ζ t = ξ t + π t . For separation of components ξ t and π t the residual deviation ξ t is simulated, based on normal distribution ξ t′ = N (0, σ ξ ) . Therefore ⎧ξ t = ϑt − ζ t , π t = 0, if ϑt − ζ t < c Sσ ξ ⎨ ⎩ξ t = ξ t′, π t = ϑt −ζ t−ξ t′, if ϑt − ζ t >= cSσ ξ Stochastic deviation ϑt is illustrated in Fig. 2.16. Here we can see that the residual deviation of load ξ t is found for each time period (hour), and the values of peak deviation of load π t appear from time to time. This algorithm is suitable for short-term forecasts where large load deviations are left out and ξ t = 0 .

Mathematical Model of Load

Figure 2.16 Peak deviation (1) and residual deviation of load (2)

The peak component values may also be used. For example, when removing the peak component from load values P ' (t ) = P (t ) − R (t )π (t ) the result may be interesting for analyzing electricity market conditions. Residual and peak components of the load are reflected in load distribution function, which is needed for assessing minimum and maximum load values. Possible deviations are assessed as ΔP = csσ where σ is load standard deviation and cs is the confidence factor. The value of confidence factor comes from given probability α and from load distribution function cS

∫ f ( p)dp = α

−∞

where f is distribution density and p = ΔP / σ is normalized deviation of load. Normal distribution is often considered, but it generally does not apply to electrical network loads. Also, the shape of distribution function depends on how the deviation of load ΔP is defined. Attention is paid mostly to the load possible deviation from its mathematical expectation,

Mathematical Model of Load

Figure 2.17 Histograms of loads in first case

Figure 2.18 Histograms of loads in second case

which is found with the help of the load model as time variable. If such a model is not used, then the load is considered as an average value for a longer time period (e.g., year), and deviation is calculated in relation to that. Also, short-term forecast deviation may be considered in relation to load conditional mathematical expectation, which considers actual load 40

Mathematical Model of Load

progress in the recent past and also possible temperature influence. In Figs 2.17 and 2.18, deviation histograms, which are found according to the load's constant average value (first case) and in relation to time variable mathematical expectation (second case), are shown. On these figures normal distribution is also presented. For assessing maximum load, attention is paid to the tail part of the histograms, which are magnified in Fig. 2.19 for the first case (in the second case the tail part of the histogram is analogous). We can see that loads have considerable probabilities of large deviations, which are practically impossible in case of normal distribution. The differences are more recognizable if the given probability is large. In Table 2.1, the actual confidence factors are shown, found for loads in order of values 10 MW. In the case of smaller loads the situation is even more contrasting.

Figure 2.19 Fragment of histograms Table 2.1 Confidence factor dependency on given probability

Probability Îą

0.9

0.99

0.999

Confidence factor corresponding to normal distribution csn

1.28

2.33

3.10

Real confidence factor cs

1.30

4...6

10...15

Mathematical Model of Load

Residual and peak deviations of load together form the so-called peaknormal distribution. Let us consider that a random variable X has peaknormal distribution, when among normally distributed values sharp deviations appear from time to time – i.e., peaks, which do not conform to normal distribution. The frequencies of positive and negative peaks may be different (including zero). So value X, with peak-normal distribution, consists of normal component X 0 and peak component X p . The peak component consists of positive X 1 , negative X 2 , and zero component Q, respectively. X = X0 + X p X p = X1 + X 2 + Q Distribution density of normal component is −

f 0 ( x0 ) =

( x0 − μ0 )2 2σ 02

2π σ 0 If frequencies of positive and negative deviations of the peak component are λ1 and λ2 , respectively, then, considering that these deviations exclude each other, we get f Π ( x Π ) = λ1 f1 ( x1 ) + λ2 f 2 ( x2 ) + λ0 Here λ0 = 1 − λ1 − λ2 . Presuming that distribution of deviations is lognormal, we may write f k ( xk ) =

1 2π σ k xk

−

(ln xk − μk )2 2σ k 2

, (k = 1, 2)

So peak-normal distribution is described by 8 parameters: μ 0 ,σ 0 , μ1 ,σ 1 , λ1 , μ 2 ,σ 2 , λ2 and it consists of normal distribution, lognormal distribution, and Poisson distribution. The last two can also be substituted with some other suitable distributions. Figure 2.20 illustrates a positive peak component histogram with approximating lognormal distribution. Distribution density of value X can be obtained with convolution f ( x) = ∫

∞

−∞

f 0 ( x0 ) f Π ( x − x0 )dx0

Here f 0 and f Π correspond respectively to distribution density of the normal and peak component. The nature of value x0 depends on how the load deviation ΔP is defined. By considering load deviation in rela-

Mathematical Model of Load

Figure 2.20 Histogram of positive peak component and distribution

tion to average value E , mathematical expectation or long- and shortterm forecasts of the load, deviation may be expressed as follows 1.

ΔP (t ) = E (t ) + R (t )γ (t ) + S (t )[ζ (t ) + ξ (t ) + π (t )] − E

ΔP (t ) = R (t )γ (t ) + S (t )[ζ (t ) + ξ (t ) + π (t )]

ΔP (t ) = S (t )[ζ (t ) + ξ (t ) + π (t )]

4. ΔP(t ) = S (t )[ξ (t ) + π (t )] As values γ (t ) , ζ (t ) and ξ (t ) are actually with normal distribution, then the equation in square brackets is in all cases normal distribution convolution with peak component. The load distribution will be achieved with linear conversion, which takes into account the deterministic functions E(t) and R(t) f P ( P) =

1 t2 − t1

⎛ P (t ) − E (t ) ⎞ 1 dt ⎟⎟⋅ S (t ) ⎠ S (t )

∫ f ⎜⎜⎝ t1

where (t1, t2) is the observed time period. Here the mathematical expectation E(t) is considered only at the first load deviation definition. Distribution function convolution and linear conversion are realized only numerically. In Figs 2.21 and 2.22, examples of load distribution curve are shown, whereby the load deviation is found according to load average value (case 1) and in relation to long-term forecast (case 2). Normal distribution density is also given for comparison. When adding the loads, the sum of normally distributed components is also normal. As a result of convolution, the role of peak component will

Mathematical Model of Load

Figure 2.21 Load histogram and peak-normal distribution at first case

Figure 2.22 Load histogram and peak-normal distribution at second case

decrease until it practically disappears. By summation of large number of loads, the result will be normal distribution like the central limit theorem states. The addends, however, shall be rather large (hundreds). Practically speaking, only transmission grid busloads may be considered as normally distributed. This is valid only for load relative deviations, which are presented above in square brackets. If the distribution is found for a longer time period than the rate of load – and especially because of the change of mathematical expectation – the distribution will remain asymmetric. If, for example, the load deviation is considered in relation to mean value, then the load distribution will remain as illustrated in Fig. 2.21, independent from the number of summed loads. However, the truth is that such type of distribution “tail” is significantly shorter than in the case of peak component, whereby the assessment of maximum load based on normal distribution does not cause any large mistakes. 44

Realization of the Mathematical Model

3 Realization of the Mathematical Model Main expressions of the load mathematical model may be used as they are, but in most cases these expressions need to be expanded and adjusted, depending on the character of load and on the accuracy of modelling needed. Anyhow, many practical problems can be solved by model application.

3.1

Hierarchy of Model Parameters

The load mathematical model consists of a large number of parameters (in the range of 1,000). The principle of modelling is that all of these parameters are somehow determined for each individual load. Simplified models, as in the case of missing load data or for other reasons, are not considered. Model parameters may be estimated for each load individually. Often it is appropriate to find some of the parameters common for certain amount of loads. For example, in a distribution network where the amount of load is large and randomness is high, we may increase the reliability of the parameters in this way. Another problem may be the lack of initial data, which may occur by missing measuring data or if the character of the load has changed and existing data is no longer suitable. If the initial data does not enable estimation of the entire model, then only a part of the parameters will be evaluated on the basis of existing data. The remainder of the parameters will be derived from the type model – from some earlier estimated load model, whose character is similar to the given load. Model parameters may be classified according to the following hierarchy: group of load – vector functions M(h) and N(t) – co-ordinate functions class of load – matrices G r – shape co-ordinates type of load – parameters a lr , blr and clr – shape factors individual load – parameters a, b, c and other factors of the trend – level factors. The co-ordinate functions and shape co-ordinates are shortly called model co-ordinates and type and level factors are called model factors (Fig. 1.2). 45

Realization of the Mathematical Model

The hierarchy of model parameters may be used diversely. As a rule, network loads are classified, and model co-ordinate functions, shape coordinates, and factors are estimated according to respective load groups, load classes, and load types. Only loads that have enough initial data and changes of which are regular (typical) are treated. After that, an appropriate type of model will be set in accordance with the rest of the load, and respective load level factors will be determined. The minimum requirement for the initial data is only one value (e.g., yearly energy) on the ground for which level factor a will be estimated. If the amount of reliable data is large enough, all factors and even co-ordinate functions (i.e., all model parameters) may be estimated for each load individually. The hierarchy of parameters does not require the application of type models, but it supports it.

Figure 3.1 Mathematical expectation of the loads, which is acquired on the same co-ordinate functions of the model

The concept of the type model is principally different from the concept of the traditional type curve of the load. For example, when changing the shape factors of the model, it is possible to achieve most different shapes of the load curve (Fig. 3.1). When only the shape factors are changed, especially the factor a, then the situation is like application of the type curves. It is also possible to displace the vector functions. It enables one to move the daily curve peaks to an earlier or later time, applying the 46

Realization of the Mathematical Model

same co-ordinate functions. The type model also specifies the rate of temperature dependency and standard deviation of the load, whereby, besides change of the load curve level and shape, the temperature dependency and dispersion of the load also change. The model parameter estimation is described in more detail in Chapter 6. The initial estimation of the parameters takes place during the load research, which is done at the beginning of load monitoring. In time, the properties of the load change; therefore, during the load monitoring process, the model parameters should be revised. It is generally adequate to adjust the model factors when the model co-ordinates stay the same.

3.2

The Changes of Load Level

The trend, presented in the mathematical model as a quadratic function, defines the level of the load. For longer period (many years), such presentation of the trend may be insufficient. Some rapid changes of the load may also occur. Jointly with the trend, other properties of the load may also change, such as, the shape of the load curve, temperature dependency, standard deviation, etc. 3.2.1

Trend of Load

The presentation of load trend as a quadratic function is acceptable if the considered time period does not exceed 2…3 years. In longer time periods increment of the load may vary. For example, the sum of the load is dependent on the growth of economy in a certain region, which may change from year to year; network busloads will be dependent on large utilities working order etc. Over longer time periods, better results can be achieved if the trend is presented as a linear fractional with specific time periods. In that case, the trend consists of linear sections a0 [1 + α (t − t0 )] where a0 is trend value at the beginning of time period at t0 , and α is the factor of increase. A suitable time period could be one year, which may correspond to the calendar year. Figure 3.2 illustrates the region's total load values normalized by temperature, mathematical expectation, and trend over 12 years.

Realization of the Mathematical Model

Figure 3.2 Load normalized values (1), mathematical expectation (2), and trend (3) for years 1992â&#x20AC;Ś2004

Figure 3.3 Forecast of the load trend for years 2003 and 2004. Load normalized values (1), mathematical expectation (2), and trend (3)

The terminal part of the actual trend curve in Fig. 3.2 is missing. Here trend and load mathematical expectation growth are found, respectively, 48

Realization of the Mathematical Model

based on the previous year trend factors. If specialists have different views on load increase (considering economic growth and other aspects), the trend factor may be changed respectively. Also, it is possible that there are step-like changes in the trend if, for example, a large consumer appears. This kind of step may occur at any time. According to Fig. 3.3, the increase of the load in 2003 is forecasted at 1 %, which is slightly less than the real growth of 1.3 % in 2002. In 2004, the load growth is 20 % (11 MW), from which 8 MW takes place rapidly on 01.08.2004. The linear fractional representation of trend and possible step-like changes do not require any changes of the basic expressions of the mathematical model. These changes can be realized by program applications. 3.2.2

Load Cases and Scenarios

The step-like changes of busloads of the electrical network may be caused by switchings in lower level networks, start-up or closing of large consumers (e.g., factories), switchings of reactive power compensation equipment, etc. These situations may be called load cases, to which different parameter values of the mathematical model correspond. Figure 3.4 describes the transmission grid busload where step changes are due to switchings in distribution network. We can see two load cases, to

Figure 3.4 Electrical network busload (1), mathematical expectation (2), and standard deviation (3)

Realization of the Mathematical Model

which different levels of mathematical expectation and standard deviation correspond (Fig. 3.5 and Fig. 3.6). These model versions may also be applied in the future for load forecast or simulation if load cases are known or given.

Figure 3.5 Electrical network busload (1), mathematical expectation (2), and standard deviation (3), first case

Figure 3.6 Electrical network busload (1), mathematical expectation (2), and standard deviation (3), second case

Realization of the Mathematical Model

In calculating power network operation, it is appropriate to join load cases into load scenarios, where each load is presented with certain cases because not all load case combinations are feasible. Figure 3.7 describes three busloads of electrical network. Here the switchings, done for equipment maintenance, decrease the load in one substation and increase it in the other substations.

Figure 3.7 Three busloads of an electrical network

Handling of the trend changes and load cases may be carried out by considering different sets of model parameters, each of which is valid for certain time intervals. First of all, the level factors change, but when, with the alternating of load cases, the shape of the load curve varies, then the shape factors of the model must also change. Alternation of load cases and changing of the trend are realized by means of the program, whereby basic expressions of the model do not need to be changed.

3.3

Temperature Dependency and Stochasticity of the Load

When considering load temperature dependency and stochasticity of the load, time changes of the model must be taken into account. The level of temperature dependency and other parameters of the model change seasonally â&#x20AC;&#x201C; but also weekly and daily.

Realization of the Mathematical Model

In the mathematical model Γ (t ) = R (t )γ (t ) time changes of the level of the temperature dependency are expressed by the rate R(t). Changes of the delay and nonlinearity of the temperature dependency may be considered by different sets of component γ (t ) parameters, corresponding to specific time periods. The temperature dependency component γ (t ) model

[

(1 − ϕB )γ t = B m ψ 0 (Tt − E[Tt ]) + ψ 1 (Tt − T1 ) T <T + ψ 2 (Tt − T2 ) T >T t

]

has seven parameters m, ϕ , ψ 0 , ψ 1 , ψ 2 , T1 and T2 . If dividing one year into 4 periods (spring, summer, autumn, winter), a week into 3 periods (working day, Saturday, and Sunday) and a day into 4 periods (morning, day, night first hours, and the rest of the night), we need 48 parameter sets to describe the time changes of the temperature dependency component. Smoothing of changes can be achieved by applying interpolation of annual parameters of the model. The stochastic submodel of the load consists of residual and peak components (Chap. 2.4). There are eight parameters included in both component models – ϕ 1 ,ϕ 2 ,ϕ 24 ,ϕ 168 ,ψ 1 ,ψ 2 ,ψ 24 ,ψ 168 and μ 0 ,σ 0, μ1 ,σ 1 , λ1 , μ 2 , σ 2 , λ2 . The time changes of the peak component need not be considered. Table 3.1 Combinations of the day types

IDay

Current day

0 1 2 3 4 5 6 7 8

0 1 1 1 1 2 2 2 2

Previous day

Previous week

0 1 1

2 2

When describing the residual component, which expresses the stochastical after-effect (autocorrelation), the problem is not time changes in

Realization of the Mathematical Model

general, but the possible combinations of the day types, (e.g., the day types before the present day). It is obvious that the load deviations on workdays are not correlated to the load deviations on weekends and vice versa. The after-effect of load deviations and the optimal values of the model parameters depend on which day type was one day – and one week ago – from the given day. The possible combinations of day types are presented in Table 3.1. Here 1 and 2 represent work and holiday, respectively. 0 shows that the day type is not checked, and a space means that the data is not used (the corresponding parameter in the model is zero). The parameters, corresponding to the type IDay = 1, are applied when there is a workday and the previous day – and the day a week ago – are also workdays. The application of the remaining combinations is also understandable. The parameters of the type IDay = 0 are averages. They can be used when the types of the day are not distinguished.

3.4

Special Days and Periods

Depending on the calendar (country), the number of special days (e.g., national holidays, days before and after holidays, etc.) may reach up to 10 % or more of the total number of days. Since the nature of the load on special days may differ from that on regular days, the special days must also be considered in the mathematical model of the load. Sometimes it is necessary to notice special periods (e.g., summer holidays, turn of the year, etc.) where the load deviates from its regular values on ordinary days for one or more weeks. The special days are represented in the mathematical model with day type l, the value of which is 8 or more (the first seven-day type corresponds to regular days of the week from Monday to Sunday). Hence, when the corresponding number of the special day is known, the appropriate shape factors are found among the model parameters and the computations are normally continued. When estimating the special day shape factors, the load on a special day is compared to the load of the corresponding reference day that is most similar to the special day. The reference days are typical days of the week. For example, Sunday for a holiday (l = 7), Friday for a pre-holiday workday (l = 5), Monday for a post-holiday workday (l = 1), etc. As in the mathematical model, it is possible to assign the type of day l to any date, then the mathematical expectation and temperature dependency of the load, which are calculated on the base of the reference day, are rather 53

Realization of the Mathematical Model

exact enough for a special day in most cases. In a simpler case, the special days are observed as similar days of the week. In that case, the number of day types in the model equals 7. For more precise modelling of the load, the ratios between the loads on special days and mathematical expectations of corresponding reference days are observed. E (t , h, l ) = λl ( h ) E (t , h, l B ) where λl (h ) is the relation function between mathematical expectation of the special day l and reference day lB. For example, in Fig. 3.8 the relation curves of loads on the first of January, normalized by temperature P ′(t ) = P(t ) − Γ (t ) and mathematical expectations of the corresponding reference day (Sunday), are presented for five different years. On the base of the mean relation curve (curve 1 in Fig. 3.8), the shape factors alr could be approximated. The shape factors blr and clr, representing standard deviation and rate of temperature dependency, respectively, are assigned with the same values as those for the reference day.

Figure 3.8 The load relation curves on New Year's day

The estimation of the relation function λl (h ) is difficult due to a large number of special days, which is caused by the fact that certain special days (for example, Christmas) may occur on different days of the week. Therefore, to some extent, the relation functions λl (h ) for the holidays, 54

Realization of the Mathematical Model

and especially pre-holiday and post-holiday days, are different. In the model, those days should be handled with different types. Since those types of special days do not occur often (if ever), the relation functions are estimated interactively. The specialist decides if the result is trustworthy or not. When the estimation results cannot be trusted – or the given special day type is missing – then λl ( h ) ≡ 1 , e.g., the reference day values are attributed to the type factors of the special day. There may also be some difficulties with approximation of the shape factors alr for special days for the initial value λl ( h ) E (t , h, l B ) to be presented with sufficient accuracy. If the load curve of the special day is significantly different from the reference day load curve (the relation function λl (h ) is considerably different from one), the model co-ordinates may not enable adequate presentation of the required daily load curve on given time of the year t. In that case, the rank of the model coordinate functions should be raised, which could be inconvenient. Then it is rational not to approximate the relation function at all and instead place them into the model as a particular special day component. The load characteristics computation program can be developed as universal – when the necessary day type l is not found among the shape factors, then the corresponding special day component is looked for. If that also fails, then the shape factors of the reference day are used. The special day calendar, where the dates and according type numbers of special days are presented, must be created by the end user, since different countries have different calendars which can also change at times. Unfortunately, dependency of special day types on various days of the week makes the preparation of the special day calendar difficult. To help the end user, an additional table is added to the model, where the possible special days are marked with a simple number, and the necessary day type l is found based on a current weekday. Such a table is composed by experts in the load research process. For example, in Table 3.2 there is a sample of special day marks. Here the special day number SpecDay on Christmas Eve is 12, and on Christmas Day it is 13. Depending on the day of the week WeekDay, the reference day RefDay for Christmas Eve is Friday, Saturday, or Sunday. The appropriate day type DayID, based on which the model parameters are found, is 38, 39, or 40. The reference day for the Christmas Day is always 7 (Sunday), but the type of the day also has three values: 41, 42, and 43 noticed that, when the holidays are

Realization of the Mathematical Model

on the weekend, consumersâ&#x20AC;&#x2122; consumption is somewhat different (lower), compared to when the holidays are on some other day. Table 3.2 Special day marks

SpecDay 12 12 12 12 12 12 12 13 13 13 13 13 13 13

WeekDay 1 2 3 4 5 6 7 1 2 3 4 5 6 7

RefDay

DayID

5 5 5 5 5 6 7 7 7 7 7 7 7 7

38 38 38 38 38 39 40 41 41 41 41 41 42 43

Comment Christmas Eve, Mon Christmas Eve, Tue Christmas Eve, Wed Christmas Eve, Thu Christmas Eve, Fri Christmas Eve, Sat Christmas Eve, Sun Christmas Day, Mon Christmas Day, Tue Christmas Day, Wed Christmas Day, Thu Christmas Day, Fri Christmas Day, Sat Christmas Day, Sun

It is not essential to model all possible combinations of the special days and days of week separately. Depending on the amount of initial data and accuracy of the model required, the holidays, for example, could be handled as one case or distinguished, whether the holiday is on a weekday or on the weekend. For more precise modelling, it is possible to survey five cases: the holiday is on Monday, on some middle workday, on Friday, on Saturday, or on Sunday. In the special day calendar, the dates of the special days and numbers SpecDay are given. Ordinary non-special days are not represented in that calendar. The load on special periods, like the turn of the year and summer holidays, should also be surveyed on a daily basis as special days. Practically, however, for those periods, the change of daily load curve is not to be considered as much as the changes of the level of the load. Therefore, it is rational to add the according component to the load trend. Relatively short (a week or two) dips of load may be added to the linear fractional representation of the trend. Location, extent and level of dips are determined by parameters tk, Tk, and Î˛ k as follows: 56

Realization of the Mathematical Model

⎛ ⎡ ⎛ t − t ⎞2 ⎤ ⎞ k ⎟⎟ ⎥ ⎟ . A(t ) = a0 ⎜1 + α (t − t0 ) + ∑ β k exp ⎢− ⎜⎜ ⎜ T ⎢⎣ ⎝ k ⎠ ⎥⎦ ⎟ k ⎝ ⎠ The dips of the load, which may also be positive, are applied only on workdays. In Fig. 3.9, for example, the linear fractional representation of the load trend for five years is presented where the dips during summer holiday periods are noticeable. Although the dips are only a small percentage of the load level, it is obligatory to consider them when accurate modelling is needed.

Figure 3.9 Load trend for years 2001...2005

When estimating special periods, the expert should determine the location and the extent of dips, which correspond to parameters tk and Tk. The level of the dips β k is found computationally. Consequently, it is necessary to match the special days to the reference days – regular weekdays, for which the load-changing pattern best fits the observable special days. For more precise modelling, the components, describing the special days and periods, are added to the mathematical model. It is essential to prepare a special day calendar where the reference days are fixed. Since it is possible to compose the reference day calendar for many years in advance, the end user should improve the calendar only when the list of the special days changes.

Object-Mode Presentation of the Mathematical Model

4 Object-Mode Presentation of the Mathematical Model When realizing the mathematical model as a computer program, it is appropriate to use the object-mode approach. Compared to the traditional manner of programming, the object-mode approach enables the program code to be structured more clearly, and code and data to be combined and treated as unit. From the viewpoint of load handling, it is important that objects can be joined into code components that can be integrated with computer application programs without having to develop any autonomic load treatment programs. The properties and methods of the program objects are accessible in the application programs. It is essential that the load handling process have the buffer files. Initial data, model parameters, and instances properties of the objects are saved there. Binary structured buffer files are handled by methods, not directly by application programs. In this chapter, the objects associated with the mathematical model, which belong to the classes ElmoNet, OLoad, OTemp, and Model, are described. The object classes EModel and Disco are observed in the following chapters.

4.1

Load Net Object

Loads are generally treated as groups â&#x20AC;&#x201C; for example, as a set of several electrical network busloads. There may also be other reasons for grouping the loads. In the load's monitoring system, these groups are named load net. There are not any special criteria for loads to belong to the same net. Only the numbers used for load identification must be unique within the net. The object 1 ElmoNet properties, methods, and buffer file corresponding to the load net are presented in Table 4.1. The properties are displayed in the left column and methods in the right column. Only properties in which every object instance is saved into file ElmoNet.elc are presented.

Here and afterwards the term â&#x20AC;&#x153;objectâ&#x20AC;? can be used, for short, for the object class or object instance. The exact meaning becomes evident from the context.

Object-Mode Presentation of the Mathematical Model Table 4.1 Object ElmoNet

ElmoNet

Other properties are also used, but they are evaluated only during the program working time.

The object name Name and the commentary characterizing the object Comment do not participate in the data handling process and are therefore freely changeable. In Table DirNet\ElmoNet.elc 4.1 these types of properties are marked rw (read-write). Because the load net identification number NetID must be unique, it may only be evaluated with the method Save. The corresponding sign in the table is ro (read-only). The load net directory DirNet indicates the location of the file LoadNet.elc, where it is possible to save the properties for many observable load nets. Name rw NetID ro DirNet rw Comment rw

Make Save Loadids

The method Make reads the necessary properties from the file and forms the running object ElmoNet. Method Save enables the properties to be saved into a file. The arguments of the method are load net identification number, and directory: Make(NetID, DirNet) Save(NetID, DirNet)

Method Loadids(NLoad, ILoad(), Comms())

scans the load objects and data. The result is the number of the loads NLoad, belonging to the considered load net, and arrays ILoad() and Comms(). In the array ILoad(1...8, 1...NLoad) there are properties of loads LoadID, DataID, DS1, DS2, WstID, TempID, ModelID, Density

The meaning of the properties is explained below. Comments characterizing loads are in the array Comms(1...NLoad). The object ElmoNet must be entered with method ElmoNet.Make, when the application program starts to work with component ElmoExe.dll. The application program evaluates the necessary properties NetID and DirNet. In the process, the application program and the objects of the component ElmoExe.dll can use the method ElmoNet.Loadids to obtain information about the loads treated.

Object-Mode Presentation of the Mathematical Model

4.2

Load Object and Temperature Object

From the aspect of the monitoring system, the load is an object, which is characterized by general data, load data and mathematical model. In the object, the properties describe general data. Load data and parameters of the mathematical model are saved into buffer files. Methods of load object carry out the load data handling. The properties, methods, and files of the load object OLoad are presented in Table 4.2. Only properties that are saved into the file Load.elc are shown. Table 4.2 Object OLoad As is the case with other objects, object name Name and commentary OLoad Comment are freely changeable (marked rw). Every load belongs to Name rw some load net (occasionally the net NetID ro has only one load) with identification ModelID ro Make number NetID. The mathematical LoadID ro Save TempID ro DataGet model describing the load (if it exists) BusID rw DataPut has the identification number DataDel Unit rw ModelID. The load itself is identified Scan Density ro by the specific number LoadID, which FileName rw must be unique within the load net. Comment rw The load number is complemented by the location identification number DirNet\Load.elc BusID, which could, for example, be FileName.eld the bus number of the electrical network. The temperature dependency of the load is determined by outdoor temperature, which is measured near the load with identification number TempID.

Load measuring unit Unit is not directly used by the component ElmoExe.dll. The result (load characteristics) has the same unit as for the load data. In the application programs, especially when calculating the operation of an electrical network, the unit of measurement might still be needed. By agreement, the property Unit values correspond to the following units: 0 – undefined; 1 – W, VA, A, V; 2 – kW, kVA, A, kV; 3 – MW, MVA, kA, kV

Object-Mode Presentation of the Mathematical Model Data density Density indicates data sampling frequency. It shows how many times an hour the load has been adjusted. Possible values are Density = 1...60. The data density must be the same for the period in question and match the data density in the mathematical model. If the data measuring frequency changes, then a new model must be estimated. Previously used load data have to be discarded or converted to the appropriate density. In the application program, regular load data is handled as a daily array PA(0...24, 1...Density), where the first index corresponds to the time of

the day and on value 0, when the second index is 1, it is the daily mean value. The second index of the array corresponds to the time interval in an hour. If, for example, Density = 4, then the load (mean) value of the first quarter is on index 1, the load value of the second quarter is on index 2, etc. The load data is saved on to a buffer file, the name of which is given in the load object property FileName. It is possible to save the data of more than one load into the same file. If the number of loads is large, then it is desirable to group the data into different files to avoid file sizes that are too large. The properties of the load object are saved into the file Load.elc and are read there by methods Save(LoadID, NetID, ModelID, TempID, Density, DirNet) Make(LoadID, DirNet, N, IDS()) Here inserting the object the array IDS(1...N) is formed, where all of the

load identification numbers are given, the properties of which are saved in file DirNet/Load.elc. Argument DirNet, as in the case of object ElmoNet, is the load net folder, where the properties are saved. Load data is saved into the file FileName.eld and are read there by methods DataPut(LoadID, DataID, DS, PA()) DataGet(DataID, DS, PA()) Here the data type DataID, which is a property of object Load, indicates

the physical nature of the data as follows: 1 – active power, 2 – reactive power, 3 – current, 4 – voltage, 5 – frequency. The value DS (Date Serial) corresponds to the date. It is the date number, value of which DS = 1 corresponds to date 31.12.1899. It is assumed that the load object properties Density and FileName, likewise LoadID, have already been evaluated.

Object-Mode Presentation of the Mathematical Model Method Scan(DataDS()) scans the load LoadID data in the file FileName.eld. The result is the array DataDS(1...2, 0...5), where bounds of intervals DS1...DS2 of data are presented for data types DataID = 1...5. The overall time interval is placed at the index 0, which considers all data types. Method Load.Scan is, among other things, used by the method of Loadids of the object LoadNet.

The application program saves the load data into the buffer file by method DataPut, not the load monitoring system itself. Because of the variety of manners in which load data is preserved, it would be difficult to create a universal program module for data acquisition. Application programs, on the other hand, may not even handle the load data directly. The nature of the load is determined by application, where the load values are formed based upon the measurements of power, current, and energy. The temperature data is handled simiTable 4.3 Object OTemp lar to load data. The properties, methOTemp ods, and files of the temperature object OTemp are presented in Table 4.3. Name rw The object name Name and commenNetID ro tary Comment are, like other objects, Make ModelID ro evaluated without restrictions. The Save TempID ro identification numbers NetID and DataGet WstID ro ModelID were surveyed earlier. The DataPut BusID rw DataDel TEDispl ro identification number for temperature Scan Density ro measuring place TempID must be FileName rw unique within the load net. The normal Comment rw temperature necessary for modelling (mathematical expectation of the temDirNet\Temp.elc perature) is obtainable from weather FileName.elt stations with enough available data (30 years). If the nearest weather station, (identification number WstID) does not coincide with the location where temperature is measured, possible systematic deviation of the temperature from the weather station data is presented by the property TEDispl. The weather station number is supplemented by number BusID, which can, for example, be a number assigned to the weather stations according to the geographic location. 63

Object-Mode Presentation of the Mathematical Model Due to the relatively slow change of the temperature, and because of its dependency inertia, it is sufficient to measure the temperature once an hour or even more rarely â&#x20AC;&#x201C; for example, once every three hours. Since the temperature data is often fixed together with the load data, the density Density of the temperature data may be greater, but it does not have to be equal with the density of the load data. If the temperature data density is less than the load data density, they are interpolated. However, the minimal data density is 1. If the temperature is measured less often than once an hour, then some of the data values are marked as missing. The properties of the object OTemp are entered and saved by methods Make(TempID, DirNet) Save(NetID, ModelID, TempID, WstID, Density, DirNet)

Temperature data is saved into buffer file DirNet\FileName.elt and read there with methods DataPut(TempID, DS, TA()) DataGet(TempID, DS, TA())

The array TA(0...24, 1...Density) is similar to array PA(), which is used to save the load data. It is presumed that the properties of the temperature object Density and FileName have already been valued. Method Scan(DataDS())

scans the temperature TempID data in the file DirNet\FileName.elt. The result is an array DataDS(1...2), where there are fixed dates of the data interval DS1...DS2.

4.3

Model Object

Object Model is the main object of the monitoring system. With its help, it is possible to find the load characteristics necessary for the application programs. 4.3.1

Object Structure

The properties, methods, and files of the model ModelID are listed in Table 4.4. The object name Name and commentary Comment could be evaluated freely. The identification number NetID shows net, to which the load model belongs. The identification number of the model ModelID must be unique within the load net. The files attached to the object Model may consist of an unlimited number of different sets of model parame64

Object-Mode Presentation of the Mathematical Model ters; therefore, the number of the loads is also not limited. However, the data density Density and special day calendar must be the same for all load models presented. The properties of object Model, shown in Table 4.4 are saved in the file DirNet\Model.elc. Other properties are also used during the application program's working time, but they are not saved in the buffer file. The object properties are entered and saved using the methods Make(ModelID, DirNet) Save(NetID, ModelID, Density, DirNet)

Table 4.4 Object Model

Model

Name rw NetID ro ModelID ro Density ro DirModel rw Comment rw

Make Save PESR TES DPI DPC Char AGroupGet AGroupPut ...

The load characteristics are calculated by the methods PESR, TES, etc. These methods, which are observed beneath, offer different ways to compute the DirNetl\Model.elc load's characteristics but require DirModel\AGroup.elm further knowledge about the model. ... For regular usage, the method Char (described in Chapter 5.4), which is based on the above-mentioned methods, is better suited.

The parameters of the load model are grouped and saved into buffer files by the components shown in Table 4.5. All model parameters belong to the folder DirModel, which is a property of the object Model. The parameters are saved and read with methods of the object Model, which, in the case of annual co-ordinates, for example, are as follows: AGroupPut(GroupID, NAC, MB, ARCrd()) AGroupGet(ModelID, GroupID, ARCrd()) Here GroupID is the identification number of the load group, and NAC and MB are factors determining the rank of the component. The parameters are handled as the array ARCrd(1...24, 1...NAC). Although the meth-

ods covered are accessible by the application programs, handling of the model parameters takes place with methods of the object Model by the computation of the load characteristics. Therefore, the saving of parameter is not further observed here.

Object-Mode Presentation of the Mathematical Model Table 4.5 Model parameters in buffer files

Notation

File

Component of model

N(t )

AGroup.elm

Annual co-ordinates

M(h )

DGroup.elm

Daily co-ordinates

Class.elm Level.elm

Shape co-ordinates Shape factors Level factors

Spec.elm

Special day component

DepTemp.elm

SDCalendar.elm

Temperature dependency component Voltage- and frequency sensibility component Stochasticity component Special day signs Special day calendar

N T (t )

ATemp.elm

Temperature year co-ordinates

M T (h )

DTemp.elm

Temperature day co-ordinates

Wst.elm

Temperature shape co-ordinates

Gr alr, blr, clr a, b, c, ... λ (h )

Type.elm

DepVF.elm Stoch.elm SpecDay.elm

GTE, GTS 4.3.2

Regular Changes

The regular changes of the load are expressed with mathematical expectation, standard deviation, and rate of the temperature dependency. Those values are almost always needed as an independent load characteristic or with combination with other characteristics. The mathematical expectation, standard deviation, and rate of temperature dependency are calculated with the method PESR(TypeID, DayID, LoadID, DataID, CaseID, DS, PESRJ()) Here, the arguments TypeID...DS are input values. The results – load characteristics for one day – are allocated in the array PESRJ(0...24, 1...Density, 1...3). Here the third index of the array points to the load characteristics in the above-mentioned sequence. The argument Density

represents the data density.

Object-Mode Presentation of the Mathematical Model

The arguments of the method PESR, load-identification number LoadID, data type DataID, and date DS were examined previously. The arguments TypeID, DayID, and CaseID are the identification numbers of the model components, which are automatically determined in the working process. When calling the method, these arguments act like marks, which enable different combinations of the characteristics to be found as shown in Table 4.6. Table 4.6 Different modes of methodPESR

Mode

TypeID

DataID

CaseID

TypeID

–1

–DataID

–2

DataID

0 2

CaseID

0 3

DataID CaseID

Result Typical curve according to TypeID Typical curve according to load LoadID Trend according to date DS

Trend according to load case CaseID Characteristics according to date DS Characteristics according to load case CaseID

Option Mode points to the mode of the method PESR, each of which has two options. If Mode = 1, the results are typical curves – i.e., the values of mathematical expectation, standard deviation, and rate, if trend A(t) = B(t) = C(t) ≡ 1. The argument TypeID is the identification number of shape factors. If, in input, TypeID > 0, then the typical curves of load are calculated according to those factors. If TypeID = –1, then the shape factors, which respond to given load with identification number LoadID, DataID, are used. If Mode = 2, where TypeID = –2, the output value is trend, given according to the load case CaseID which is given as an argument to the method or is valid on the date DS. Finally, if Mode = 3 (input TypeID = 0), the characteristics are found on their base form – i.e., both typical curve and trend are calculated and multiplied. It is also possible to vary the computation of the characteristics according to given or current load case. 67

Object-Mode Presentation of the Mathematical Model

Possibilities of the method PESR are still not exhausted. In regular computations, the day type DayID is found according to the current date DS, whereby the input mark DayID = 0. If in the input DayID = 1...7, then the results correspond to the pointed weekdays (from Monday to Sunday). Moreover, if in the level factor entry, the load type number TypeID = 0, then we deal with a trivial model. In that case, the output gives the trends of the observed characteristics. Hence, the method PESR has a range of different possibilities. Those possibilities are used in other procedures of the monitoring system – for example, when estimating the model parameters. In application programs, the method Char (Chap. 5.4) is better suited for calculating the load characteristics. Mathematical expectation and standard deviation of the temperature can be found with the method TES(WstID, DS, Density, TESJ())

Here the results are given as an array TESJ(0...24, 1...Density, 1...2), where there is mathematical expectation at the index 1 (the third index) and standard deviation at the index 2. The argument WstID is the weather station identification number, and parameter DS indicates the date. Density of the temperature in this model is 1. If the argument of the method Density > 1, then the results are interpolated. This is necessary to achieve the same data density as in load characteristics. 4.3.3

Temperature Dependency

In the mathematical model, the temperature dependency of the load is considered as Γ (t ) = R(t )γ (t ) where R(t) is the rate and γ (t ) is the relative component of temperature dependency. Rate R(t), which is calculated together with the mathematical expectation and standard deviation by the method PESR, expresses temperature sensitivity – the load increases when the temperature rises 1 ºC. In a simpler case, the component γ (t ) equals temperature deviation in relation to the normal temperature γ (t ) = ΔT (t ) . In more precise modelling, details of the temperature dependency are considered, starting with the delay of the temperature influence. The relative temperature dependency γ (t ) for given date DS is found by the method 68

Object-Mode Presentation of the Mathematical Model DPI(TypeID, TempID, WstID, DS, ISim, DPIJ())

Here TypeID is the identification number of shape factors, which also identifies the sub model of the temperature dependency. Thus, the relative temperature component is found according to the load type. The shape factors and the identification number TypeID are usually in accordance with the load. That does not exclude the possibility of estimating the model parameters for number of loads. In that case, the corresponding values of the relative temperature dependency factors are equated. Temperature deviation fixed in the measuring point, identification number TempID, is found in relation to the normal temperature, which is inherent to a weather station with identification number WstID. The normal temperature (temperature mathematical expectation) is found using the method TES. Possible systematic deviations from the weather station's normal temperature are considered with the property TEDispl of the object OTemp. The temperature dependency influence is found according to the data saved in the file for temperature data. It is also possible to simulate the temperature in two ways. If the value of the mark ISIM is between â&#x20AC;&#x201C;30...30, then it is considered as temperature deviation for the given day. If ISIM > 1900, then some previous year's temperature values are used, which, of course, have to exist in the temperature data file. If ISIM = 100, then the temperature dependency is calculated based on observable data from the current year. The temperature data file may also include the meteorological temperature forecast, which is replaced by the measured data when they arrive. The method DPI does not differentiate between the real data and the forecast. If the necessary temperature data are missing, then the temperature dependency component is reduced (considering inertia) to zero. The method DPI results are given in the array DPIJ(0...24, 1...Density, 1...3), where, in the place of 1 (the third index), there is the value of the relative temperature dependency component. In the indices 2 and 3 there are, respectively, temperature values and computed temperature deviation values, which are used on load analyzing process, especially in temperature simulation, when the current temperature changes are not used. From the point of view of the application program, the temperature data is fixed and saved with a proper sampling rate. In general, the temperature data density may be between 1â&#x20AC;Ś60. Some of the data values may be 69

Object-Mode Presentation of the Mathematical Model

absent if the temperature is fixed less frequently. In the mathematical model, it is assumed that the data density Density = 1. If the density of raw data is higher, then the necessary congruity is obtained when treating the data by method DPI. The necessary density of data Density in an array DPIJ(), which has to be equal to the load data density, is achieved through interpolation. Because of inertia (inherent to the temperature dependency of the load) and the recursive nature of the ARIMA-model, intermediate results are needed in addition to the current temperature data. Those results are saved, for further use, in buffer file DirModel/TempVar.elm. Since this file is used only by method DPI, it is not examined here. 4.3.4

Stochasticity

The stochastic component of the load is presented as relation Θ (t ) = S (t )[ζ (t ) + ξ (t ) + π (t )] where S(t) is the load standard deviation, ζ (t ) is expected deviation, and ξ (t ) and π (t ) are residual deviation and peak deviation of load, respectively. The expected deviation is needed in applications, since it enables the short-term forecast of the load to be found. The residual and peak deviation may also be used. The standard deviation of the load is found by the method PESR. Other parameters are calculated on given day DS with the method DPC(DS, PD(), DPCJ())

Here in the array PD(0...24, 1...Density) are the values of the relative deviation of the load θ (t ) = [P(t ) − E[P(t )]] / S (t ) as initial values of the method DPC. In the array DPCJ(0...24, 1...Density, 1...3), there are the results of the method, which, according to the third index, are expected deviation, residual deviation, and peak deviation. The intermediate results, in accordance with the ARIMA model, are saved into the buffer file for further handling, as in the case of temperature dependency. For the method DPC the results are saved in the file DirModel/StochVar.elm.

Object-Mode Presentation of the Mathematical Model 4.3.5

Special Day Calendar

The nature of the load on special days may vary considerably from the typical weekday load. It is important that the special day calendar, which is the base for load characteristics computation, is up-to-date. Since calendars are not similar in different countries, the end user must ensure the accuracy of the calendar. During preliminary load research, it is possible to compose a special day calendar for many years in advance. Nevertheless, from time to time, the calendar may need to be corrected. The base for a special day calendar is seen in Table 3.2, where the following parameters of special days: number SpecDay, weekday WeekDay, reference day RefDay and day type DayID are assigned. This table is saved in the file DirModel\Specday.elm. It is read and changed with the methods SpecDayGet(ModelID, SpecDay, WeekDay, RefDay, DayID, Comment) SpecDayPut(SpecDay, Week, Day, RefDay, DayID, Comment) When reading the input parameters are ModelID, SpecDay, and WeekDay or DayID; when saving all the method parameters may be

used. The special day calendar consists of pairs of data â&#x20AC;&#x201C; date DS and special day number SpecDay. The calendar is saved in the file DirModel\SDCalendar.elm. The file is read and upgraded with the methods SDCalendarGet(ModelID, DS, SpecDay, WeekDay, RefDay, DayID) SDCalendarPut(DS, SpecDay) When reading, the input parameters are ModelID and DS; when saving,

both method arguments are used. In conclusion, it should be mentioned that the model object Model buffer file, defined by identification number ModelID, may consist of an unlimited number of parameter sets, and thus the corresponding number of loads is not limited. But there can be only one special day calendar. Hence, if the load net LoadNet contains the loads of the electrical network, which are extended to different countries, the loads must be grouped with countries and their models, and special day calendars must be allocated to different model objects.

Load Monitoring

5 Load Monitoring The concept of load monitoring means load forecasting, imitation, and analysis. Monitoring does not comprise active procedures for load changing, like load management. The success of the monitoring process is based on an adequate mathematical model, which describes the load changes in the observable period, for the past and future times.

5.1

Load Characteristics

The mathematical model describes load, but it does not give practically needed values directly, like load forecast, for example. These values (i.e., load characteristics) can be found, based on the model. Load characteristics may be divided into primary and derived characteristics. Primary characteristics are directly obtained from the mathematical model. Derived characteristics are achieved with combinations of primary characteristics by simple arithmetic relations. When necessary, it is possible to upgrade the primary characteristics by a certain widening of the mathematical model. Sometimes, in application programs, simplified load models are used. As the parameters of such models may be derived based on a given load model, then simplified models may be also considered as load characteristics. 5.1.1

Primary Load Characteristics

Primary load characteristics are mathematical expectation of the load E(t), standard deviation S(t), temperature influence R (t )Îł (t ) , etc. The actual values of load and temperature, P(t) and T(t), may also be considered as primary load characteristics (Table 5.1). Sometimes, additional parameters, which are not included in the mathematical model, are required to find primary load characteristics. For example, when simulating the temperature, the conditions of simulation must be given. The identifier CharID in Table 5.1 is used to compose derived characteristics.

Actual load data A[P] is used in load analysis. These may also be used for finding other characteristics, e.g., the deviation of the load. The need for load data restoration occurs if some of the actual load values are missing or are not reliable. Restored data AR[P] will be found on condition 73

Load Monitoring Table 5.1 Primary load characteristics

Mark Additional in the Name parameters equation PA A[P] P(t) Actual load PRE AR[P] PRE(t) Restored load IPRE, cS Mathematical expectation of PE E[P] E(t) load PS S[P] S(t) Standard deviation of load Rate of temperature dependPR R[P] R(t) ency PET TP[P] A(t) Load trend θ (t ) PD D[P] Normalized deviation of load S (t )ζ (t ) Load expected deviation PC C[P] S (t )ξ (t ) Load residual deviation PX X[P] S (t )π (t ) Load peak deviation PPX P[P] R(t )γ (t ) Temperature dependency PI I[T,P] Simulated temperature PIZ I[Z[T],P] ISIM dependency ΔP(t ) Loss of power PDL DP[P] c0, c1, cS TA A[T] T(t) Actual temperature TZ Z[T] Simulated temperature ISIM Mathematical expectation of TE E[T] temperature Standard deviation of temTS S[T] perature TD D[T] Deviation of temperature

CharID

Notation in the text

P(t ) ⎧ PRE (t ) = ⎨ ⎩ E (t ) + R(t )γ (t ) + S (t )[ζ (t ) + ξ (t )] According to that condition, missing load data will be replaced with short-term forecast to which simulated value of residual deviation of load ξ (t ) is added. Restored load data can be handled with standard statistical methods. For example, where summing up, reliable values of daily or monthly energy can be found even if some of the hourly data is missing. When finding the restored load, the additional parameters are mark IPRE and confidence factor cS. If cS = 0, then only the missing load 74

Load Monitoring

data is substituted. If cS = 2...10, then the load values, deviation of which is too large, are also substituted. For example, if [P(t ) − E (t ) − R(t )γ (t ) − S (t )ζ (t )]/ S (t ) > cS The restored load is issued as load characteristic AR[P] or, if the mark IPRE > 0, then as characteristic A[P], i.e., together with the actual load. Load mathematical expectation E[P], standard deviation S[P], rate of temperature dependency (temperature sensitivity of load) R[P], and load trend TP[P] are achieved from the mathematical model directly. Normalized deviation of load D[P] is found according to the equation P(t ) − E (t ) − R (t )γ (t ) θ (t ) = S (t ) Residual deviation X[P], expected deviation C[P], peak deviation P[P], and temperature dependency I[T,P] of the load result directly from the mathematical model. Their values are normally presented in rated units, i.e., multiplied with load standard deviation or with the rate of the temperature dependency, respectively. In order to find temperature dependency, the actual values of the temperature or meteorologically forecasted data are used. If we simulate the temperature data, then it is possible to find the influence of the simulated temperature I[Z[T],P]. It is possible to simulate the temperature in two ways. If the value of the mark ISIM is between –30...30, then it is considered as temperature deviation for the given day. If ISIM > 1900, then some earlier year's temperature values are used, which, of course, have to exist in the temperature data file. Loss of power DP[P] expresses the approximate value of distribution network losses, which is be found with following empiric equation 2 ⎡ ⎛ P (t ) ⎞ ⎤ ⎟⎟ ⎥ ΔP (t ) = Pmax ⎢c0 + c1 ⎜⎜ ⎢⎣ ⎝ Pmax ⎠ ⎥⎦ where P(t) and Pmax are the total load of the distribution network and its maximum value, respectively, and c0 and c1 are factors. Typically c0 = 1...2% and c1 = 2...6% . If the actual load is considered, then maximum load is found from the actual load data. In the case of load forecast the following equation is applied: CS Pmax = max[E (t ) + c S S (t )]

Load Monitoring

where E (t ) and S (t ) are mathematical expectation of the load and standard deviation, and cS = 2.5...5 is confidence factor. Actual temperature data A[T] is used for temperature dependency calculation. Also the meteorologically forecast temperature may be conditionally considered as actual data. Simulated temperature Z[T] is found if influence of the simulated temperature is calculated. Mathematical expectation of temperature E[T] is the same as normal temperature in a given region (weather station). Standard deviation of temperature S[T] and deviation from the normal temperature D[T] can also be observed. Loss of power DP[P] represents the load characteristic, which is found on the basis of the mathematical model, with additional expressions. Similar primary characteristics may also be added when needed. For that purpose, the computer program must be complemented with appropriate segments, which do not change the structure of the mathematical model or the main principles of the basic procedures. 5.1.2

Derived Load Characteristics

Derived characteristics are achieved by combining primary characteristics. Possible relations for load forecast are, for example, as follows: E[P] – long-term forecast at normal temperature E[P] + I[T,P] – expected value of load (long term) E[P] + I[Z[T],P] – long-term forecast E[P] + I[T,P] + C[P] – short-term forecast Here the expected value of load belongs to the load analysis since it is assumed that the outdoor temperature is known. Examples of other characteristics for load analysis are: A[P] – I[T,P] – normalized load A[P] – I[T,P] + I[Z[T],P] – simulated load Normalized load, where temperature dependency is removed from the actual data, corresponds to normal temperature. Simulated load corresponds to the given temperature in simulation conditions. Load characteristics for practical applications may be derived from primary characteristics in various ways, including the use of multiplication and division as well as addition and subtraction. More sophisticated relations are also possible. The equation for computation of the derived characteristics may be made up by the application program 76

Load Monitoring

end-user. Also, the interpretation of the characteristics is the responsibility of the end-user. As an unusual example, where the load peak component is subtracted from the actual values, “trimmed load” could be given. A[P] – P[P] The result is load values, which correspond to the condition where large deviations could not occur. That characteristic may be used for electrical network planning or for operation in the electricity market. More regular load characteristics for load monitoring are closely observed in Chapter 5.2 – 5.3. 5.1.3

Simplified Load Models

In traditional application programs for calculating electrical network operation, the load of the network is treated simplistically, based on pattern curves. Pattern curves are actually load mathematical expectation of day types (working day, Saturday and Sunday), which are presented, for example, by months. When, in addition to mathematical expectation, standard deviation pattern curves are also given and the equation to assess temperature dependency is added, then it is possible to speak about a simplified load model. The model observed in this book as counterbalance, may then be named as a normal model of the load. If the normal model is developed, then it is simple to derive the simplified model on the basis of that. Load temperature dependency is presented in the simplified model, for example, as the following equation:

ΔP (t ) = c (T (8) − E[T(t)]) E[P(t )] here ΔP(t ) – influence of temperature c – parameter T (8) – mean value of temperature of the given day and previous day at 8:00 a. m. E[T(t)] – normal temperature (here temperature monthly mean value) E[P(t )] – mathematical expectation of load. An example of normal temperature for Finland is shown in Table 5.3. The alternative for temperature dependency calculation is ΔP (t ) = c(T (t ) − E[T (t)]) E[P (t )]

Load Monitoring

where T (t ) is the previous day's average temperature and E[T (t )] its mathematical expectation, which equals several years average temperature (rate of temperature) according to time t of the region. Table 5.3 Normal temperatures

Month Temp. Month Temp.

1 –8.7 7 16.8

2 –8.9 8 14.8

3 –5.4 9 9.6

4 1.3 10 3.8

5 8.1 11 –0.8

6 13.5 12 –0.4

If pattern curves are given monthly, the simplified model of the load consists of the following arrays: EP(1..24, 1..12, 1..3) and SP(1..24, 1..12, 1..3) – load mathematical expectation and standard deviation (24 hours, 12 months, 3 types of day) CT(1..12, 1..3) – temperature dependency factors (12 months, 3 types of day) ET(1..12) or ET(1..365) – normal temperatures (12 months) or rates of temperatures (365 days of a year). In electrical network information system Xpower, the average load of some group is presented as index series ΔE k Qki qki Pki = ⋅ ⋅ 8736 100 100 where ΔE k is k-order yearly average energy consumption of load group and Qki and qki are the outer- and inner-index in time period i. The outer-index corresponds to load seasonal changes in two-week periods, and inner-index corresponds to daily changes (24 hours) separately for each two-weeks period and type of day (working day, Saturday, Sunday). Load standard deviation S ki is presented through inner index rki , which, in previous equation, replaces index qki . Temperature dependency is applied to outer index Qki′ = Qki + ck (Ti − Ti ) where ck is constant factor and Ti and Ti average temperature and its rate for a given two-week period.

Load Monitoring

5.2

Load Forecasting and Simulation

From the above mentioned, it is worth remembering that load forecasting does not directly belong to the mathematical model of load. Rather, the load model describes the regularities of load changes, which, in fact, have been in force in the past. For future prognosis, additional conditions must be submitted, which could change the formation of the load. Only in trivial cases (for example, short-term forecasts) are simulation results received directly from the primary load characteristics. Anyway, additional conditions are present here, too – the lead-time for forecasting the load and meteorological forecasting of temperature. In long-term forecasting (lead-time of a week or more), additional conditions are set directly. First of all, the temperature can be given (simulated). For a longer period, load trend is given. It is possible to vary other conditions as well. In this approach, it is rather conditional to distinguish between load forecasting and simulation. The most probable value of the future load may be called a forecast and other possible variants a simulation. If the lead-time of the forecast is a year or more, then it is advisable to specify the load trend. On the fractional linear form of trend (Chap. 3.2.1), it means changing the growth factor α for the observable year. Otherwise, for the following years, the trend is calculated using the statistically specified value of the growth factor. The experts’ vision is the base for evaluating the trend, considering economic growth and technical conditions. It should be mentioned that, in the mathematical model, the trend expresses the so-called clean growth of the load, where the randomness of the load and temperature dependency, as well as the change of the workday and holiday ratio in different years, have been eliminated. The ratio of workdays can change, for example, when certain holidays fall on workdays or weekends in different years. Since the trend belongs to the mathematical expectation, on estimation of which, in addition to the special days also other exceptional loads, which may be caused by accidents, strikes etc., are excluded, then also those phenomena are not expressed in the value of the trend. Only a mathematical load model can determine such a “clean” load trend, and the results may be a surprise even to experts. The experts should somehow conform to using the mathematical model of the load when determining the trend for the future. The exact determination of the trend may be necessary for larger loads, which are more precisely handled. For smaller loads (distribution net79

Load Monitoring

work busloads, e.g.), specifying of the trend is practically not possible or even necessary because of their noticeable stochasticity. For busloads, it is more substantial to attend load cases (Chap 3.2.2), which are conditioned by switchings in lower-lever networks, irregular work of the factories, etc. The possible load cases are fixed when estimating the mathematical model. Only occasions, that can be determined by the experts and which may occur again in the future are considered as load cases. Accidents and other random events do not belong here. In the mathematical model, only some of the model factors are estimated according to different load cases. When switchings in the electrical network are predicted, it is possible to change the model parameters at the appropriate time for forecasting. Those changes are practically executable in the part of the electrical network that is operationally observable, where the switchings are planned and their time is known. In the operationally unobservable electric network, the time of the load case change is not known, but the consideration of the load cases may be useful here also. For example, there are irregularly working factories, which may work on weekends but not on workdays. If the local network maintenance is planned for the weekend, it must be remembered that such a factory may be working. Once again, only cases, that are technically defined and repeatable are considered as load cases to the extent that is possible and necessary to consider by the planning and managing of the electrical network. When monitoring the load, special days and periods must be considered. While installing the monitoring system, a country's special day calendar is formed years in advance. But the calendar may change, for example, concerning national holidays. One-time changes are also possible â&#x20AC;&#x201C; for example, when a holiday is announced as a workday or vice versa. It should be remembered that the base for considering special days is the reference days â&#x20AC;&#x201C; ordinary weekdays, the nature of the load which is most similar to the special day load. That is why the mathematical model describes the load with considerable precision even when the observable special day is new and the inherent model component is missing. The special day calendar works equally well for larger and smaller loads. However, special periods, which belong to the load trend, are appropriate only for large loads. Mathematical expectation corresponds to the normal temperature. The actual temperature deviation of outdoor temperature causes load increase 80

Load Monitoring

or decrease. While forecasting the actual temperature values are missing. They are substituted by the meteorological forecast or simulated. The meteorological forecast is practically reliable for some days ahead. When the lead-time is longer, the temperature must be simulated. As mentioned above (Chap. 5.1.1), two possible means of simulation were presented, which are based on supplying a certain deviation from normal temperature or by using temperature data from previous year. More possible temperature simulating methods could be added. On load shortterm forecast, it is possible to vary the temperature forecast deviation in order to determine the possible limit values of the load. The short-term forecast of the load is obtained by adding the temperature influence and the expected deviation to the mathematical expectation. The basis for calculation of expected deviation is the load data normalized according to the temperature for the last 10 days. For that, the actual load and temperature data are necessary. For forecast periods, where the actual values of the load are missing, the expected deviation decreases when lead-time increases, practically becoming zero after 7â&#x20AC;Ś10 days. In addition to the above-mentioned variability of the temperature forecast, it is possible to evaluate the possible load deviation compared to the ordinary deviation in order to detect the confidence interval of the forecast. According to the model of the stochastical component of load (Chap. 2.4), the load deviation consists of residual deviation with normal distribution and peak deviation with lognormal distribution. The standard deviation of the residual deviation is in the range of 0,5â&#x20AC;Ś1 compared to the standard deviation S(t) of the load. When necessary, that ratio can be specified according to ARIMA-model. The peak deviations occur on smaller loads (less than 10 MW) and reach up to ten times of standard deviation values of the load or even more. However, large deviations of the load occur only rarely and in the case of larger loads, they do not occur at all. Figure 5.1 provides an example of load expected value and short-term forecast. Expected value corresponds otherwise to long-term forecast but the temperature influence is calculated on the basis of actual data. The short-term forecast is acquired by adding the load's expected deviation to the long-term forecast. In Fig. 5.2, in addition to the short-term forecast, the actual and normal temperature (which is scaled into right axis) is presented. In Fig. 5.3, the mathematical expectation (curve 1) and 81

Load Monitoring

simulated values are presented. Curve 3 corresponds to the temperature deviation of â&#x20AC;&#x201C;10ÂşC of the normal temperature. Curve 2 is generated using temperature data from a previous year, when there was a cold winter.

Figure 5.1 Actual load (1), expected value (2), and short-term forecast (3), hourly values

Figure 5.2 Load short-term forecast (1), actual load (2), actual temperature (3), and normal temperature (4) hourly values

Load Monitoring

Figure 5.3 Mathematical expectation of the load (1) and simulated values (2) and (3) on daily level

Load forecasting, as a practical implementation of the mathematical model, is generally an interactive activity from our point of view. Experts evaluate possible future situations and formulate forecasts according to the available information. Experts must know the possibilities of the monitoring system but not necessarily the structure of the mathematical model. The application program must have a user interface suitable for the experts handling forecast procedures. Monitoring systems make necessary changes to the model parameters and carry out calculations.

5.3

Load Analysis

Load analysis, like forecasting and simulating, is not definable by the mathematical model but rather by the end-users of the monitoring system. On the basis of the mathematical model, it is possible to find different load characteristics. The purpose of using them is determined by the end-user. The formal indication of analysis is that the actual values of the load are mostly known. One practical purpose of load analysis is observation of the regularity of load principle changes, and the other is to ensure that the load data accords with the mathematical model. The latter is necessary for clari83

Load Monitoring

fying the need for model adjustment and for discovering large measurement errors (mistakes). When observing the reasons for load changes, the temperature influence is often considered to be the most interesting one. The data, normalized by temperature, are obtained if the temperature influence is subtracted from actual data. If the simulated temperature influence is added to the result, it becomes evident what the load could have been like on a cold winter, for example (Fig. 5.4).

Figure 5.4 Load actual (1), normalised (2), and simulated (3) values on daily level

The features of the load curves mostly appear via observation of the load data without even using the mathematical model. With smaller loads, the load stochasticity can be so large, that regular changes of the load are not noticed. In that case, it is possible to determine the load character through mathematical expectation. In Fig. 5.5, the private household load (curve 1) is comparatively irregular. Based on model, it appears that the mathematical expectation of the load (curve 2) is noticeably larger late in the evenings and in the first hours of the night, apparently caused by electrical heating.

Load Monitoring

Figure 5.5 Load value (1), mathematical expectation (2) and standard deviation (3)

The restoration of the data (Chap 5.1.1) may be necessary to make various conclusions about the load. If some data is missing, averaging of the loads (energies) and calculating the sum loads is not possible. Fig. 5.6 is an example of the load where some data is missing (curve 1). Necessary

Figure 5.6 Measured (1) and restored (2) load

Load Monitoring

daily average values can be calculated on the basis of restored load values (curve 2). The mathematical model's suitability to the load can be determined by comparing the load data to the short-term forecast, taking the lead-time Ď&#x201E; = 1 . Close accordance of those characters, as in Fig. 5.1 and Fig. 5.2, indicates the preciseness of the mathematical model as well as the reliability of the load data. For extensive measurement systems, like SCADA system, some measurement transducers or other equipment may fail without even being reflected on system work. If, for example, a load is supplied through two transformers and one transformer transducer fails, then it would appear that the observable load reduces the amount of that value. Because that kind of a load reduction is not noticed by SCADA, the failure of a transducer may go unnoticed for a long period of time. However, the monitoring system of the load allows one to discover that kind of deviation. Experts, of course, determine the reasons for large load deviations. Those changes can also be load cases, which need appropriate handling.

Figure 5.7 Load value (1), mathematical expectation (2), and standard deviation (3)

Based on load characteristics it is possible to make conclusions about consumers, too. It is possible to determine the regularities of factory work, etc. An exotic example is presented in Fig. 5.7, where a fragment 86

Load Monitoring

of load characteristics from a private household is presented for six days (from Saturday to Thursday). The load peak in this case is explained by use of the sauna electricity stove twice a week – on Saturdays and on Wednesdays. The mathematical expectation of the load and standard deviation, which are estimated on the basis of data for many years, indicates that the usage of the sauna in that manner is regular.

5.4

Computation of the Load Characteristics

The primary load characteristics necessary for monitoring may be found by methods of the object Model described in Chapter 4. The derived characteristics in the application program may be calculated using the primary characteristics. The practical implementation of the methods requires some basic knowledge about the possibilities of the mathematical model. Also, these methods examine load characteristics only one day at a time. That is why the general method Char, which is much easier to the custom user, has been added to the object Model. Method Char is based on the methods described in Chapter 4. The method Char computes the characteristics of one load with properties LoadID, DataID for time interval DS = DS1...DS2 Char(LoadID, DataID, DS1, DS2, Formula(), IChars())

The argument Formula(1...NFChar) of the method Char is a string array, which includes equations for derived characteristics based on primary load characteristics (Chap. 5.1.2). The equations consist of the identifiers CharID (Table 5.1) of the primary characteristics and of possible numerical constants, which are bound with operations like adding, subtracting, multiplying and dividing. Dividing can only be done with standard deviation and rate, since those numbers are never zero. In array IChars(1...NFChar) the indexes corresponding to the equations are presented. It is considered that, in an application program, it is rational to make the array Formula() reusable, whereat on one call to method Char, only part of the characteristics are found – those, indexes of which in array IChars() are larger than zero. On repeated call on method Char, other characteristics are calculated. If the same characteristics are calculated, similar characteristics are marked with different index KChar. Examples for equations of the load characteristics were presented in Chap. 5.1.2. Here, an example of usage of numerical values is shown. If a confidence interval of load change is wanted, which corresponds to condition ±3 σ , then the load characteristics must be found 87

Load Monitoring PE â&#x20AC;&#x201C; 3 * PS and PE + 3 * PS

Load characteristics in question are saved into the buffer file DirModel\Char.elm when they are calculated. The characteristics are read from the file by method CharGet(LoadID, DataID, IChar, KChar, DS, Char())

where IChar = 1...NFChar is the index of the equation of the characteristic, DS is the date, and Char(0...24, 1...Density) is the array consisting of values for one day. The load characteristics, found with method Char, are saved for so-called immediate usage, since the computational equation is not saved, and the current index IChar may respond to different load characteristic by next call of method. If the above-mentioned possibility is followed, that the equation array Formula() is formed as suitable for all computations in the application program, then disarrangement would not happen even when the characteristics are saved by various computations. The load characteristics may be read from the file and used in the application program immediately after they are found. If the method Char is repeatedly called, then it is possible to get an overview of the file DirModel\Char.elm with method CharScan(NChar, CharInf())

which delivers a number of characteristics NChar and an array CharInf(1...6, 1...NChar), with the attributes LoadID, DataID, IChar, KChar, DS1, and DS2 for each load characteristic saved into the file. Here DS1...DS2 present the time interval, to the extent that data for the observable characteristic is available. On repeated use of the method Char, the size of the file DirModel\Char.elm may become too large, and some of the characteristics or the file should be deleted. The characteristics are deleted with method CharDel(LoadID, DataID, IChar, KChar)

which work as: if LoadID > 0, DataID > 0, IChar > 0 and KChar > 0, then one of the characteristics is deleted if LoadID > 0, DataID > 0 and IChar = 0, then all the characteristics according to data type DataID of the load LoadID are deleted if LoadID > 0 and DataID = 0, then all of characteristics according to the load LoadID are deleted if LoadID = 0, then the file DirModel\Char.elm is completely deleted. 88

Load Monitoring

When deleting, the characteristics are not physically deleted, except in cases where the file is deleted entirely, but are marked as missing, and can be restored later. If in method CharDel set LoadID < 0, then the characteristics restoration is performed similarly to the deletion process. For some primary load characteristics computation, additional parameters must be provided (Table 5.1). For example, when simulating the temperature influence, the simulation constant ISIM should be marked. Values of the mark ISIM between â&#x20AC;&#x201C;30...+30 are added to the normal temperature as simulated deviation. If ISIM > 1900, then the temperature dependency is found using the corresponding year's data. The parameter ISIM and other necessary additional parameters must be saved into the file DirModel\CharPar.elm before calling the method Char. Additional parameters are saved with the method CharParPut(LoadID, DataID, CharID, CPar())

where CharID is the identification number of the characteristic. The array CPar(1...12) consists of the necessary (maximum 12) additional parameters. When saving the character number CharID = 0 or DataID = 0, then the given additional parameters apply to all loads or data types. The additional parameters are read from the file by method Char, if needed. For supervision purposes they can also be entered into the application program, using the method CharParGet(LoadID, DataID, CharID, CPar())

Since the actual values of the load and temperature also belong to the load characteristics, it is sufficient to use only the method Char of the object Model for monitoring the load in the application programs. Additionally, the method DataPut of the objects OLoad and OTemp must be used to save the actual load and temperature data into a buffer file. Thus, methods OLoad.DataPut, OTemp.DataPut, Model.Char, Model.CharGet, and Model.CharDel are sufficient for load treatment in the application programs. Other possibilities of the objects are used only rarely.

Model Estimation

6 Model Estimation The structure of the mathematical model is formed according to the physical nature of the load with engineering direction concerning the purpose and accuracy necessary for the model. The structure of the model is the same for all loads. The model parameters must be estimated on the particular load data to describe the specific load. The main phase of the estimation, the initial estimation, is performed during the load research, where, for example, a certain electrical network load is operated. In the load research, only loads for which enough hourly load data is available (at least one year) and changes of which are regular, are estimated. The result of the initial estimation is the type models (the model components and the typical sets of factors) for the electrical network. The main components of the model are co-ordinate functions, and shape co-ordinates, and the sub models describe the temperature dependency and stochasticity of the load. In the second phase of estimation, the model editing, the appropriate type model is attached to every load. In order to form load models for a certain load, a proper number of model parameters corresponding to the available load data is estimated. At least one parameter must be estimated for every load. If there is enough load data, it is possible to estimate the unique model for every load. When the load changes are irregular, the trivial model is used. During the estimation, all available quantitative and qualitative load data is used. As a result of the estimation, all models will have the same structure. Therefore models for all loads are equivalent in practical applications. The accuracy of the model, however, depends on the amount and the quality of the data used for estimation.

6.1

Estimation Principles

The components of the mathematical model are estimated using the hourly load data, which have been fixed at certain data density â&#x20AC;&#x201C; i.e. one or more times per hour. Hourly data and temperature data of the load for the entire year are needed to implement the estimation process. There should be considerably more data available for several years, in order to obtain reliable results. However, over a longer period, the load characteristics may change, which prevents more accurate results from being achieved. For counterbalance, it is possible to estimate the model co91

Model Estimation ordinates and other components jointly for a group of loads. Poor determinacy between the mathematical expectation and the temperature dependency of the load is also a problem in the estimation process. Sometimes it is difficult to determine whether load deviations mean the change of the load character (mathematical expectation) or if it occurs due to the influence of the temperature. 6.1.1

Model Co-ordinates

The estimation of the model components originates from the evaluation of the mathematical expectation. The traditional evaluation of the mathematical expectation (based on mean value) is not possible to use due to load changes. Another way is to smooth the load values with plain curves. Here, it is also difficult to find the appropriate load data for smoothening because, for example, the load daily changes occur too quickly. It is possible, though, to find the loads, change in time of which is smooth. Let us name the values of load data, fixed in weekly time over the course of a year as characteristic vector. One characteristic vector consists of 52 load values (the possible 53rd value is not considered). The total value of the characteristic vectors in a year equals the number of weekly hours (168) multiplied by the data density. The characteristic vectors are smoothened by a lower order (4…5) Fourier series MB ⎡ ⎛ 2π ⎞ ⎛ 2π ⎞⎤ Si (t ) = ai 0 + ∑ ⎢aij′ sin⎜ jt ⎟ + aij′′ cos⎜ jt ⎟⎥ ⎝ T ⎠ ⎝ T ⎠⎦ j =1 ⎣ where MB is the series order and T is the yearly period. The least-squares method is used for smoothening using the criterion Wi = ∑ [Pik − S i (t k )] 2 = min k

where Pik is the load value at the time t k . The observed criterion gives us the linear system of equation to determine the parameters a0i , aij′ , aij′′

∑ Sk STk Ai = ∑ Pik Sk k

where Ai and Sk are vectors A i = [ai 0 , ai′1 , ai′′2 ,..., ai′,MB , ai′′,MB ]

⎡ ⎞⎤ ⎛ 2π ⎞ ⎛ 2π ⎞ ⎛ 2π ⎞ ⎛ 2π tk ⎟, cos⎜ tk ⎟,..., sin⎜ MBtk ⎟, cos⎜ MBt k ⎟⎥ S k = ⎢1, sin⎜ ⎝ T ⎠ ⎝ T ⎠⎦ ⎝ T ⎠ ⎝ T ⎠ ⎣ 92

Model Estimation

The adding up is performed above the data (k ≤ 52), appropriate with the given characteristic vector. The elimination of inappropriate data is a problem when smoothening the characteristic vectors. They correspond to the non-characteristic network state (including special days) or are errors of the data acquisition system. Some data may also be missing. The data corresponding to special days may be fixed, based on the special day calendar. Missing data is marked with an appropriate sign in the initial data. In order to determine other inappropriate data, the load Pik and smoothening function values are compared. The value of the load is valid, when Pik − Si (tk ) < c Sσ [Pi ] where σ [Pi ] is the standard deviation of the characteristic vector and cS is the confidence factor. The standard deviation is found using criterion 1 σ [Pi ] = ∑ [Pik − Si (tk )]2 n −1 k where the addition is performed above the appropriate values of the load. Since not all inappropriate data is previously known, an iteration process is needed where the data corresponding to special days and missing data is declared as inappropriate. The status of other data can change in both directions; the values can be marked as inappropriate, and the values

Figure 6.1 Load characteristic vector and smoothening curve

Model Estimation

previously declared as inappropriate can be restored later. Such an iteration process converges if the confidence factor cS is large enough. The best results are obtained when the confidence factor is cS = 2.7. An example of smoothening the characteristic vector is presented in Fig. 6.1. Here the appropriate data is marked with a plus, the square corresponds to the special day data, and the data determined to be not applicable in the smoothening process is marked with cross. The smoothening curves of the characteristic vectors represent the mathematical expectation of the load. These are modified into the predetermined form of the model afterwards. The modification process, where the model co-ordinates and factors are found, takes place using the methods of applied mathematics, which are not observed here. If the number of estimated loads is large (e.g., electric network busloads), then the co-ordinate functions and shape co-ordinates are estimated according to the load classes, respectively. Hereby, the load classification, which was performed, considering the nature of the consumers, is elaborated. The mathematical expectation and the standard deviation of the temperature must also be modelled. The normal temperature, which usually is found in weather stations as a 30-year average value measured every hour, is the base for evaluating the temperature mathematical expectation. The estimation of the mathematical expectation principally means approximation of the normal temperature in a form suitable for the model. Weather stations do not usually have the temperature standard deviation values. The standard deviation can be evaluated on the ground of the measured temperature data. 6.1.2

Temperature Dependency Model

The temperature dependency of the load is expressed by the rate R(t) and relative temperature dependency component γ (t ) (Chap. 2.3) as Γ (t ) = R(t )γ (t )

[

(1 − ϕB )γ t = B m ψ 0 (Tt − E[Tt ]) + ψ 1 (Tt − T1 ) T <T + ψ 2 (Tt − T2 ) T >T t

]

The rate of the temperature R(t) is estimated at the same time as the mathematical expectation and standard deviation (Chap. 6.3). The model of the component γ (t ) consists of seven parameters m, ϕ , ψ 0 , ψ 1 , ψ 2 , T1 and T2 . The estimation is aggravated because parameters change over. The nature of the marginal components (the last two addenda of the 94

Model Estimation

component) must also be identified. The principal problem in the observable period is the particularity of the weather type for available load data (sufficient time for rotation of weather types is considered 30 years) and the poor determinacy between the mathematical expectation and the temperature dependency of the load. The component γ (t ) may be estimated according to load classes to increase the model's reliability. (Rate R(t) is always inherent to a load). The similar response to the temperature deviation is one of the bases in load classification. The parameters of the component γ (t ) are estimated by characteristic time interval to consider such changes. It is necessary to divide the year into at least four periods (spring, summer, autumn and winter). For more precise modelling, it is possible to observe the three periods of the week (work day, Saturday, and Sunday) and four periods of the day separately (morning, day, the first hours of the night, the rest of the night). The model parameters of temperature dependency are estimated in two phases due to the above-mentioned difficulties. First, the nature of the load temperature dependency is investigated, and the structure of the model (time periods, need for marginal components, etc.) and compatible loads are determined. In the second phase, model editing, the model parameters are estimated conclusively. The indication curves – functions of the residual of the normalized deviation of the load dependency from temperature Δθ (t ) = f(T ) – can be used as one medium to research the temperature dependency. The residual is expressed as P (t ) − E (t ) − γ (t ) R (t ) where the component of temperature dependency γ (t ) is calculated on given model parameters. The residual value does not depend on temperature (the indication curve is practically horizontal) if the model parameters are appropriate for given conditions. Δθ (t ) =

Figure 6.2 and Figure 6.3 give examples of indication curves in the winter for loads, which belong to the same class (residence, direct electrical heating) and to the same weather station. At first, the temperature dependency is not considered (Fig. 6.2). The situation in the second case corresponds to the temperature dependency model with parameters m = 5, ϕ = 0,91 (H0 = 5 and H = 10 hours), ψ 0 = −0,06 , ψ 1 = 0,03 ,

Model Estimation

ψ 2 = 0 , T1 = −20 and T2 = 0 . The marginal components of the model T1 = −20 and indicate that when the temperature goes below –20 °C,

then the load increase decelerates.

Figure 6.2 The initial load indication curves

Figure 6.3 The load indication curves when considering the temperature dependency

Model Estimation

Based on indication curves it is possible to find the necessary number of time periods and their limits, determine the purpose of the marginal components, and inquire which loads can be considered as belonging to the same class. Afterwards the model parameters can be estimated conclusively, using the methods of the applied mathematics. Although the number of simultaneously estimated parameters is only seven, problems arise because of poor determinacy of the model. For example, when the temperature deviation ΔTt is on the same level for a longer period, then the delay of temperature influence (which is described by parameters m and ϕ ) does not have any significance. From the viewpoint of the estimation, it means that the estimates of those parameters can have large errors. It is especially difficult to distinguish the parameters m and ϕ using statistical methods. The evaluation of the marginal components is aggravated due to the small amount of data where the temperature deviations are large. An algorithm is needed, where the model parameters are partially estimated. Firstly the parameters ψ 0 and ϕ – the level of the temperature dependency and delay – and then both marginal component parameters ψ 1 , T1 and ψ 2 ,T2 are evaluated. The parameters, describing the delay m and ϕ , are attached to each other vϕ m = vH = 1−ϕ The ratio v is determined via special research. The estimation takes place according to a given time interval and other conditions, which are determined in the research phase of temperature dependency.

The mathematical expectation of the load principally corresponds to the normal temperature. Therefore, estimation of mathematical expectation is based on the data normalized concerning the temperature P ′(t ) = P (t ) − R (t )γ (t ) As a temperature dependency model is initially missing, an iteration process must be arranged where the mathematical expectation and the rate of temperature dependency are adjusted after evaluation of the component γ (t ) . Practically speaking 2…3 iterations are needed. Still the result is not always reliable. Poor determinacy between mathematical expectation and temperature dependency arises. For example, when the observable time period is two years and both winters are mild, it is statistically difficult to find what the load level could have been on a normal or cold winter. Consideration of the daily temperature depend97

Model Estimation

ency changes may also falsify the change of the mathematical expectation curve. Because of insufficient data, we must agree upon a simplified temperature dependency model (i.e., simplified estimation of the model). There are no problems if the temperature dependency of the load is small. 6.1.3

Other Model Components

Appropriate estimation of the mathematical expectation and temperature dependency of the load is crucial from the viewpoint of model adequacy. Other model components, that express the load's stochasticity and change on special days, are estimated later, without any reaction to the mathematical expectation and temperature dependency. Principally, the load voltage and frequency sensibility also have an effect on the load. However, it is assumed that voltage and frequency deviations are relatively short and that they are smoothened in the estimating process of mathematical expectation and temperature dependency. Furthermore, the voltage sensitivity has some significance only in the case of electrical network busloads, where the model's preciseness is inevitably lower. The normalized value of the load stochastic deviation is determined on the ground of the mathematical expectation, standard deviation, and temperature dependency as 1 [P(t ) − E (t ) − R(t )γ (t )] ϑ (t ) = S (t ) in which the load residual and peak component appear. The residual component ξ t is presented, using Box-Jenkin`s model Ψ( B ) ξt Φ( B) where operators Φ (B ) and Ψ (B ) are given as

ϑt =

Φ ( B ) = (1 − ϕ1 B − ... − ϕ MF B MF )(1 − ϕ M B M )(1 − ϕ N B N ) Ψ ( B ) = (1 − ψ 1 B − ... − ψ MP B MP )(1 − ψ M B M )(1 − ψ N B N ) Here, the daily displacement factors MF and MP are actually in the range of 1…2. M = 24 and N = 168 mean the consideration of the daily and weekly after-effect, respectively. Hence, 8 model parameters are to be estimated: ϕ 1 ,ϕ 2 , ϕ 24 ,ϕ 168 , ψ 1 ,ψ 2 ,ψ 24 and ψ 168 .

Model Estimation

The estimation of the residual component is applicable with the methods of computational mathematics with the criterion of the minimum load short-term forecast error. The day type combinations, which form nine versions of the model, are presented in Table 3.1. If the load changes are regular enough or the short-term forecast does not offer interest, then it is possible to limit only with one version of the model, where the day types are not distinguished. The model structure's correspondence to a real situation may cause some difficulties in the estimation process. If one of the model parameters proves to be redundant, then the estimation process may not converge or give wrong results. Therefore, it is necessary to conduct the estimation for many different cases, starting with the simplest. Estimation of the load peak component is difficult due to poor determinacy between normal and lognormal distribution belonging to the peak normal distribution. Since the load distribution is rarely needed in practice, the estimation of the peak component is not observed here. When handling special days, the reference day (which can be some appropriate regular weekday) is attached to every observable special day type; for example, holiday â&#x20AC;&#x201C; Sunday, pre-holiday workday â&#x20AC;&#x201C; Friday, etc. When estimating the relation curves, loads are first normalized by temperature and the mathematical expectation is calculated according to the corresponding reference day. Then, the mean relation curve is approximated to find the reference day type shape factors. The approximation may also be overlooked, in which case the mean relation curves are used later in their original form. A small number of days of the observable type may be a problem when finding the relation curves. Therefore, the relation curves are evaluated interactively, at which the expert decides which point curves are useable and which are not. In Fig. 6.4, the relation curves on a Midsummer Day for 8 years are presented. The reference day is Sunday. One can see that the load is a little lower than the corresponding mathematical expectation found on the reference day (i.e., Sunday), but there are exceptions also. The mean value (red curve) in the figure is found using the blue relation curves. The green relation curve is not used, because the expert considers them not to be reliable.

Model Estimation

Figure 6.4 The relation curves corresponding to the Midsummer Day

The special day components (relation curves) may also be evaluated on the type bases, as united for many loads, in order to increase the reliability of the estimation. It is also possible to quit the estimation if there is not enough data for the special day type given. Then the special day component is identified as one, which means that the mathematical expectation is found according to the corresponding reference day. The load's standard deviation and rate of temperature dependency are always found according to the reference day.

6.2

Load Research

Based on previously described principles, the estimation of the mathematical model may seem complicated. From the viewpoint of engineers engaged in the network's operation, planning and management, and other practical power-engineers, it is true â&#x20AC;&#x201C; deeper particularization into load modelling would require too much of their time and abilities. Actually, the model estimation may be performed in two phases. First, in the load research phase, the principal questions of estimation are handled. Next, in the model editing phase, the models are adjusted, upon consideration the changed load formation conditions. The load research is mostly necessary only when starting the monitoring, and because of this professionals can perform it as contract work. The model editing is possible on 100

Model Estimation

the engineering bases, which does not require more intimate knowledge of the modelling principles. The application program user interface must be on necessary level. If the number of loads handled is large (for example electrical network busloads), then the procedures, belonging to the load research, need to be conducted later also. However, the number of such procedures is limited. Although the main purpose of load research is to estimate type models, the entire task is much larger. The load research includes: defining the purpose of the load research composing the load list forming and storing of the load data developing the special day calendar modelling temperature estimating model co-ordinates and temperature dependency estimating the special day component and stochastical component of the model editing the model verifying the model. The depth and the content of the load research depend on the situation. In the simplest case, when only one load is treated with adequate data, and in an observable region where the load research has been conducted before, the load research simplifies to some semi-automatic execution of typical procedures. On the other hand, when the number of the loads is high, the amount and quality of the data is insufficient and the nature of the loads is uncertain, then more extensive work should be performed. The purpose of using load models and where they will be implemented is also important. Although the model structure is the same for all the loads, the means for estimation are not. They depend on the available data and the required accuracy of the model. For example, if the volume of data is insufficient, then the type models are used; if the temperature dependency of the load is small (for example, industry load), the relative temperature dependency component is not estimated, etc. If the systematic acquisition of load data starts only when the load monitoring system is implemented, the load research should be conducted in different stages, according to the data obtained as well as to the increase of the experience.

101

Model Estimation

The first step of the research is to compose of the load list, which includes the general data (name, type, connection point of load, etc.). Depending on the situation, the necessary amount of work may be trivial or quite large. It is possible to estimate a unique model for every load if enough data is available for all observable loads. If the number of loads is high (hundreds or thousands), then it is still rational to estimate the model co-ordinates, sub models of the temperature dependency, etc., as typical for a certain number of loads. That demands research concerning the nature of the loads and classification according to the load groups and load classes. If there is not enough data available for some kind of loads, then the appropriate type model must be found or an estimation of the type model should be accomplished. In the verification process, if it turns out that the choice of the type model has failed, then the classification of the loads must be renovated, and the model estimation process must be repeated. The amount and the quality of the load data used in the estimation of the mathematical model may be different. Necessary load data rarely coincide with the network measurements. In most cases, there are sums of measurements, which may include losses also. For example, in the case of electric network busloads, transformer losses may or may not be included in the busload, depending on how the network operation is calculated on the higher level of the network. The density of the load data must be determined – how many times per hour the data is fixed. If the data is obtained from the dispatch system (SCADA), where, for example, the data is fixed once a minute, then it is possible to average them into hourly data, half-hourly data, etc., depending on how the load characteristics (load forecast, etc.) are to be implemented. Possible missing data may be a problem in the process of calculating. The data may not be added if even one of the addenda is missing – for example, because of an error in the measurement system. For larger number of addenda, the added load forms may be excessively “holey” – only a part of the data can be found. A solution in that situation is to use restored data (Chap. 5.1.1), but that requires modelling of the added loads. The temperature data must not be forgotten. Temperatures must be measured various times a day at allocation near the loads. The number of fixed temperature data during 24 hours must be sufficient for being interpolated to the load data density. Daily average temperatures are not suitable for that purpose.

102

Model Estimation

In order to manage load data, the special day calendar must be developed, and the mathematical expectation and standard deviation of the temperature must be modelled. The special day calendar, containing national holidays, pre-holiday and after-holiday days and other systematic deviations from regular weekdays, is national and may be composed many years in advance. The ability to change the calendar (e.g., if some workday is declared as holiday or vice versa), can be accommodated in the application programs. The mathematical expectation of the temperature, which is used to find the temperature deviation, is estimated based on normal temperatures that represent the last 30 years of average temperatures every hour. Normal temperatures are available from weather stations that have been functioning long enough. The possible systematic deviation between temperature measurements and normal temperature is added to the mathematical expectation when finding the temperature dependency of a certain load. This is because the location of those weather stations may not coincide with the place at which the temperatures were measured. Sometimes the standard deviation of the temperature may also be needed. The standard deviation may be evaluated on the basis of temperature measurement data, since the standard deviation of hourly data generally does not exist even in weather stations. The number of observable weather stations need not be large. For example, it is considered sufficient when the temperature is modelled from 6 weather stations in Estonia and 10 in Finland. Developing the load model starts with estimating the co-ordinate functions and shape co-ordinates. Typically the load data, normalized by temperature, should be used as initial data. At first, the temperature dependency model does not exist, and the iteration process, which proceeds with non-normalized data, must be arranged. However, the checking calculations show that the temperature influence on the model co-ordinates is not considerable. For the following evaluation of the shape factors, which enable the mathematical expectation and the temperature dependency of the load to be found, the data should be still normalized. Proceeding from the non-normalized data, the preliminary temperature dependency model is obtained, on the base of which the data is normalized and the estimation of the model is repeated. Practically, 3â&#x20AC;Ś4 iterations are sufficient.

103

Model Estimation

For model adequacy, estimation of the mathematical expectation and the temperature dependency is crucial. Further estimation of the stochastic component and the special day component does not present any considerable problems. Computational difficulties may occur when estimating the parameters of the stochastic component, because of its non-linearity. The special day components are estimated interactively, as observed above (Chap. 6.1.3). In the load research, only loads, for which enough hourly data is available (at least one year) and changes of which are regular, are estimated. The result is the type components of the model â&#x20AC;&#x201C; co-ordinate functions, shape co-ordinates, and sub models describing the temperature dependency and stochasticity. For estimation of type models, the loads must be categorized into groups, classes, and types. Traditionally, the loads are classified in order to constitute the pattern load curves. In the case of load models, such classification is not necessary, because it is possible to compile (estimate) a unique model for every load. Nevertheless, classification is a rational means to find common model components for a certain group of loads and therefore increase their estimation reliability. The classification is necessary when modelling loads with incomplete data. At the final phase of the load research, it is reasonable to examine the adequacy of the mathematical model. It is possible to compare the deviation of the load data with the short-term and long-term expected value of load. If the deviations are too large, then the cause must be investigated, and the model estimation procedure should be repeated. The list of load research activities has turned out to be rather long. Nevertheless, one should pay attention to the matter of researching the nature of the load, not getting acquainted with some formal mathematical method, as in the case of forecast models. All knowledge, obtained in the research process, is used later, when solving practical exercises. The more complete the knowledge of the load's nature, the more successful are the results of the applications. Professionals can perform the load research and present the results to the operating engineers, who do not have to spend their own time and resources doing so.

104

Model Estimation

6.3

Model Editing

In the load research process type models are estimated first, i.e., models for which there is enough initial data and change regularly. Other models are estimated later, on the basis of type models. When a sufficient amount of high-quality load data is available, it is feasible to estimate a unique model at once for all observable loads. However, as the load's nature changes over time, the model will need to be adjusted from time to time. In connection with type models, it is possible to distinguish model components and factors. The components include model co-ordinates, relative temperature dependency model, stochasticity model and special day components. The shape and level factors form the model factors. In model editing, the process may be limited by evaluating mostly the shape factors or only the level factors. The factors may be estimated on the basis of current data, as only a small part of the model parameters belong to the factors. Thus, model editing in general means evaluating the factors, provided that the model components are known. There is no difference if the components are found based on observable load data or not. The model editing is crucial if the observable load was not a part of the estimation process of the model components. In that case, the model components are determined as typical, and the model factors are found based on the existing load data. The need for model elaboration for all loads arises when new load data is gathered. In order to specify the long-term forecast, it is possible to find the factors for the future also. For example, it is possible to evaluate the load trend in the future or add new load cases. The model parameters observed as shape factors are alr, blr, and clr (Chap. 2.2), which determine the changes of the shape and level of the mathematical expectation, standard deviation, and the rate of the temperature dependency. As normalized alr ⇒ a ⋅ alr , blr ⇒ b ⋅ blr , clr ⇒ c ⋅ clr the mean values a, b, and c present the level factors, which represent the level of the observable load characteristics, and the normalized shape factors represent the form of the curves.

105

Model Estimation

The shape factors can be found using the least-squares method. For the mathematical expectation 2

⎡ ⎤ ∑ ⎢MT (h )∑ (alr G r )N(t ) − Pthl′ ⎥ = min alr th ⎣ r ⎦ ⎡ ⎤ ∑ ⎢MT (h )∑ (alr G r )N(t ) − Pthl′ ⎥ × MT (h )∑ G r N(t ) = 0 th ⎣ r r ⎦ The result is the linear system of equation in relation to factors a lr (r = 0...NSC) ⎡ ⎤ ∑ ⎢MT (h )∑ (alr G r )N(t )⎥ × MT (h)∑ G r N(t ) = Pthl′ MT (h)∑ G r N(t ) th ⎣ r r r ⎦ where Pthl′ = Pthl − Rthl γ t are the load values normalized in relation to the temperature. The shape factors of standard deviation and the rate blr and clr are found with similar equation systems, but instead of the normalized value of the load Pthl′ there are corresponding deviations Pthl′ − E[Pthl ] or ratios ( Pthl − E[Pthl ]) / B k (Tthl − E[Tthl ]) . This ratio is found for the appropriate interval of the temperature deviation ΔTmin ≤ ΔT ≤ ΔTmax . Displacement B k (e.g., k = 10) considers the inertia of the temperature influence. The required factors must be found iteratively, because the necessary rate of the temperature dependency R(t) (needed for data normalization) also needs to be adjusted or is missing at the beginning at all. Inappropriate data must be also eliminated. The criterion is the same as for smoothening the characteristic vectors (Chap. 6.1.1). After the estimation, the shape factors are normalized and the results are the level factors. The level factors may also be estimated separately. The need for that would first arrive when there is not enough data for finding the shape factors. The evaluation of the level factors is also necessary for adjustment of the load trend and determining load cases. The level factor a for the mathematical expectation may be found according to condition 2

⎡ ⎤ ∑ ⎢a MT (h )∑ (alr G r )N(t ) − Pthl′ ⎥ = min a thl ⎣ r ⎦ wherefrom ⎡ ⎤ ∑ ⎢aMT (h )∑ (alr G r )N(t ) − Pthl′ ⎥ × MT (h )∑ (alr G r ) G r N(t ) = 0 thl ⎣ r r ⎦ 106

Model Estimation

a = ∑ Pthl′ M T ( h )∑ (alr G r )N(t ) thl

∑ MT (h)∑ (alr G r ) N(t ) thl

The parameters b and c, representing the level of the standard deviation and the temperature dependency, can be found analogically, by applying the above-shown deviations or ratios instead of values Pthl′ . The level factors may be connected to certain time intervals, which enables the trend to be presented as a linear fractional, including rapid changes of the load (Chap. 3.2.1). The rapid changes of the load may be observed as load cases, which are used for load analysis and forecasting future load levels. In Fig. 6.5, there is an example of editing results on rapidly changing load. Here, three load cases can be distinguished. The corresponding mathematical expectations and standard deviations are presented in Fig. 6.6, Fig. 6.7, and Fig. 6.8.

Figure 6.5 Load actual value (1), mathematical expectation (2), and standard deviation (3) on the weekly level

When adjusting the model currently, the mathematical expectation factors are mostly edited, whereby the standard deviation and rate factors stay unchanged. If the shape factors of the rate of the load are still edited, the relation between mathematical expectation and the temperature dependency must be considered. The dependency of temperature may turn out to be wrong if the estimation interval is too short or the weather conditions are exceptional

107

Model Estimation

Figure 6.6 Load actual value (1), mathematical expectation (2), and standard deviation (3) in case TypeID = 1

Figure 6.7 Load actual value (1), mathematical expectation (2), and standard deviation (3) in case TypeID = 2

When finding the shape factors for the mathematical expectation, the observable time interval must not be too short either, because the mathematical expectation would be adapted too much to that time interval and therefore be unacceptable elsewhere. The appropriate time interval should be one or more years. If there is less load data available 108

Model Estimation

Figure 6.8 Load actual value (1), mathematical expectation (2), and standard deviation (3) in the area of case TypeID = 3

then additional constraints may be set. For example, when estimating and using winter data, the summer load curve should remain unchanged. If the model corresponds to the load, then significant changes in the shape of the load curve do not occur. Fig. 6.9 and Fig. 6.10 are examples of the mathematical expectation curve before and after model editing.

Figure 6.9 Normalized load data (1) and mathematical expectation (2) before model editing

109

Model Estimation

Figure 6.10 Normalized load data (1) and mathematical expectation (2) after model editing

Besides the regular load data (time series) single data may also be used when editing the model. For example, the model level factors could be found on the basis of check measurements such as the load value at certain hours of a night and a day, both in winter and summer. The yearly mean value (yearly energy) can be used as well. In that case, it is only possible to evaluate the value of the level factor a (trend) for the mathematical expectation. Other factors do not change, or, if needed, they are estimated on the basis of the type model. The load's hourly data is required for estimating the model shape factors. Observable time intervals may be fragmentary. It must be noted that the above-mentioned excessive adaptation of the model does not occur in certain periods of the year. If the character of the load does not change in the observable network, then the load research does not have to be renovated, since the decrease and increase of single loads, the appearance of new loads, and the shape changes of the load curves are treated by model editing steps, which can be accomplished during the load monitoring. As in the editing process, only model factors â&#x20AC;&#x201C; not the components â&#x20AC;&#x201C; are changed so there is no need to repeat the load research. Editing the mathematical load model belongs to the load monitoring process and is achievable by the workers carrying out the network 110

Model Estimation

operation. Although some questions about model structure may arise here as well, they can be foreseen during load research, and necessary proposals and instructions may be developed. The operational personnel proceed with engineering aspects only and do not need to understand the model structure more comprehensively.

6.4

Estimation Object

The object EModel, which belongs to the component ElmoExe.dll, is used in the estimation of the mathematical model. The methods of object EModel enable previously deTable 6.1 Object EModel scribed activities needed for load research and model editing to be EModel realized. Although the methods of the object EModel are available for Smooth the application programs, their use GroupID ro CrdFun ClassID ro ShapeCrd requires a certain amount of experTypeID ro Shape tise. Therefore the object EModel LoadID ro DepTemp should first be applied with the IndTemp DataID ro autonomic program ElmoSet, as Stochastic ... shown in Fig. 1.3. There is no need SpecData to implement the estimation procedures into the application DirModel\AGroup.elm programs, and it is possible to use DirModel\DGroup.elm the program ElmoSet only, where ... the load changes regularly and the need for model editing is small. However, in large networks (especially electricity market operations when handling the small consumers), where the loads “come” and “go”, the estimation operations and the object EModel may play a significant role. Properties, methods, and files of the object EModel are listed in Table 6.1. The object EModel is principally the continuation of the object Model. The EModel differs from other objects – in that only one working instance is formed in the program and its properties are not saved. Also, the files belonging to the object EModel contain the load data and model parameters are the same as in the object Model. The initial estimation of the model starts with developing the co-ordinates (Chap. 6.1.1). At first, the characteristic functions of the 111

Model Estimation

load are smoothened with the method Smooth. Afterwards, the results are treated with the methods CrdFun and ShapeCrd, which are forming co-ordinate functions and shape co-ordinates, respectively. Both single loads and load groups may be treated. The classification of the loads into groups and classes is observed by saving the co-ordinate functions and shape co-ordinates into the buffer file according to the load group and class, as marked by the identification numbers GroupID and ClassID, respectively. The next phase is to estimate the shape and level factors of the mathematical expectation and standard deviation using the method Shape. The load data is normalized if the temperature dependency model exists. The level factors are found by given time intervals, which enable one to find the trend as fractional linear or to observe the load cases. The temperature dependency of the load is estimated with the method DepTemp. Here the rate of the temperature dependency is estimated first,

then the parameters of the relative temperature dependency model are found. The base of estimation is the model structure and the parameter values, which are obtained by researching the temperature dependency on the base of indication curves (Chap. 6.1.2), which are found with the method IndTemp. During the research process, the loads, which can be treated simultaneously since they belong to the same load class, are identified. It is typically wise to adjust only the rate, without changing the relative temperature dependency model. The rate of the temperature dependency model is appropriate to the load. The relative temperature dependency model is found according to the load class, but it is saved according to the load type, with identifying number TypeID. The methods Stochastic and SpecData are intended for the estimation of the stochasticity model and special day components of the load, respectively. Since the special day components are estimated interactively, the method SpecData prepares only the necessary data â&#x20AC;&#x201C; load relation curves (Chap. 6.1.3). Other functions are handled by the program ElmoSet or with some of the application programs. The parameters of the stochasticity model may be estimated simultaneously with larger groups of loads.

112

Electrical Network Operation Monitoring

7 Electrical Network Operation Monitoring The electrical network operation depends considerably on network busloads, which are formed by active and reactive power of the substations. Although the number of the observable buses for calculating network operation may be large (hundreds and thousands), modern computer technology does not set any noticeable constraints on the application of the load models. It is also principally possible to consider the load cases inherent for the network busloads. The estimation of the load models offers more problems, though, since the necessary measurements are not always available â&#x20AC;&#x201C; even in the transmission network, not to mention medium-voltage, and especially low-voltage, distribution networks. A close connection between the electrical network operation and the change of network busloads prevails in distribution networks. Besides the load, the transmission network operation is determined by the management of the generation, the regulation of the interchange load flows, etc. Although the voltage regulation and switchings are also done in the distribution network, the nature of the operation is determined here by changes of the load. From the viewpoint of availability of the measurement data, the distribution network may be divided into operatively observable and unobservable parts (Fig. 7.1). In the observable part

Figure 7.1 Loads on the various levels of electrical network

113

Electrical Network Operation Monitoring of the distribution network, the data, which can be used for handling the network operation, is obtained by the dispatching system (SCADA). Similar data for the lower-level distribution network (the unobservable part of the distribution network) is not available. Measuring hourly electricity consumption of consumers for commercial purposes of the electricity market (energy meter reading, EMR) has recently expanded significantly, though. Unfortunately, the operative connection between different measuring systems is missing. However, the commercial data may also be used for estimating the load models. When even the commercially measured data is missing, load models can be estimated according to the data from the client information system (CIS). The integration of the mathematical model of the load with the procedures of the operation's computation may occur on different levels. In the transmission network, where the operation's calculation is based on bus voltage, the values of the loads occur only as initial data. In the distribution network, the busloads may be taken as independent state variables. Incidentally, it enables the distribution network operation to be estimated even when it would otherwise be impossible due to the modest redundancy of the data. It is possible to go even further â&#x20AC;&#x201C; to model the distribution network's operation variables (load flows, currents, and voltages) similarly to loads. The result will be operation characteristics, which enable one to forecast, simulate, and analyze the operation for longer periods of time without performing extensive computations of the network state.

7.1

Transmission Network

The transmission network busloads are formed in the distribution network or by large electricity consumers who are directly connected to the grid. The regularities of the busloads depend on the nature of the consumers. The large number of busloads and possible level differences do not prevent regular changes of the load, temperature dependency, stochasticity, and other regularities to be described in the mathematical model. Rapid changes of the load cause more problems. Possible changes of transmission network busloads occur, for example, due to switchings in the distribution network, which are done for equipment maintenance, optimizing the operation, etc., or due to the working irregularities of large consumers. On the reactive power side, switchings of the compensating devices and the change of the regulation mode are 114

Electrical Network Operation Monitoring possible. Changes of the load level are considered in the model as load cases. The alternative cases of the network load form load scenarios. The mathematical model enables necessary load characteristics to be found for analyzing and planning the transmission network operation – for example, actual load data in the restored, normalized, and simulated form; short- and long-term forecasts of the load; maximal and minimal values of the load, etc. Based on different load scenarios, it is possible to find, for example, the next load characteristics: AR[P] – restored load AR[P] – I[T,P] – normalized load AR[P] – I[T,P] + I[Z[T],P] – simulated load E[P] – mathematical expectation of the load E[P] + I[T,P] + C[P] – short-term forecast of the load E[P] + I[Z[T],P] – long-term forecast of the load E[P] + I[Z[T],P] + CS*S[P] – maximal load E[P] + I[Z[T],P] – CS*S[P] – minimal load. The restored load AR[P] is the same as the actual load A[P] there, where the hourly data is available and reliable. Elsewhere, the data is interpolated or simulated. The normalized load AR[P] – I[T,P] describes the actual load, from which the temperature influence is subtracted. Such a load corresponds to the normal temperature. The simulated load AR[P] – I[T,P] + I[Z[T],P] corresponds to the simulated temperature. The temperature may be simulated when finding the temperature dependency of the loads based on the temperature from another year, which may offer some interest (cold winter, etc.) in analyzing and forecasting loads. Another possibility is to add a certain deviation (for example, ±10 °C) from the normal temperature. The mathematical expectation E[P], giving the load a probable value at a specific time, may be calculated for any time period. The mathematical expectation corresponding to the normal temperature can be used for analyzing and forecasting loads. Short-term forecast of the load E[P] + I[T,P] + C[P] is found as a sum of mathematical expectation, temperature influence, and expected deviation C[P] (conditional mathematical expectation of the load deviation). The short-term forecast differs from the mathematical expectation in time, where the deviation is practically applicable (about a week ahead), and the actual values of the temperature or meteorologically forecast data are available. The long-term forecast of the load E[P] + I[Z[T],P] is computable for any time interval. The maximal and minimal load E[P] + 115

Electrical Network Operation Monitoring I[Z[T],P] Âą CS*S[P] are found by adding the possible random deviation to the simulated load. Here CS is the confidence factor, the value of which depends on the given probability and load distribution. When necessary, other load characteristics may be also observed. In order to obtain characteristics that correspond to real situations in the electric network, the load models must be estimated. The amount of initial data needed usually exists in the transmission network. However, cases where some hourly measurements of the buses are absent may also occur. The reason may be a change in the nature of the load (new large consumers are added, e.g). The estimation is accomplished in two phases according to the principles described in Chapter 6. In the first phase, during load research, the type models of the load are developed; later, in practical monitoring process, the models are edited. It is possible to estimate a unique model for every load, because sufficient data is generally available in the transmission network. The type models, which represent a load model of some appropriate consumer groups, must be applied only in exceptional cases. The transmission network busloads measurements are, in fact, duplicated by the grid's dispatching system (SCADA) and by the distribution network or large electricity consumers. It might be possible to request the integration of the information systems, but this is not necessarily needed to estimate the load models. The load models may successfully achieve the connective role. Indeed, the model co-ordinates are estimated during the load research whereby offline data from the transmission and distribution networks may be used. The changes of load and possible load cases of transmission network busloads result from distribution network or large consumers. In the load research process, the load cases are defined, and the appropriate factors are estimated. Adjustment of the model factors may also be done based on the data from the transmission network. Thus, the unification of the dispatching systems of the transmission network, distribution network, and large consumers is not necessarily needed. Changes that happen in the distribution network, which appear as transmission network load cases, may also be detected through transmission network data, the load models have been prepared beforehand as needed. However, for the (short-term) forecast of the transmission network load, the planned changes in the distribution network should also be known beforehand for the transmission network operation. The necessary communication between the networks' personnel is 116

Electrical Network Operation Monitoring self-evident. The load models are edited in the monitoring process. The editing is necessary to adjust the load models when the character of the load changes. The smooth changes of the load, which happen over a longer period of time, are periodically considered in the model shape and level factor adjustments (e.g., once a year). The sharp load changes reflect, first of all, in the model level factors. The rapid changes of the load trend or the alternation of the load cases is possible. It must be emphasized that only situations that are understandable from an engineering perspective and may occur again, in the future (switchings in the distribution network or the commutation of reactive power devices, e.g.) are considered load cases. Accidents and other uncontrollable phenomena are not expressed in the load model. The need for load model editing is discovered in the operation monitoring by comparing the actual and the expected values of the load. The control is done automatically based on the appropriate criterion. When finding the essential deviations, the expert, load manager, must decide if it is a new situation (change of trend, load case, etc.), that must be considered in the load model or not. The expected values of the load (short-term forecast) may be used to estimate for transmission network operation. If comparing the expected values of the operation parameters (calculated based on expected value of load with measurements), it is possible to notice sufficient accordance or difference between given and calculated data. If large deviations occur in some single measurement, then it is evidently a case of bad data. Mass deviations indicate unforeseen changes. Hence, the load models may be used at least for the first stage of the operation estimation, discovering the bad measurement data. In cases of unseen changes of the operation, the expected values are excluded, which does not obstruct further monitoring of the network's operation when the operation returns to the planned conditions. The information, whether or not the operation runs according to the given frame, is itself important to the operating personnel.

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Electrical Network Operation Monitoring

7.2

Observable Distribution Network

The medium voltage distribution network may be represented as a set of feeders starting from the HV/MV substation, consisting of line parts and other elements (transformers, capacitors, etc.) and ending in the distribution substations (Fig. 7.2). Switching disconnectors at the disconnection points can change the configuration of a feeder. MV/MV substations and voltage regulators may also be included. The load of a MV feeder is formed by active and reactive loads of distribution substations. The load of a distribution substation is, in its turn, formed as the sum of LV loads and losses of distribution transformers. Distribution network state parameters are operationally measured in HV/MV substations and distribution substations. Measured data may include values of active and reactive power, currents and voltages at various points of the Figure 7.2 Feeder diagram feeder. The metering frequency in the SCADA is usually one or more times per minute. When monitoring network operation, further data is aggregated and presented as average values of an hour or a part of the hour. The distribution network operation is calculated by feeders. The initial values used are the transmission substation voltage U0 and the busloads. The feeder element equivalent circuit is presented in Fig. 7.3. In the case of lines, the power losses ( Î&#x201D;P, Î&#x201D;Q ) are usually discounted. In the case 118

Electrical Network Operation Monitoring

of transformers, power losses as well as transformation ratio k must be considered. A simple, two-way iteration process is used in the Figure 7.3 Equivalent circuit of a feeder computation of the opelement eration. First, the power losses and power transmission in lines is found starting from the busloads at the end of the feeder. As an initial value of the bus voltage, the nominal voltage may be used. In the second phase, the bus voltages are calculated, starting from the HV/MV substation bus, the voltage of which U0 is considered as given. In the second phase of the calculation, the load dependency on the voltage and the effect of the voltage regulators are considered. The described algorithm usually converges in 2…5 iterations. 7.2.1

State Estimation

The goal of state estimation is to refine measuring data, but it is especially important to discover bad data – significant measuring errors and mistakes. Calculation of distribution network feeder state parameters is based on network equations through main state variables – supply voltage and bus loads Vj(U0, p1, p2, ... pn) Notation pi corresponds here both to active and reactive power. Thus the general number of loads n is equal to double the number of buses, and there are n + 1 main operating parameters U0, p1, p2...pn. ~ Supposing that, for the considered state, there are m measurements V j (P, Q, U or I), it is possible to obtain the refined state variables (estimates) Vj from the criterion 2 ~ ∑ V j − V j = min , j= 1...m

[

]

As in distribution networks, the necessary data redundancy does not exist in most cases – the number of measurements m does not essentially exceed the number of main operating variables n + 1, or is even smaller.

119

Electrical Network Operation Monitoring

To resolve this problem, additional conditions are applied, which demand that the increments of state parameters be as small as possible ∑ Δ2 pi = min , i = 1...n i

In other words, that the state of the network needs to be as close as possible to that, which is found by means of the load models. Let us see the network equations being linearized over main state variables, when marking p0 = U0 ΔV j = β 0 j Δp0 + β1 j Δp1 + β 2 j Δp2 + ... + β nj Δpn where

β ij =

∂V j

, i = 0...n ∂pi In the matrix representation Λ = BΔ where Λ = [ΔV0 , ΔV1 ... ΔVm ] is the vector of deviations of measurement data Δ = [Δp0 , Δp1 ... Δpn ] is the vector of deviations of main state variables

β10 β 20

β11 ... β1n β 21 ... β 2 n

...

β m0

...

is the sensitivity matrix (Jacobian)

β m1 ... β mn

Here the deviations Δpi are found on the basis of the values calculated by a load model (expected value of load). It is also possible to compose the mathematical model for the supply voltage U0. For simplicity, the trivial model is used, where mathematical expectation and standard deviation of voltage are constant. Let us see the extended vector of measurement increments ~ ~ ~ ~ Λ 0 = ΔV1 , ΔV2 ... ΔVm , 0, 0...0

[

]

in which the n last components are zeros. Next, matrix B0 with n + 1 columns, where on the first m rows there are elements of sensitivity matrix B and on the next n there is a zero vector and a unit matrix, is composed. 120

Electrical Network Operation Monitoring

B0 =

β10 β 20

β11 ... β1n β 21 ... β 2 n

...

β m0 0 0 ... 0

...

β m1 ... β mn 1 0 ... 0

... ... ... ...

0 0 ... 1

The above-mentioned conditions are simultaneously satisfied, if T ~ ~ Λ 0 − B 0 Δ Λ 0 − B 0 Δ = min

(

)(

)

From here, a system of linear equations over the increment vector Δ follows ~ B T0 B 0 Δ = B T0 Λ 0 The deviations vector Δ allows all state variables to be refined – it is possible to calculate both the specified measured variables (estimates) and whatever other operating parameters as well. For finding elements of the sensitivity matrix B, the deviations of the model parameters Δp0 ... Δpn must be assigned. Then the corresponding load values must be found, and the deviations ΔV j of considered state variables are calculated on the base of non-linearized network equations. As a result ΔV j β ij = , i = 0...n, j = 1...m Δpi The results of the estimation are legitimate if there is no bad data and exclusive states are not observed. If bad data is obtained, they must be removed and the estimation procedure must be repeated. It is also possible to attempt to define the exclusive states more precisely. If the exclusive state still takes place, estimation of this state is not possible, and the measurements of this moment are not used. As a result, dispersion of the short-term forecast increases slightly, but, in principle, this does not obstruct the process of estimation.

121

Electrical Network Operation Monitoring

Bad data and exclusive loads are detected on a case-by-case basis or as scenarios. Errors of measurement data may be related – for example, due to telecommunication line disturbances. Indications of exclusive situations are detected by rating the load deviations and the elements of the sensitivity matrix. 7.2.2 Model Editing

The load model expresses changes of the load. Therefore, the parameters of the model are essentially the same if the nature of the load (composition of the consumers, etc.) does not change. If changes occur, the model parameters must be adjusted. The editing is done for the model factor without changing the co-ordinates (Chap. 6.3). Let us see the bus loads pi with parameters (factors) of the model ai1, ai2...air at the moment tk pi(tk, ai1, ai2 ... air), i = 1...n where r is the number of parameters that have to be estimated. Next, the network equations Vj = Vj(U0, p1, p2, ... pn) describing the state variables via the parameters of the load model are presented Vjk = Fjk(a1, a2, ... al), j = 1...m where l = nr is the number of the parameters involving all load models. The function Fjk corresponds to the moment tk on which the load values and supply voltage depend. The parameters of the load model are calculated from criterion 2 ~ ∑∑ V jk − V jk = min

[

]

~ where V jk are the measurements at the moment k. Network equations, linearized over the model parameters, are ΔV jk = α1 jk Δa1 + α 2 jk Δa 2 + ... + α ljk Δal where

α sjk =

122

∂V jk ∂a s

, s = 1...l

Electrical Network Operation Monitoring

In the matrix representation Λk = Ak Δ where Λ k = [ΔV1k , ΔV2 k ... ΔVmk ] is the deviation vector of measurements Δ = Δa1 , Δa 2 ... Δal is the increment vector of model parame-

[

ters

]

α11k A k = ...

α m1k

... α1lk ... ... is the sensitivity matrix (Jacobian) ... α mlk

If the vector of measurement deviations is ~ ~ ~ ~ Λ k = ΔV1k , ΔV2 k ... ΔVmk then the increments for model parameters are calculated according to the criterion T ~ ~ ∑ Λ k − A k Δ Λ k − A k Δ = min

[

]

(

)(

)

that gives the system of linear equations for the vector Δ ~ ∑ A Tk A k Δ = ∑ A Tk Λ k

(

)

The base for linearizing the network equations is the load values, calculated by load models (expected value of load). Relatively to the same values metering deviations are found. Linearization is acceptable if the deviations of the measurements are not too large. Otherwise, it is necessary to use the iterative process in which the increments are added to the model parameters, and new state variables and deviations of measurement are calculated. Thereby linearization (calculation of the new sensitivity matrix) is not to be repeated, and it is possible to continue with the old matrix. The elements of sensitivity matrix are calculated as ΔV jk α sjk = , s = 1...l, j = 1...m. Δa s The number of measuring data must be larger than the number of parameters being estimated. For example, if the number of loads is 10 (5 distribution substations), parameters 10, and measurements 10, then the 123

Electrical Network Operation Monitoring

number of the required values is l = 100, so more than 10 measuring hours are needed. Therefore, it seems that daily measuring is quite enough to get the results. But the adequacy of the results is a question. It is clear that the model, estimated on the basis of the data from one or more days, is not usable for a longer period. For example, the data of winter loads is not representative for summer loads. Generally it is an adaptation problem, which can be resolved by taking into account the essential meaning of the model parameters. The degree of the equation system to be resolved is high (for the abovementioned modest example, it is 100). It means that the volume of calculations can be too large and the equation system is poorly conditioned. The way out is to use the results of state estimation, by means of which the busloads are always found independently from the structure of measured state variables. Consequently, it is possible to receive all load values, state parameters and edit the parameters of all load models.. If m = 1 and l = r, we get A k = α1k ... α rk

[

]

Δ = Δa1 , Δa 2 ... Δa r ~ ~ Λ k = ΔVk = ~ pk T ∑ A k A k Δ = ∑ ~pk A Tk

[ ] ( )

pk is the estimated value of the observed load at the moment k. where ~ The degree of the equation system is now r, or, for the above-mentioned example, it is now 10, and problems with calculation volumes are removed.

The initial form of the load model may be obtained based on results of the MV distribution network load research. The type models may also be used. To derive the MV network load models, lower unobservable network data should be preferred. This way, the more reliable results, which also consider the possible load cases, may be achieved. 7.2.3

Operation Monitoring

The distribution network operation monitoring consists of analyzing, forecasting, and simulation. Editing the load models are also to be added to the monitoring process. The network operation estimation requires a separate procedure, the purpose of which is to check and increase the 124

Electrical Network Operation Monitoring

accuracy of the measurement data. The values of the state variables and other values (for example, network losses) based on past data are found in the analysis process. The reduction of the actual loads to the normal outdoor temperature or some other exceptional weather condition is possible. The meteorologically predicted temperature is used in short-term forecasts (lead time up to one week). In long-term forecasts, the temperature conditions may be simulated. Hence, simulation belongs to the operation analysis and to the forecast process. Beside outdoor temperature, forming the load for other conditions may be simulated. Changes of the feeder configuration do not obstruct the operation monitoring if the actual or expected configuration of the network is known. The possible feeder configuration changes are fixed by the dispatching system and checked in the operation estimation process. Configuration changes in the low voltage network are not observed by the dispatching system. The results may be rapid changes of the medium voltage network busload values, which are observed as load cases. The load cases are connected only with the planned changes. The emergency cases are not observed. Just as between the transmission network and distribution network, the loads of the observable part of the distribution network are formed in the unobservable part. The possible specification of the load cases is important. The adjustment of the model factors for different load cases may also be carried out on the basis of observable network data. The efficiency of the monitoring methodology depends on the accuracy of the load model, which, in its turn, results from the regularity of the change of the loads. The trivial model may be implemented for irregularly changing loads. Hence, the practical application of the monitoring methodology requires clarification. It may be assumed that the medium voltage network operation estimation is not feasible for the purpose of increasing the accuracy of the measurement data. Nevertheless, it may be possible to determine whether or not the operation fits with the load and network model (it is a regular operation). The appearance of exceptional states may be presented as events in network dispatching, which also establishes that the network operational computation on current data is not adequate. Later (on the next day, e.g.), the exceptional states must be specified, and the load and network models should be adjusted as needed. The model adaptation algorithm and method for detecting bad data and exceptional loads need to be clear. The operation of the voltage regulat125

Electrical Network Operation Monitoring

ing equipment also complicates the monitoring methodology. Nevertheless, the monitoring methodology may be applied in both offline and online data processing. The limitations (due to the computer resources) are not foreseen, as the excessively complex computations are missing.

7.3

Unobservable Distribution Network

The calculation of the unobservable distribution network operation does not differ essentially from the calculations in the observable part of the network. The network configuration is radial, and the necessary information concerning the load is received from similar load models, as in the higher level of the distribution network. Problems could arise when obtaining the necessary load data for estimation of the model. As on the unobservable level of the distribution network, if the on-line measurement data is not available, then the operation estimation and current editing of the load models are not possible either. The necessary load data is gathered from a different information system in off-line mode. It must again be emphasized that the structure of the model is the same for all loads. If the available data is not sufficient to estimate the model parameters, then the type models are implemented. The model simplification due to the absence of the data is not observed. There are several information systems operating in the unobservable part of the distribution network through which it is possible to receive the load data. One of these is the commercial measurement system (EMR), where the loads are measured for the electricity market. For larger electricity consumers, the commercial measurements (based on the hourly data) have existed for some time. In the case of small consumers, the measurements have been performed on a monthly or yearly energy basis. Taking hourly measurements of small consumers has increased in recent years. For example, Nordic countries have set the objective to measure all consumersâ&#x20AC;&#x2122; hourly energy consumption in up-coming years, because the free electricity market balance settlement would be done more fairly. The load measurement is performed by quality measurement systems but also as special-purpose measurements to get the type load pattern or to receive the load data of some characteristic days of the year. General data about the load and possible single data (monthly or yearly energies, etc.) may be obtained from client information system (CIS).

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Electrical Network Operation Monitoring

In the unobservable network, the estimation of the load models takes place (as on the higher levels of the network) in two stages. Proceeding from the load research, the type models are estimated and edited afterwards, so that all model parameters for the observable loads are determined. The model editing is repeated in the operation monitoring according to the change of the nature of the loads and the gathering of the load data. The relevance of load research in the handling of the unobservable distribution network loads is considerably more important than on the network's higher-level. Compared to the other levels of the network, the extent of the network and according number of the busloads is a considerably higher. An essential problem is the acquisition of the load data. The number of possible information systems, the quality and the usability of the measurement data, different methods of access and data formats, etc. make the load research a very laborious job. The load cases, which are caused by the switchings, irregularities of factory work etc., have a special place in the load model estimation. Consideration of the load cases is obligatory, since the level of the load may vary considerably in different load cases. However, the load cases may be determined only for loads for which sufficient load data is available. From the viewpoint of the network's operation, the larger and more important consumers are measured through a commercial measurement system. In the case of small consumers, where hourly load data is not available, the load cases are not observed. Based on relevant estimate models, it is possible to find all the characteristics necessary for load monitoring and, on those grounds, for monitoring the distribution network operation. Difficulties may arise when determining the load cases, because an operative information exchange between different levels of the network may not occur, not to mention communication between the network and factories. Here it is possible to apply the simulation, where the load characteristics are determined and the network operation is calculated for different load cases. That way, the possible limits of the operation are achieved. When needed, the probabilistic methods or fuzzy logic, which enable different operations to be handled complexly, may be applied. Although computational difficulties do not occur when handling the distribution network load (despite their large number), the application pro127

Electrical Network Operation Monitoring

grams must be used to realize the operation monitoring. It is possible to develop the autonomic programs for load handling and transfer the load characteristics to the application programs via the data files. Integration of the load monitoring and applications in the same program is more rational. In some application programs, simplified load models are used, especially in those that handle the operation of the unobservable distribution network. The derivation of those models does not offer any difficulties based on the normal load model (Chap. 5.1.3).

7.4

Modelling of the Distribution Network Operation

Distribution network operation depends on busloads. Therefore, the network operation variables such as power flows, currents, voltages, etc. change similarly as do loads. It is possible to notice periodicity, temperature dependency, stochasticity etc. Therefore, it is possible to represent the changes of the operation variables with a similar mathematical model as for loads. Active and reactive power may essentially be measured at any point of the feeder. Therefore, based on measurements, it is possible to compose the according models. Thus, if the loads are modelled in the distribution substations, it is possible to form a task to move the model to the higher location, at which point the new model considers the loss in the observable line part. In the branching point, where several loads appear, the new model describes the total load. When continuing in the same manner, it is possible to reach the model of the supply transformer in the medium voltage network and the model of the HV/MV transformer load. The model adding, convolution, is a complex operation, where the model factors and components (co-ordinates) both change. In the estimation process (Chap. 6), the model components and factors are handled separately. The model components may be estimated simultaneously for a certain number of loads, but the factors generally correspond to single loads. When presuming (for an approximation) that all the busloads of the feeder are estimated in the same co-ordinates, then it is also possible to describe the operation variables of the feeder in the same co-ordinates, and only the model factors are to be estimated. The model factors are formed by the level and shape factors of the mathematical expectation, standard deviation, and the rate of the tem-

128

Electrical Network Operation Monitoring

perature dependency. Let us first consider the case of the mathematical expectation, which is expressed as E (t , h, l ) = M T ( h )â&#x2C6;&#x2018; (alr G r )N(t ) r

Since the co-ordinate functions M(h) and N(t) and shape co-ordinates Gr are presumably known (they are the same as for loads), then, for every observable operation, only parameters shape factors alr (l = 1...NTP, r = 0...NSC) are to be found. Here the number of the day type NTP = 7 (if it is presumed that the special day components are handled separately) and the rate of the shape co-ordinates NSC = 10...12. The data necessary for the estimation of the shape factors are obtained from the computations of the feeder operation on given load values. The extensive process may be imagined, where the loads are simulated and the operation is calculated at different moments in time and on different values of load deviations, and, after that, the mathematical expectation of the operation variables is estimated. However, when presuming that the operation variables depend on the load linearly, then the same relation also applies between mathematical expectations. Therefore, upon giving the value of the mathematical expectation of the load for some moment in time and calculating the feeder operation, then the mathematical expectation of the operational parameters at the observable moment are obtained. Hence, for finding the shape factors, only NTP Ă&#x2014; (1 + NSC) feeder state on given mathematical expectation values of the load must be calculated. The results of these calculations are input data for the estimation of all observable operation parameters. The feeder states must be found for the moments in time, where the mathematical expectation mostly depends on the shape factors. For some shape factors alr, it is a moment, where the absolute value of the derivate d E ( t , h , l ) = M T ( h )G r N ( t ) dalr is maximal. Marking the according daily time as hlk and yearly time as tlk, (k = 0...NCS), it is now possible to compose the linear equation system for finding the shape factors by day types l = 1...NTP D l A l = El Here Al and El are the vectors of the values of shape factors and mathematical expectation, respectively.

129

Electrical Network Operation Monitoring

E (t l 0 , hl 0 , l ) al 0 ... A l = ... , E l = a lNCS E (t lNCS , hlNCS , l ) Dl is a matrix, elements of which are d lkr = M T ( hlk )G r N(tlk ) It should be noticed that the matrix Dl only depends on the model coordinates, since the characteristic moments in time hlk and tlk are defined based on the co-ordinates. Therefore, the matrix Dl, which can be named as a model co-ordinate matrix, is the basis for estimating the shape factors for all at the operational parameters. The standard deviation and the rate of the operation variables are described in a model as analogical expression to the mathematical expectation, only the shape factors alr is substituted accordingly by factors blr or clr. The main matrix of the equation system Dl for finding the factors stays the same, as is the case for the mathematical expectation. In order to get the necessary absolute term, which would substitute the vector El, the specific increment is added to the load, the feeder operation at every characteristic moment in time is calculated, and the vector Pl of the operational variables is found. If the increment, added to the load, equals the value of the standard deviation or the rate of the temperature dependency, then it is possible to assume, that the increment of the operation variables Pbl – El or Pcl – El also respond to it and D l B l = Pbl − E l D l C l = Pcl − E l where Bl and Cl are the vectors of the shape factors of standard deviation and the rate on given day type l, respectively. However, the algorithm assumes linear dependence, which, in case of standard deviation, is somewhat to query. In the above discussion, the model shape factors are not in the normalized form, presenting both the level and the shape of the variables. In the normalized form, the level factors and trend component are expressed. For the mathematical expectation E (t , h, l ) = aA(t ) M T ( h )∑ (alr G r )N(t ) r

130

Electrical Network Operation Monitoring

If the model shape factors have already been estimated, then it is possible to estimate the values of the level factor a and parameter α of the trend component A(t ) = 1 + α (t − t0 ) for current adjustment of the mathematical expectation. Finding the level parameter and the trend parameter does not present any difficulties, when the value of the mathematical expectation of the load is given and the feeder operation is calculated at the appropriate time. The level factors of the standard deviation and the rate may be adjusted in the same way. In conclusion, in the modelling of the distribution network operation, it is first necessary to estimate the load models in the coordinates appropriate for the feeders. After that, the loads must be simulated at representative time moments, calculating approximately 200 states for every feeder and evaluate the level and shape factors of the operating parameters. Since the feeder operation calculation does not require a large amount of computer resources, there are no limits to model the distribution network operation. The method is applied only to the operation of the distribution network, which is found on the given conditions, such load cases, network configuration, the control point of the voltage, and reactive power regulators, etc. When changing the conditions, the modelling must be repeated and the extent of the calculations increases. It must be emphasized that this is not an abstract selection of possible options, which could lead to the rapid increase of the computations (the so-called combinatorial explosion), but rather the conditions selected on an engineering bases in order to achieve the practical objectives. The modelling offers similar possibilities of handling the operation variables, as in the case of the loads. The operation forecasting, analyzing, and simulation are possible for different given conditions. The results are obtained for any given time interval, not just single states like in traditional calculations. The modelling of the operation variables (model convolution) may be applied optionally – for example, describing the load flow on some interesting section in the feeder. The derivation of the load model in the feeder point of supply is necessary, because it corresponds to the higherlevel network busload. However, optional modelling does not provide any distinctive computational economy, because the operation calculation (necessary for receiving the initial data) gives the base for the estimation of all observable feeder operation variables. The volume of 131

Electrical Network Operation Monitoring

work of the evaluation of the model factors according to some operational parameter is noticeably smaller. The distribution network modelling methodology presented here is, of course, not exhaustive. The assumptions made here need to be checked and the details of the operation formation (e.g., influence of the voltage and reactive power regulators, etc.) need further explanation. Nevertheless, there is no doubt that the modelling described here is applicable and gives far more possibilities for planning and operating of the distribution network operation than the traditional approach.

7.5

Distribution Network Object

The code component ElmoExe.dll, formed for the load handling, is also comprehensively usable when calculating the distribution network operation. First of all, the network busloads need to be treated â&#x20AC;&#x201C; saving the data, estimating the load models, and calculating the characteristics. The same procedures must also be used in the case of operation variables, when they are modelled. The only specific procedure connected to the operation is the calculation of the feeder operation on a given network configuration and load values. For calculating the distribution network operation, the object class Disco, which belongs to the code component ElmoDisco.dll, is developed. It should be emphasised that the electrical network operation is a completely separate phenomenon, which is planned and operated, considering its reliability, electric quality, and economy purposes. Although the distribution network operation is determined by the loads, different operating conditions and purposes, which do not belong to the theme of this book, should also be considered. Up to now the component ElmoDisco.dll and object Disco are meant only for checking the ideas presented in this chapter. Therefore they are not further observed here.

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Summary

8 Summary The electrical network load is the determining factor in planning and operating an energy system operation. Knowledge about the load on different levels and forms is required. In applications, the mathematical model presents the load.

8.1

Load Monitoring

Monitoring is understood as load forecasting, simulation, and analysis. Monitoring does not involve any active procedures for changing the load (e.g., load management). Success of monitoring is guaranteed by an adequate mathematical model, which describes the changes of the load during the entire observable period, both in the past and in the future. In the modelling process, the load is observed as an individual object, which has a name, connection point, characteristic consumer composition, and other attributes. Active and reactive power and current characterize the load quantitatively. The structure of the mathematical model is formed in accordance with the physical character of the load and engineer's understanding of the purpose and necessary accuracy of the model. The determination of the regularities and quantitative description of the load are set as a purpose when developing the model. Possible quality (sampling frequency, etc.) and volumes of the load data are not to be taken considered. Indeed, the nature of the load does not depend on how it is measured. The practically needed characteristics of the load (short-term or long-term forecast, etc.) are also initially left for later. It is only considered that the model is applied for treatment of network operation dynamics neither for transients nor for long term planning of power system. In the model, load regularities such as regular changes, dependency on outdoor temperature and state variables, stochasticity, and controllability are described. The regular changes are daily, weekly, and yearly periodicities; trend; and the character of the load on special days. The role of temperature dependency in cases of electrical heating or air conditioning is rather large. In the model, inertia, non-linearity, and timely changes of the influence of the outdoor temperature are considered. Dependency on state variables is expressed by voltage and frequency sensibility of the load. The stochasticity is especially noticeable in lower, distribution 133

Summary network loads. The ratio of standard deviation to the mathematical expectation in such loads is comparatively high. From time to time, large deviations, which do not match the normal distribution, may occur in smaller loads. The load is mostly managed indirectly, through tariffs. Direct management by electrical network operating personnel also occurs. Changes of loads may also happen rapidly caused, for example, by switchings in the lower network, start-up or closing of large consumers (factories), switchings of the reactive power compensating devices, etc. Rapid changes of the load may be observed as load cases, to which the according parameters of the mathematical model correspond. When handling electrical network loads, it is suitable to incorporate the load cases into load scenarios, where each load is represented with a specific case. On one hand, the observation of single cases for all loads is too labourintensive. On the other hand, there is no need for all possible combinations of load cases. The structure of the mathematical model is the same for all loads. In order to describe particular loads, the model parameters must be estimated according to the load's data. The estimation of the model parameters is of crucial importance for practical application of the mathematical model. The principle is that all the model parameters for each load must be estimated. The simplified models are not observed due to the absence of load data. Depending on the amount and quality of the load data, and the needed accuracy of the model, different ways of estimation are possible. The purpose of estimation is to achieve correspondence of the model to the observable load. During the estimation, no attention is paid to the nature of the practically needed load characteristics (short-term or long-term forecasts etc.). As a result of the estimation, a similar model is obtained for all the loads. Therefore, in practical applications, all load models are equivalent. However, the accuracy of the model depends on the quantity and quality of data used in the estimation. The main phase of the estimation, initial estimation, takes place in the load research, where, for example, a certain electrical network load is handled. During the load research, loads that have sufficient hourly data (in the range of at least one year) and that change regularly are estimated first. The result of the initial estimation may be used as type models of 134

Summary the observable electrical network, which means typical sets of model parameters. First of all, parameters that belong to the components of the model are observed as typical. The main components of the model are co-ordinate functions, shape co-ordinates, and sub models, which describe temperature dependency and stochasticity. Model components may be estimated simultaneously for several loads of similar nature. The remaining parameters, called model factors, are more pertinent to certain loads. In the second phase of estimation, when editing the model, a suitable type model is attached to every load, and a number of factors for every observable load are estimated (based upon the amount of initial data or suitable because of other considerations). At least one factor must be estimated for every load. It is possible to estimate a unique model for each load if enough load data is available. Besides estimating of type models, the load research consists of determining and classifying loads, clearing up the availability and quality of the load data, etc. In the load research process, loads of the observable electric network are divided into groups, groups into classes, and classes into types. The group and class components of the model are estimated, then model factors are evaluated for all loads. This process of estimation is necessary, for example, for regularly (through dispatching system) non-measured loads, for which there is not enough initial data. But the application of the type components is practical, even if the necessary data exists, because it enables the reliability of the model to be increased. The structure of the model is composed so that the model components, involving the majority of parameters, are relatively stable. Therefore, it is not necessary to update them later, even when the character of loads changes in time. The model factors may be adjusted (edited) on the basis of recent load data. To a certain extent the model editing can also be automated by adaptation. All available quantitative and qualitative data concerning the load is used during the estimation. The availability of necessary data may vary. On the higher level of the electrical network, the data is received through SCADA. In the lower part of the distribution network, the operatively unobservable level, such data is not available. Here the data, which is obtained through the commercial measurements for the purpose of energy markets (EMR), may be used. The load is also measured in the electricity quality measurement systems, and special-purpose measure135

Summary ments carried out in order to receive load patterns. The general data and single data (monthly or yearly energies, etc.) of the load may be obtained from the client information system (CIS), when necessary. The load models may be the uniting factors of the different information systems. The load research is performed on the basis of off-line load data that is obtained from all available information systems. The operative handling of the loads (including editing of the load model) is done separately within the limits of each information system. To forecast loads accurately, it is only necessary to perform engineering-type notifications between different levels of the electrical power networks and large consumers concerning possible changes of electricity consumption. The mathematical model describes the load but does not give the practically needed values â&#x20AC;&#x201C; for example, load forecasts. Those values, load characteristics, may be found on the basis of the model. For that, the necessary load characteristic must first be defined and then determine how to find them on the base of the model. Conditions corresponding to the parameters must be determined. For example, when the load forecast is requested in standard conditions (normal temperature etc.), then the mathematical expectation of the load, which can be found at any time period, corresponds to it. When the conditions are changed â&#x20AC;&#x201C; for example, giving the real or simulated temperature and adding the influence of the temperature dependency to the mathematical expectation â&#x20AC;&#x201C; then corresponding load characteristics are obtained.

Figure 8.1 Load monitoring acts

136

The procedures of analyzing, forecasting, and simulation belonging to the load monitoring process may only be distinguished conditionally. For example, when the actual load data is available, the according procedures is analysis; if not, it is forecasting. Simulation, especially when considering the simulation of temperature influence, belongs equally to analyzing and forecasting. This situation is illustrated in the Fig. 8.1,

Summary where the editing of the load model is also considered as a monitoring procedure. Indeed, the models are edited in the current handling of the load, together with the other monitoring procedures. As a result, the model factors â&#x20AC;&#x201C; and therefore the calculated load characteristics â&#x20AC;&#x201C; may change. The practical application of the mathematical model of the load may seem rather complex. But it must be noted that the application procedures result from the necessity to clear up the existing conditions. Loads must be defined, the availability of the load data must be established, the nature of the necessary load characteristics and conditions of determination must be clarified, and much more. Such engineering problems are inevitable in electrical power networks and other places where loads are handled. The mathematical model offers a systematic base and, when correspondingly programmed, a necessary means for handling loads. However, the load model is not a magic formula that automatically solves all problems, but rather a tool for experts. The skill and experience of the engineers is still primary for resolving the electrical network's operational and planning problems. The load model enables the engineers' vision to be achieved more effectively via corresponding application programs. Beside the engineers' vision, the load model also reflects theoretical principles of the regularities of load formation and changes that are discovered as a result of long-term scientific research. As a result of the principles of load modelling, new practical possibilities of the comprehensive monitoring system may emerge even for experts in the field. Hence, the many possible uses of the load monitoring system also presumes the efforts of the system's end users. The development of the principles of the mathematical modelling of the load and the corresponding monitoring system are far from exhausted in this book. It is possible to discover new principles of load modelling as a result of theoretical investigations. The major requirements for refining the model and further development of the applications accrue from the needs and conditions of practical problems.

137

Summary

8.2

Monitoring of the Electrical Network Operation

The operation of the electrical network depends on busloads, which are formed by the active and reactive power of the substations. When calculating the network operation, the number of the observable buses may be large (hundreds or thousands); however, the modern computational possibilities do not limit the applications of the load models. Principally it is also possible to consider the load cases connected with rapid changes of the network busloads. The estimation of the load model presents more problems, because the necessary measurement data is not available even in the transmission network, not to mention medium-voltage and lowvoltage distribution networks. The close connection between the operation of the electrical network and changes of the busloads takes place in the distribution network. The transmission network operation is beside the load, influenced also by the management of power generation, regulating the interchanges, etc. Although the voltage regulation and switchings are also done in the distribution network, the nature of the operation is mainly determined here by the changes of the load. The integration of the mathematical model of the load with procedures to calculate the electrical network operation may be realized on different levels. In the transmission network, where the bus voltage is used as a base in the operation computations, the load values are treated as initial data. In the distribution network, the busloads may be taken as independent variables. It also enables the distribution network operation to be estimated even when it would be impossible due to the slight redundancy of the measurements. It is possible to go further â&#x20AC;&#x201C; the distribution network operation variables (power flows, currents, and voltages) could be modelled in the same way as loads. The result is the operation characteristics similar to the load characteristics, which enable the operation to be forecasted, simulated, and analyzed for longer periods of time without mass computations. The mathematical model enables one to find the necessary load characteristics â&#x20AC;&#x201C; for example, load data in restored, normalized and simulated forms; long-term forecast; maximal and minimal values of the load; etc., for analyzing and planning the transmission network's operation. The load characteristics may be found based on different load scenarios.

138

Summary Distribution network operation is calculated by feeders, where voltage of the primary substation and busloads are used as initial data. In the calculation process, the simple two-way iteration process is used: first the power losses and power flows in the lines are calculated, starting from the busloads at the end of the feeder. Then, the bus voltages are calculated starting from the buses of the primary substation. If the distribution network operation is estimated on the basis of load models and the model factors are edited (and especially if the operation variables are modelled), then the model factors become the main variables of the operation. It must be noted that the load values correspond to the state but the model factors correspond to the operation â&#x20AC;&#x201C; it is possible that the model factors do not change over a longer period. On the operatively observable level of the distribution network, the measurement data from SCADA can be used for operation monitoring and editing of the load models. It is necessary to estimate the network operation, first of all to detect incorrect measurements (mistakes). Since the necessary data redundancy generally does not exist in the distribution network, the load model may be used. However, exceptional operations may occur that cannot be estimated this way. The exceptional measurements are not used when continuing the estimation process. As a result, the dispersion of the short-term forecast increases to some extent, but it is not typically an obstacle for continuing the estimation. On the other hand, recognizing the exceptional states may be important when analyzing and managing the network operation. The calculation of the operatively unobservable distribution network operation does not principally differ from the calculations in the observable part of the distribution network. The network configuration is radial, and the necessary load information is received from similar load models as on the higher level of the distribution network. Because of missing online measurements on the unobservable level of the distribution network, the estimation of the operation and editing of the load models are accordingly not possible. Necessary load data from different information systems is obtained off-line. The estimation of the load models takes place in two phases. In the first phase (connected to the load research), the type models of the load are formed, and later, during the practical monitoring of the operation, the models are edited. If there is sufficient load data, which often happens on the operatively observable level of the network, a unique model for 139

Summary every load may be estimated. In the unobservable network if enough offline data is available, it is also possible to estimate the model according to the load. The load cases, which are caused by switchings, irregular work of factories, etc. have an essential place in model estimation. However, load cases may be determined only for loads for which enough load data is available. Fortunately, the larger and more significant loads (from the perspective of network operation) are measured by SCADA or by a commercial measurement system. Load cases for small consumers, for which the hourly data of the load is not available, are not observed. Possible configuration changes of the network are fixed by SCADA, and it is checked in the operation estimation process. The changes on the lower level of the network are not observed by the dispatching system. Here it is possible to implement the simulation process, where the load characteristics are found and the network operation is calculated for different load cases. Thus, the possible changing limits of the operation are acquired. When handling the loads of the unobservable distribution network, the importance and volume of load research work is greater than on the higher levels of the network. Already the extent of the unobservable level of the network and the number of the busloads is considerably higher than other levels. Gathering load data is a problem in itself. The possible variety of information systems, the quality and the usability of the data, and different means of access to the data make the load research difficult. Based on well-estimated models, it is possible to find all the necessary characteristics for load monitoring as well as for monitoring the electrical network operation. However, the electrical network operation is a phenomenon in itself, the possibilities and purposes of which do not belong to the field of load monitoring. In addition to the principles of the electrotechnics (for example, network equations), the load monitoring system offers an energetic base â&#x20AC;&#x201C; load models, for treatment of the network operation.

140

Summary

8.3

The Component Presentation of the Monitoring System

For the practical application of the mathematical model of the load, appropriate computer programs must be developed. The possible variety of applications makes the development of the programs complex. However, the autonomic program for load handling is generally not needed. The information about the loads is used as initial data for calculating the electrical network, for operating in the electricity market, etc. The handling of the load should take place in the corresponding application programs, not in autonomic programs. The component technology of the programming is the proper way to treat the problem. In this book, the necessary procedures to realize the mathematical model are aggregated into the ActiveX-type code component ElmoExe.dll. The code component ElmoDisco.dll is developed to calculate the distribution network operation. By using the component ElmoExe.dll, it is possible to perform all the specific operations belonging to the handling of the load data, model estimation, and calculating the load characteristics. General activities, connected to the user interface, do not belong here. Because the structure of the mathematical model is the same for all loads, the program component ElmoExe.dll can also be used in all applications. Therefore, it is not important whether the loads of a large electrical system (up to gigawatts) or the load of a private residence (up to kilowatts), active or reactive power are considered, or short-term or long-term forecasts are needed, etc. â&#x20AC;&#x201C; in all cases, the necessary actions are done by the same code component. The application programs that use the component ElmoExe.dll, their purposes, and user interfaces are different. The component ElmoExe.dll involves some object classes, from which the necessary instances are formed in the application programs. Programs communicate with the objects through its properties and methods. The voluminous data (hourly load measurements, the parameters of the mathematical model, etc.) are retained in the data buffer consisting of binary files, which are intended for handling by objects of the monitoring system. The objects do not handle other data that are located in external databases or in public files. The load data is formed and saved

141

Summary into buffer files by the application programs using the means of the component ElmoExe.dll. The component ElmoExe.dll can also be used in calculating the distribution network operation. Here, first of all, it is necessary to handle the busloads â&#x20AC;&#x201C; save the data, estimate the model and calculate the load characteristics. To calculate the operation, the code component ElmoDisco.dll exists. The application of the load monitoring system proceeds from load research, where the initial estimation of the model parameters takes place. Thereafter, in the monitoring process, the model parameters are edited periodically. The necessary means for estimating the parameters and for editing are offered by the component ElmoExe.dll. The procedures themselves may be realised in the application programs. Since the procedures connected to the load research and editing the model are somewhat specific, then autonomic program ElmoSet is supplemented with component ElmoExe.dll, which permits realization of the procedures. In that case, only the calculation and use of the load characteristics for specific purposes occur in the application programs.

142

B bad data 119, 122 Box-Jenkins model → ARIMA model buffer file 17

C characteristic vector 92 class of load 45 component ElmoExe.dll 18, 60 confidence factor 38, 42 co-ordinate function 26, 45

D data density 62 date serial 62 day type 24 dependency on frequency → frequency sensensitivity of load dependency on voltage → voltage sensensitivity of load distribution network, observable 113, 118 distribution network, unobservable 113, 126

E estimation oject 111 exclusive state 121 expected deviation of load 14, 22 expected value of load 23, 36, 76

F forecasting model 8 frequency sensensitivity of load 25

L lead time 23, 79 level factor 14, 105 load analysis 83 load case 15, 49, 107 load characteristic 14, 73 load characteristic, derived 76 load characteristic, primary 73 load distribution function 39 load forecasting 79 load manager 19 load model → mathematical model of load load monitoring 14, 73, 133 load net object 59 load object 61 load research 12, 91, 100 load scenario 15, 51 load simulation 79 long-term forecast 14, 76, 79 load trend 47

M mathematical expectation of load 11, 21, 23 mathematical model of load 21 model co-ordinate matrix 130 model co-ordinates 45, 92 model component 13, 107 model editing 13, 91, 105, 122 model estimation 12, 91 model factor 13, 45 model object 17, 59 model of load, normal → mathematical model of load model of load, simplified 9, 77 model of load, trivial 8, 31 model of temperature 26

indication curve 95

143

N normal temperature 11, 22 normalized load value 28, 76

O operation characteristic 114 operation monitoring 124 operational dynamics 133

P peak deviation of load 22, 38 peak-normal distribution 41

R rate of temperature dependency 22, 32 reference day 53 relation function of special day 54, 99 relative temperature dependency 22, 33, 52 residual deviation 22, 38 restored load 73

S simplified load model 9, 77 simulated load 76 shape co-ordinates 45 shape factor 45, 105

144

short-term forecast 14, 22, 81 special day 24, 53 special day calendar 55, 71 special day component 55 special period 53 standard deviation of load 22, 23 state estimation 119 stochastic component 22, 36, 52 stochastic submodel â&#x2020;&#x2019; stochastic component

T temperature dependency 11, 22, 31, 51, 94 temperature model 26 temperature sensitivity â&#x2020;&#x2019; rate of temperature dependency temperature object 63 transfer function of temperature dependency 33 transmission network 114 trend of load 25, 47 trivial model 10, 31 type model 12, 46 type of load 45

V voltage sensensitivity of load 25