Selection of the Reconstruction Options for Industrial Power Supply System

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

Selection of the Reconstruction Options for Industrial Power Supply System under Uncertainty Conditions on the Basis of the Game Theory Criteria Alina Iuldasheva1,a, Aleksei Malafeev2,b 1 PhD Student of Department of Industrial Electric Power Supply, Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation 2 Candidate of Engineering Sciences of Department of Industrial Electric Power Supply, Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation a

alinayuldasheva1@gmail.com

b

malapheev_av@mail.ru DOI 10.13140/RG.2.2.34252.41609

Keywords: production risks, game theory, decision criteria, reliability assesment, uncertainty, damage.

ABSTRACT. Research objective: The reliable power supply at a reasonable cost is a fundamental for the development of any country. Special attention should be paid to the power supply system of industrial enterprises. In the designing, the operation and the mode planning of this systems it is required to account not only the power supply reliability, but also the risks associated with operation interruptions. The task of risk assessment is complicated because of such characteristic feature of industrial power supply system as the uncertainty of information of possible emergency modes, operational loads, etc.

Methods. The combination of two methods: the sequential network reduction and the Newton's method is proposed for the calculation of equivalent reliability indices of complex systems. On the basis of reliability calculation the damage from power supply interruption is determined. The game theory criteria are proposed to use for the decision making in case of uncertainty. The Wald, Minimin, Hurwicz, Bayes, Hodge-Lehmann, Savage, Laplace, Multiplication, GermeierHurwitz criteria are calculated and analyzed.

Scientific novelty and practical significance. Proposed algorithm for reliability evaluation allows to determine the probability of no-failure, failure intensity and recovery time. The algorithm can be used to evaluate the reliability of an existing distribution system and to provide useful planning information regarding improvements to existing systems and the design of new distribution systems. This algorithm wit the Iron and Steel Works in Russia. Application of game theory criteria allows toselect the optimal strategy for the power supply system development and to compare the different variants of normal and repair maintenance schemes of network in uncertainty conditions.

Introduction.In terms of modern market economy, the financial impact of unreliable power supply is of great importance. Particular attention should be paid to the reliability of power supply systems of the energy-intensive industries such as Iron and steel works. Power supply system of a large Iron and steel work has a number of features: the high level of redundancy, chosen on the design stage; the significant transformer power that considers development of production; the combination of explicit and implicit redundancy at all voltage levels; insufficient statistical information on certain elements outages in 35-220 kV networks. So the accounting of reliability of power supply systems is nessesary in both the design and the operation of power supply systems of industrial enterprises.

The operation of power supply system of large industrial enterprises with their own power plants and meshed distribution networks of 110-220 kV, is accompanied by operational risks caused by the technical condition of electric power grid and substation equipment, the work of relay protection, the mode management solutions for grid and stations, the investment and energy saving policy. If the risks associated with equipment failures can be assessed by methods of reliability theory, the risks

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associated with the decision-making of operational and administrative technical personnel should be assessed taking into account the specific psychological aspects.

In recent years, the economic crisis at the Iron and steel works has led to an extreme tightening of energy-saving policy, and therefore the solutions for energy efficiency are often made without taking into account the reliability of power plants. Reason for that is the relatively low frequency of outages of any particular consumer. The volume of statistical information on emergency outages is extremely insufficient to determine the outage probability and the damage. Therefore, the approach allowing to account the uncertainty should be used for quantitative risk assessment.

1. Review of the literature related to the risk evaluation.With the rapid increase of energy demand, correct risk evaluation of power supply systems is of immediate interest. Power systems behave probabilistically because of random nature of load variations and element outages so the risk assessment in real time is challenging. An effective risk assessment model should provide quantitative risk indices to represent system reliability [1]. Usually, only failure statistics are used in power system Significant works have been dedicated to probabilistic risk evaluation of power supply system and substation configurations. Commonly used framework for power system risk assessment was reviewed in [1-3], where the method, use and economic cost were considered in detail. But in this traditional methodology failure risks of elements, such as circuit breakers and transformers, were not studied. Generally, the elements risk assessment in substations is made independently [4 6]. As a result, there is a lack of a mechanism to transform element operation conditions into failure risks in the traditional framework.

The risk assessment model of a combinative system of a transmission network and substations was presented in [4]. Proposed method allows to evaluate system risks considering both transmission networks and substations by assessing new load limitations at load points for every failure state. As an improvement, substations are no longer observed as a transmission node and substation configurations and individual elements, such as breakers and transformers, are linked to system risks by analysing the statistical data of substation elements. However, the component failure data are still based on historical statistics and the effect of online element operation conditions cannot be integrated in risk evaluations. A multi-objective risk assessment framework was presented in [7] and probabilistic indices for assessing real-time power system security levels were derived. But the operation risks of elements still were not considered. Failure probability model, which can demonstrate the influence of surroundings on failure probabilities, based upon the Evidential Reasoning (ER) theory was developed in [8] for overhead lines. Nevertheless element outage rates were set as a fixed value, which were not connected to operating conditions of elements. Contingency identification method of components was presented in [9] and based on the ER theory and the functional group decomposition principle. In that work, element conditions, such as operating conditions and monitoring data of power transformers were not considered, and elements were just observed as part of transmission lines. The ER algorithm [10] is developed for combining evidence with a firm mathematical foundation, which can be employed to aggregate diagnosis information and deal with uncertainties. Proposed in [11] method employs ER for component risk assessment and the Monte Carlo (MC) simulation for system state selections and considers not only historical failure statistics of transmission systems but also operation failure risks of system components. The ER approach is used to evaluate element conditions and connect such conditions to failure rates using upto-date element operation data (online and offline data).

Different approaches for risk management in renewable energy projects are presented in [12]. In this work two ways of risks evaluation are proposed: qualitative and quantitative. Qualitative approaches deal with the evaluation of single risk issues, while quantitative approaches deal with the evaluation based on expertise, so the results is presented in descriptive (risk register) or graphical (risk mapping) formats. The risks were identified through a Delphi process and assessed in terms of probability, MMSE Journal. Open Access www.mmse.xyz

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impact and affected parameter (CAPEX, OPEX, revenues, etc.). The analysis of risks in power supply systems with renewable energy sources is presented out in [13]. The operational risk of consumers power supply (ORCS) is used to account the effect of the random nature of weather conditions on the electricity generation. Proposed method of risk calculation is based on the economic theory of portfolio analysis. The ORCS determines the probability of consumers required power limitations due to the random nature of weather changes. The ORCS calculation is required when choosing the optimal combination of renewable energy sources for the local power grid.

In the paper [14] risk evaluation method is proposed for power grid renovation project in power market. It presents the risk variables affecting performance of power supply company in power grid investment activity, and builds the probability distributing functions according to the variables' physical characteristic, which changes the method fixing power sale quantity, power sale/purchase price, power supply reliability and loss rate in traditional power grid renovation project technology and economics evaluation. The method builds risk evaluation model with increment principle and quantifies power grid investment risk.

In [15-18] authors have developed risk-based measures for various types of security assessment and various components of the power system. These measures of risk consider only a predefined set of contingencies, but do take into account the probability of these events, the uncertainty affecting the future load and other system parameters. Paper [19] states that power system and risk-based approach to security assessment provides considerably more information to base operating decisions, then the traditional N-1 security criterion. Authors argues that risk should be evaluated in terms of expected outage costs to the consumers and the risk calculation should factor in the real probabilities of outages leading to load interruptions. This paper illustrates how to compare the cost and benefit of relaxing operating limits with the adaptive deterministic security boundaries application.

In addition to the N-1 criterion, probabilistic risk indices have been developed and used for power system planning [1, 20, 21]. However, the this risk indices used in selecting planning alternatives do not include long-term voltage instability risk. In current planning practice, the deterministic longterm voltage stability analysis [22] for planning alternatives is performed only for N-1 contingencies without considering probabilities of occurrence of contingencies. Some probabilistic voltage stability risk indices have been presented in the past years: system-wide risk indices [15, 23, 24] and local risk indices [25]. The system-wide risk index functions are used to judge whether or not the voltage instability risk of a system in an objective year meets the requirement in planning, while the local risk functions are used to identify weak location, where a strengthening is needed to increase system voltage stability. However these risk indices cannot be used for the planning purpose, because they do not consider various possible pre-contingency states and have to be recalculated once load level changes. The paper [26] analyses the long-term voltage stability for the power system planning. The combination of system-wide and local risk index functions with load level changes are proposed in [26] to evaluate voltage instability risk for planning alternatives during a whole planning period. The presented risk index functions are calculated by integrating a quadratic optimisation model into one single Monte Carlo simulation process. The optimisation model can find the maximum load level for voltage stability and the Monte Carlo simulation does not need to be repeated during the whole planning period for a given planning alternative.

A lot of works is dedicated to risk assesment, however non of them could allow to evaluate risks of industrial power supply system with accounting of all it's characteristic features. Thus, in conditions of uncertianty the task of development the method for risk assesment for the complex industrial power supply system is very important today.

2. The methodology for reliability evaluation of industrial enterprise power supply system. The review of reliability evaluation methods for the power supply system had shown that their application to the complex power supply system of large industrial enterprise is difficult. The existing methods suggest different criteria for reliability evaluation and individual detailed analysis of each scheme.

Such methods as table-logical and logic-probabilistic inapplicable for this task because of the extreme MMSE Journal. Open Access www.mmse.xyz

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complexity of table of expected logical connections or resulting fault tree. Thus, the development of method for reliability evaluation of large industrial enterprises and its application in the task of reliability analysis for the power supply systems is of immediate interest.

The the method of sequential network reduction [21] was proposed for equivalent reliability indices calculation for the meshed network. According to it the block diagram, which represents an analog of real elements connections of power supply scheme: transformers (T), circuit breakers (CB), overhead lines (PL), generators (G) and cables, is composed. Possibility of power flow direction accounting on the network elements is implemented in the algorithm; thus the part of scheme which is not involved to the electricity transmission to particular consumer will be excluded from the equivalent reliability indices calculation.

Approach based on the sequential network reduction method [27] for calculation of power supply systems modes is proposed to determine the reliability indices. Block diagram representing an analog of real elements connections of power supply scheme (transformers (T), circuit breakers (CB), overhead lines (PL), generators (G)) is composed on the basis of the power supply scheme. Each element of the block diagram is represented as a multi-beam star, which form is determined by the number of element connections. The simplification algorithm is based on the sequential elimination of elements with the replacement of the n-beam star (fig. 1) by the n-gon (fig. 2) with diagonals polygon; this operation reduces the number of elements at the each stage of transformation by one.

For the excluded element a set of equations, linking the probability of no-failure of the star p1, p2, p3, p4 and the sides and diagonals of the polygon p12, p13, p14, p23, p24, p34 , is composed. The set of equations for the considered example has following form:

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(1)

The Newton's method [28] was chosen for solving set of equations (1) as the most efficient numerical iterative method for finding roots of systems of nonlinear equations. Obtained values of the probability of no-failure combined with existing ones in the scheme according to the rules of seriesparallel reduction[29]: for series connection of 2 elements: pekv =p1p2; for parallel connection: pekv = p1+p2 p1p2.

Before the reliability indices evaluation the steady-state mode calculation is carried out. The results of mode calculation are used for the consideration of power flow direction. This procedure allows to exclude from the scheme for reliability indices calculation the part of scheme which is not participate in electricity transmission to the selected consumer. On the basis of the developed calculation created. It allows to calculate probability of no-failure pekv the failure flow parameter recovery time TRekv. The calculation algorithm in more detail is described in [31], [32].

ekv

and the

3. Application of the reliability evaluation methodology. According to the modernization project of industrial enterprise the reconstruction of production workshop is planned, that will result to the increment of load. To meet the new power needs of workshop there is a project of construction of the power plant which will provide an extra total capacity of 37 MW. The fragment of the power supply system scheme of workshop is presented on Fig. 3. For the approbation of proposed algorithm the calculation of the power supply reliability indices of the workshop in different operating conditions is carried out.

The calculation results are presented in Table 1.

Table 1. Results of calculation of reliability indexes.

Type of accident Normal operating mode I Short circuit on PL "Ss-85 Ss-62", planned maintenance of PL "Ss-60 Ss62" II Planned maintenance on PL "Ss-85 Ss-62", short circuit on PL "Ss-60 Ss62" III Short circuit on PL "Power Plant G2", planned maintenance on PL "Ss-60 G-2"

Probability of nofailure, pekv 0.9976

Failure intensity, ekv, (1/h) 0.047

Restoration time,TRekv, (h) 0.0511

0.99294

0.071

0.0994

0.99189

0.075

0.1081

0.99437

0.059

0.0954

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Fig. 3. The fragment of the power supply system scheme of production workshop.

Results had shown that the considered fragment has a high reliability in normal operating mode, but in some modes, for example when the emergency (short circuit) happens in the process of repair mode in close areas the recovery time increased by 2 times what leads to the significant economic and technological damages. In order to reduce the possible damage caused by power supply interruption the measure for reliability improvement is proposed installation of redundancy feeders. Fig.s 4, 5, 6 and Table 2 show the proposed redundancy options at voltage of 110 kV.

Fig. 4. Redundancy option A1.

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Fig. 5. Redundancy option A2.

Fig. 6. Redundancy option A3.

Table 2. Characteristics of the proposed redundancy options.

Strategy

Redundancy option

L, km

K, (mln.RUB)

A1

PL "Ss-62-PowerPlant"

2.8

2.76

0.1

0.06

3.6

2.10

A2 A3

HVL branch line PL "PowerPlant -G-25 W" Ss-62 PL "Ss-62

Ss-30"

4. Damage from power supply interruption. In modern industrial enterprises the equipment of the first category of the power supply reliability is dominate, as a consequence in tasks of mode planning the damage control is important. So in planning of industrial power supply system modes it is

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proposed to make decision not only on the basis of the equivalent reliability indices assessment, but also taking into account the expected damage from unreliability.

To assess the effects of the most serious outages the VaR-method (Value at Risk) is used [7], the value of the membership function (confidence level) = 0.02. Damage is represented as a unilateral fuzzy number with a boundary in the form of the Cauchy curve. The value of the technological damage DTECH for level:

,

(2)

where DAV average damage; DB

boundary damage.

This approach is similar to the popular delta-normal method, which is based on the use of normal distribution fractile and volatility of risk factors, which is acts as its coefficient of variation [8]. The In case of sufficient statistical information about outages in power supply system for particular enterprise the average damage DAV could be determined on the basis of calculated values of recovery time:

,

(3)

where D0 steel industry D0=18.3 TR recovery time, h; P consumer`s load limitiation, kW. The boundary damage in this case can be calculated approximately on the basis of the coefficient of variation CV of outage time resulting from processing information about emergency events in the power supply system: ,

where (

(4)

the quantile of the normal distribution relevant to the confidence probability of 0,95 = 1,67).

The characteristic feature of the large industrial enterprises is the installation of power plants on their territory. In this regard in case of stopping the power plant equipment, additionally to the technological damage there will occur the damage caused by underproduction of electric power DSt by the enterprise power plants, due to the need to purchase electricity at a higher price: ,

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(5)

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where (Cg Cp) the difference between the cost of electricity generated Cg plants and purchased electricity Cp Pst

underproduction of power by local power plant during TR , kW.

For considered redundancy options (Table 2, Fig. 4, 5, 6) damages for different types of possible accidents are calculated. Type of units installed at the power plant is taken into account for damage calculation. There are two gas reciprocating units installed according to the reconstruction project.

For example, a steam-turbine plant can stop only in case of auxiliary services supply interruption or in case of loss of stability. But when the the gas reciprocating units are installed the short circuit in network can lead to their shoutdown by relay protection and that will result to technological damage and the damage caused by underproduction of electricity.

The results of damage calculation are presented in Table 3.

Table 3. Damage calculation.

II

1.2 1.4

Type of Workshop acciden load value t Initial load P III

1.2 1.4

R,

1 8 1 8 1 8

DTECH

DTECH

h

DTECH

Initial load P

DSt (G-2)

1.4

DTECH+ DSt (G-2)

1.2

1 8 1 8 1 8 1 8 1 8 1 8

DSt (G-2)

I

h

No damage

Initial load P

R,

The damage caused by power supply interruption, (mln.RUB) A1 A2 A3 0.82 0.82 6.53 6.53 0.98 0.98 7.83 7.83 1.14 1.14 9.14 9.14 0.89 0.82 7.10 6.53 1.06 0.98 8.44 7.83 1.22 1.14 9.78 9.14 The damage caused by power supply interruption, (mln.RUB) A1 A2 A3 0.05 0.05 0.05 0.4 0.4 0.4 0.05 0.05 0.05 0.4 0.4 0.4 0.05 0.05 0.05 0.4 0.4 0.4 DSt (G-2)

Type of Workshop acciden load value t

5. The optimal strategy selection in uncertainty conditions. The task of the optimal strategy selection in mode planning for power supply systems can interfered to game with nature. There are two types of basic tasks in the games with nature: 1) the problem of decision-making in risk conditions, when the probabilities with which nature takes every possible state are known; 2) the

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problem of decision-making under conditions of uncertainty, when it is not possible to get the information about the probabilities of possible nature state.

The study of games with nature, as well as strategic should begin with composing of a payoff matrix, which is essentially the most time consuming step in the decision making process.

There are several criteria for the optimal strategy selection in theory of games with nature.

1. The Wald criterion (maximin) is a pessimistic approach. The strategy is chosen accordingly to the condition

and coincides with the lower price of the game. This criterion appeals to the

cautious player who looks for ensurance that in the event of an unfavourable outcome, there is at least a known minimum payoff. This approach may be justified because the minimum payoffs may have a higher probability of occurrence or the lowest payoff may lead to an extremely unfavourable outcome.

Thus, the measure of efficiency Wi of strategy Ai according to the Wald criterion is minimum gain of player A: . Price of the game according to the Wald criterion: .

2. The criterion of Minimum is an pessimistic and it is selected from the condition of

.

3. The Hurwitz criterion adheres to the intermediate position, taking into account the possibility of the worst as well as the best behavior of nature. It recommends a strategy defined by the formulas below.

-

for positive-flow payoffs (profits, revenues): .

-

(6)

for negative-flow payoffs (costs, losses):

.

(7)

where A the optimistic coefficient (ranges from 0 to 1). The value of A depends on the player's responsibility: the higher it is, the closer A to the 1. A cautious player will set A = 1, which reduces the Hurwicz criterion to the Wald criterion. An adventurous player will set A = 0, so the Hurwicz criterion may be replaced by maximax criterion.

The optimal strategy will have the maximum value Max(H(Ai)) for positive-flow payoffs, and minimum value Min(H(Ai)) for negative-flow payoffs.

4. The Bayes criterion. At the primary stage of calculation the measure of efficiency for the each n

strategy Bi is determined as: B( Ai )

Q j a ij where Q1, ..., Qn - distribution of probabilities of nature j 1

states. The optimal strategy will have the maximum value Max(B(Ai)) for profits, and minimum value Min(B(Ai)) for losses.

5. The Hodge-Lehmann criterion is based on the Wald and the Bayes criterion. The efficiency indicator for each strategy HL(Ai):

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(8)

where q quantitative measure of the player confidence degree to a given distribution of probabilities qi = p (Pj).

Price of the game on the criterion of the Hodge-Lehmann: Max(HL(Ai)) for profits, Min(HL(Ai)) for losses.

6. According to the Savage criterion (minimax) the strategy that does not allow excessively high losses is optimal. The regret matrix is used. It elements reflects the player losses in case when for each nature state the best strategy will not be chosen. Decision rule is defined as: 1. Transform the payoff matrix X ={Xij} into an regret matrix R ={Rij}.

-

for positive-flow payoffs (profits, income):

.

(9)

for negative-flow payoffs (costs) where Rij is the payoff (reward) for row i and column j of the payoff matrix R:

.

(10)

2. The efficiency indicator for each strategy is determined as the maximum from regret matrix Max(R(Ai)).

3. The optimal is the strategy with minimum efficiency indicator.

7. The Laplace criterion is stated that to none of the possible nature states Sj, j = 1, ..., n, can not be given the preference. All of the nature states are considered as of equal probability. This principle is called the principle of Laplace "insufficient reason". Therefore, if there are n outcomes, the probability of each is 1/n. The strategy price is determined as:

.

(11)

L(Ai) revenues) and min for negative-flow payoffs (costs).

-flow payoffs (profits,

8. The Multiplication criterion. The measure of efficiency Pi of strategy:

.

(12)

Price of the game: Max(P(Ai)) for profits, Min(P(Ai)) for losses. For the multiplication criterion the positivity of all the probabilities of nature state and of all player wins is essential.

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9. The Germeier-Hurwicz criterion as well as the Wald criterion is a criterion highest pessimism of player, but in contrast to the Wald criterion player makes decision with highest possible circumspection, and takes into account the probabilities of the nature states Q. The measure of efficiency:

.

(13)

Price of the game determined as maximum of measure of efficiency - Max(GH(Ai)).

6. Application of the game theory criteria for the selection of strategy for power supply system development. To determine the optimal variant of redundancy for the above considered power supply scheme the payoff matrix is made. Elements of matrix represent the cost of measures to create a reserve for each strategy taking into account the damage. As uncertain information were considered: the restoration time TR, load growth and the type of accident. The possible types of accidents are modelled using a scenario approach. Thus, the payoff matrix of the game include 3 uncertain values (Table 4). As optimality criterion for strategy selection the costs (including damage value) are selected.

Table 4. The payoff matrix.

Type of accident

Workshop load value Initial load P

I

1.2 1.4 Initial load P

II

1.2 1.4 Initial load P

III

1.2 1.4

R,

(h)

1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8

The cost of measures to create a redundancy for strategies, (mln.RUB) A1 A2 A3 2.76 0.87 0.87 2.76 6.58 6.58 2.76 1.04 1.04 2.76 7.89 7.89 2.76 1.20 1.20 2.76 9.19 9.19 2.76 0.95 0.87 2.76 7.16 6.58 2.76 1.11 1.04 2.76 8.50 7.89 2.76 1.28 1.20 2.76 9.84 9.19 2.81 0.11 2.15 3.16 0.46 2.50 2.81 0.11 2.15 3.16 0.46 2.50 2.81 0.11 2.15 3.16 0.46 2.50

Graphically considered strategy shown in Fig. 7.

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C, mln.RUB 10 8 Strategy A1 Strategy A2 Strategy A3

6 4 2

Nature state

0 0

5

10

15

20

Fig. 7. The cost of measures to create a redundancy for the considered strategies.

It can be seen from the Fig. 7 that none of the strategies is dominant, therefore the mentioned above criteria are used for the optimal strategy selection.

1. he pessimistic player will choose. The decision maker prefers the highest value of bad conditions. However, according to Wald's criterion, he should select the maximum of the row minima. So the strategy A1 is selected as optimal (Table 5).

Table 5. The co Strategy

Minimum value, (mln.RUB) 2.76 0.11

A3

0.87

2. The strategy A2 is selected as minimal of the row minima (Table 5) under Minimin criterion.

3. According to , the player is between pessimistic and optimistic choice.

Hurwicz criterion value is calculated according to (7). Minimum and maximum values of each strategy has been multiplied by optimistic coefficient (A = 0.6). The lowest calculated value is selected, so the optimal strategy is A1 (H(Ai) = 2.92 (mln.RUB)).

Strategy

A3

Minimum value, Maximum (mln.RUB) value,(mln.RUB)

Hurwicz criterion value, (mln.RUB)

2.76

3.16

2.92

0.11

9.84

4

0.87

9.19

4.20

4. The characteristic feature of selection the optimal strategy by the Bayes criterion is the consideration of nature states probabilities. In considered example it will be the distribution of probabilities of operating with different power load (Table 7). Elements of payoff matrix (Table 8) are multiply by the probabilities (Table 7) and the measure of efficiency for each strategy will be determined as a sum of values. The strategy A1 with a minimum sum is optimal (Table 9).

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Table 7. The probabi Q1 (for Initial load P)

Q2

Q3

0.45

0.3

0.25

Table 8. The payoff matrix costs for average value of recovery time.

Strategy 2.76 3.73 3.73

I 1.2P 2.76 4.46 4.46

1.4 2.76 5.20 5.20

II 1.2P 2.76 4.81 4.46

2.76 4.05 3.73

1.4 2.76 5.56 5.20

2.98 0.28 2.32

III 1.2P 2.98 0.28 2.32

1.4 2.98 0.28 2.32

Table 9. The computation results of Bayes criterion.

Stra tegy

Value of costs multiplied on probabilities, (mln.RUB) I II III Q1.P 1.24 1.68 1.68

Q2.1.2

Q3.1.4

0.83 1.34 1.34

0.69 1.30 1.30

Q1.P 1.24 1.82 1.68

Q2.1.2

Q3.1.4

0.83 1.44 1.34

0.69 1.39 1.30

Q1.P 1.34 0.13 1.04

Q2.1.2

Q3.1.4

0.90 0.08 0.70

0.75 0.07 0.58

Bayes criterion value, (mln.RUB ) 8.50 9.26 10.95

5. To determine the optimal strategy by the Hodge-Lehmann criterion the values of the Wald (Table 5) and the Bayes (Table 9) criteria are used. For each strategy themeasure of efficiency was calculated as (8) for different values of q (Table 10). The strategy with minimum H(Ai) selected as optimal.

Table 10. The computation results for Hodge-Lehmann criterion.

Hi , (mln.RUB) Stra tegy

W, (mln.RU B)

B, (mln.RU B)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

HodgeLehmann criterion, (mln.RU B)

7.9 7.3 5.6 5.0 4.4 3.9 3.3 6.78 6.21 7.93 3 5 3 6 8 1 3 8.2 7.3 4.6 3.7 2.8 1.9 1.0 0.11 9.14 6.43 5.53 8.24 4 3 2 2 2 1 1 9.9 8.9 5.9 4.9 3.9 2.8 1.8 0.87 10.95 7.93 6.92 9.94 4 4 1 1 0 9 8 6. The Savage criterion focuses on avoiding the worst possible consequences that could result when making a decision. It views actual losses and missed opportunities as equally comparable. For decision making the payoff matrix (Table 4) is onverted to the regret matrix (Table 11), using formula (10), and the minimax rule applied to the regret matrix. The optimal strategy is A1.

2.76

8.50

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Table 11. The fragment of Regret matrix and computation results for Savage criterion.

Regret matrix, (mln.RUB) Savage Str I II III criterion, ate (mln.RUB 1.2P gy ) 1 8 1 8 1 8 1 8 1 8 1 8 1.7 1.5 1.8 1.5 2.7 1.89 0 0 0 0 0 2.70 2.7 2 6 9 6 0 3.8 5.1 6.4 0.0 4.1 0.0 6.8 0 0 0 0 0 6.87 2 3 3 5 9 6 7 3.8 5.1 6.4 3.8 6.4 2.0 0 0 0 0 0 2.04 6.43 2 3 3 2 3 4 7. The Laplace criterion can be interpreted as a transition model between the probability/risk model of decision theory and game theory in that it suggests that in the absence of any probabilities which could potentially differentiate the payoffs, equal probabilities should be assigned. The value of game for each strategy according to (11), was calculated. The optimal strategy according to this criterion is with the minimum measure of efficiency value A1.

Table 12. The computation results for Laplace criterion.

Laplace criterion, Strategy Sum (mln.RUB) 51.02 2.84 57.32 3.18 67.47 3.75 8. The Multiplication criterion. The measure of efficiency P(Ai) for each strategy is calculated according to (12) (Table 13). Strategy A2 corresponding to the minimal measure of efficiency is optimal.

Table 13. The computation results for Multiplication criterion.

Value of costs multiplied on probabilities Strateg I II III . . . . . y Q1 Q2 1.2 Q3 1.4 Q2 1.2 Q3 1.4 Q2.1.2 Q1.P Q1.P P 1.2 4 0.83 0.69 1.24 0.83 0.69 1.34 0.90 1.6 8 1.34 1.30 1.82 1.44 1.39 0.13 0.08 1.6 8 1.34 1.30 1.68 1.34 1.30 1.04 0.70

Q3.1.4

Multiplicatio n criterion, (mln.RUB)

0.75

0.45

0.07

0.01

0.58

3.59

9. The Germeier-Hurwicz criterion as well as the Wald criterion is a criterion highest pessimism of player A, but in contrast to the Wald criterion player A makes decision with highest possible circumspection. The measure of efficiency GH(Ai) for each strategy is calculated according to (13) (Table 14). Strategy A1 corresponding to the minimal measure of efficiency is optimal.

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Table 14. The computation results for Germeier-Hurwicz criterion.

I Stra tegy

Q1.

P

II

Q2.1.2

Q3 .1.

4

1.2 4

0.83

1.6 8 1.6 8

III Mi n

Max

Germeier -Hurwicz criterion, (mln.RU B)

0.75

0.6 9

1.34

1.08

0.08

0.07

0.0 7

1.77

1.09

0.70

0.58

0.5 8

1.68

1.24

Q1 P

Q2.1. Q3.1.

2 4

Q1 P

Q2.1. Q3.1.

2 4

0.69

1.24

0.83

0.69

1.34

0.90

1.34

1.30

1.82

1.44

1.39

0.13

1.34

1.30

1.68

1.34

1.30

1.04

.

.

Results of calculation of all listed above criteria and corresponding to them optimal strategies are presented in summary Table 15.

Table 15. The optimal strategies and corresponding measures of efficiency.

Strateg y

Criteria values, (mln.RUB) Wal d

Mini min

Hurwic z

Bayes

HodgeLehmann

Savage

Laplace

Multipli cation

Germeier - Hurwitz

2.76

2.76

2.96

8.50

7.93

2.70

2.83

0.45

1.08

0.11

0.11

4.87

9.14

8.24

6.87

3.15

0.01

1.09

0.87

0.87

5.03

10.95

9.94

6.43

3.75

3.59

1.24

2.76

0.11

2.96

8.50

7.93

2.70

2.83

0.01

1.08

Optima l strategy Value of game

Thus, the A3 strategy is not optimal strategy on any of the criteria. The A2 strategy is the optimal strategy for the minimin criterion and multiplication criterion. According to the most of criteria (7 out of 9) A1 strategy is optimal, despite the fact that this redundancy option correspond to the highest capital costs.

Summary. In the article the methods of calculation of equivalent reliability indices and damage from the power supply interruption for the large industrial enterprice in different operating modes are presented. Different options of normal and repair schemes could be compared with the help of proposed algorithm on the basis of calculated reliability indices. Accounting of the power flow direction allows to calculate the planned operating conditions, in view of the operational configuration schemes and taking into account changes in load.

The method based on the use of the game theory criteria is used to determine the optimal strategy for the power supply system development in conditions of initial information uncertainty. The options for redundancy and different types of emergency modes were analyzed. On the basis of game theory criteria the optimal option for redundancy with the purpose of reliability increment was chosen. The MMSE Journal. Open Access www.mmse.xyz

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developed method allows to select options for the power supply system development, taking into account the operation of its power plants, the values of operational risks, the reliability indices.

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Programme for the computer, database, TIMS 2015 [31] A. Iuldasheva, A. Malafeev, Reliability Evaluation for Electric Power Supply Management, -12 [32] A. Malafeev, A. Iuldashev , Accounting for power flow direction in the problem analysis of structural reliability of power supply systems // Proc. of Higher Education. Russian Electromechanics, -40 Cite the paper Alina Iuldasheva & Aleksei Malafeev (2016). Selection of the Reconstruction Options for Industrial Power Supply System under Uncertainty Conditions on the Basis of the Game Theory Criteria. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.34252.41609

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