Optimum Replacement Policies of Conventional Energy Sources by Clean Ones Subject to Current Electri

International Journal of Energy Science (IJES) Volume 4 Issue 1, February 2014 doi: 10.14355/ijes.2014.0401.01

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Optimum Replacement Policies of Conventional Energy Sources by Clean Ones Subject to Current Electricity Demand Maryam Parsa1, NasirUddin Ahmed*2, Mustapha C.E. Yagoub3 School of Electrical Engineering and Computer Science, University of Ottawa, ON, Canada mparsa@purdue.edu; *2ahmed@site.uottawa.ca; 3myagoub@site.uottawa.ca

1

Abstract Developing optimum decision policy for gradual replacement of conventional energy sources by clean ones is an important field of current research and the main concern of this article. Eight main energy sources are considered and the decision policies are formulated with the objective of minimizing the implementation and environmental costs while meeting the electricity demand during the entire plan period. The selected energy sources are Coal, Petroleum, Natural Gas, Hydropower, Wind, Solar, Geothermal and Biomass.

A nonlinear dynamic Lotka‐Volterra model, first introduced in Miah, Ahmed and Chowdhury [1] and later extended in Parsa, Ahmed and Yagoub [2], is used for modeling the dynamic changes in the level of electricity generation from each of the eight available energy sources. Optimal control theory is used to find the optimum decision policies for integration of renewable energy sources into the national power gird of any country.

As a case study for our numerical results, official released data of United States Energy Information Administration website is used for the level of electricity generation from each of the energy sources mentioned above. Different scenarios are considered for the electricity demand. These range from U.S prediction for twenty years plan period to two percent annual growth rate for different plan periods of twenty and thirty years. The proposed methodology is general enough and hence applies to other energy problems with slight modifications based on the plannerʹs objectives.

Keywords Mathematical Models for Energy Systems; Optimum Decision Policy; Electricity Demand and Environmental Concerns

Introduction With the increase of world population, and growth of the industrial complex in developing countries, electricity demand continues to rise. Associated with the rising demand, is a significant increase in the level of Green House Gas emissions due to electricity

generation from polluting energy sources such as Coal and Petroleum thereby threatening health of the general population.

In this paper, we consider electricity generation from all the eight main energy sources and use the Pontryagin Minimum Principle to develop the optimum decision policies for the gradual replacement of conventional energy sources by clean ones. The decision policies for this transition are found while satisfying the electricity demand during the plan period and keeping the implementation and environmental costs as low as possible. Coal, Petroleum, Natural Gas, Hydropower, Wind, Solar, Geothermal, Wood and Biomass are the eight main energy sources that are considered in our model.

In a recent paper of the authors [2], a similar model and method was proposed for different energy problems, satisfying the desired levels of electricity generation from each of the energy sources while minimizing the environmental and investment costs.

In contrast, in this paper integration of renewable energy sources into the power grid is accomplished while the demand during the plan period is met and the environmental and implementation costs are kept to a minimum.

Use of the Lotka‐Volterra model in energy systems and optimal control theory was proposed for the first time in [1]. There energy sources were classified into two categories: polluting and nonpolluting. Here we consider all the eight principal energy sources some of which are conventional and some renewable and present optimal policy of electricity generation from each of the sources.

This article is mainly a continuation of our previous paper [2], with emphasis on satisfying the electricity demand during each time interval of the plan period.

The official released data of U.S. Energy Information

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Administration website [3] is used to validate the model. A set of nonlinear differential equations based on Lotka‐Volterra type interaction is presented ([4], p24) as the dynamic model for our energy generation and regulation problem. Optimal policies are constructed using the Pontryagin Minimum Principle of Optimal Control Theory ([1,2,4]).

compared to demand, environmental cost (in terms health and climate) and the cost of investment for program implementation. All these concerns are taken into account in the following (cost) function (2):

tf

Rest of the paper is organized as follows. The dynamic model of the system and the proposed cost function are presented in section II followed by the system identification in section III. The problem of determining the optimum decision policies and the corresponding results for different scenarios are discussed in section IV. Conclusion and future work of the paper are presented in sections V and VI.

The nonlinear dynamic model based on Lotka‐ Volterra system is utilized in this article. This model is very similar to the one presented in [2]. For convenience of the reader, this model is also reproduced here briefly.

j1, ji

x x j , i  1, 2,...,8

ij i

(1)

2

F(xi (t))  vi xi (t) , v  0

where xi denotes the level of electricity generation from energy source i, ui represents the percentage growth rate of energy source i, and the βij stands for the interactions between the level of electricity generation from energy source i and j. In other words, βij denotes the impact of the level of electricity generation from energy source j on the level of electricity generation from energy source i.

(3)

According to experimental data given in [5], for the same level of electricity generation, Petroleum produces 1.23 times more CO2 than Coal. This relation is used for the weights νi in our numerical results section. The factor νi (> 0) denotes the penalty (proportional to the cost of public health and climate degradation) associated with the production of electricity using the polluting source i.

In this model, the variables ui are considered as the decision policies, that take values from the set U={u: |ui|≤ 1, i = 1,…, 8}, where ui = 1 means a hundred percent rate of increase and ui = ‐1 means a hundred percent rate of decrease in the level of electricity generation from energy source i.

The expression (2), as shown above, consist of three components. The first component is dependent on mismatch between demand and supply of electricity aimed at minimizing the difference between the level of electricity generated and the actual electricity demand during the plan period. The second component representing {Environmental Cost} is aimed at minimizing the electricity production from the polluting energy sources (Coal and Petroleum). The third term represents {Implementation Cost}, and this is

Cost Function The cost function must represent all the concerns of the planner. In this paper the planner is concerned with the cost of insufficient production of energy

2

(2)

8

In general, F is any suitable monotonically increasing function of its argument representing the damage caused to the environment due to production of greenhouse gasses. For our numerical computations we have chosen F as follows:

The dynamic system is given by a system of eight nonlinear interconnected differential equations as follows:



tf

Here D(t) stands for the electricity demand during the plan period and the quantity given by the sum represents the total energy actually produced. In this article we consider two different profiles for the electricity demand. One is based on the U.S. EIA [3] prediction and the other is based on the assumption of two percent annual growth rate. In fact the planner can choose any demand function based on projections of population and industrial growth. In Equation (2), Γ (> 0) is the weight (or importance) given to a mismatch between the supply and demand. The factor qi (> 0) represents the investment cost per unit change of existing infrastructure by new ones.

Non‐Linear Dynamic Lotka‐Volterra Model

8

8

1 1 { F(xi (t))} dt   { qi ui (t))2 } dt  2 t0 i1 2 t0 i1

Problem Description

xi  ui xi 

8 1 f {(D(t)   xi (t))2 } dt   2 t0 i1 t

J(u) 

International Journal of Energy Science (IJES) Volume 4 Issue 1, February 2014

Optimum Decision Policy

aimed at keeping implementation costs as low as possible given the limited resources.

Consider the system Equation (1), where all the fifty‐ six unknown β‐parameters are identified and given in Table 2. For this system, we introduced the objective function given by Equation (2). Optimal control theory, in particular, the Pontryagin Minimum Principle [1, 2, 4, 7], is used to determine the optimum decision policy.

The minimum principle minimizes the cost function given by the expression (2) subject to the dynamic constraint (1).

The objective is to determine the decision policies ui(t) , that minimizes the cost function J(u), expressed by Equation (2).

System Identification There are fifty‐six unknown parameters β = {βi,j} in the model expressed by Equation (1). Finding the optimum decision policies would be impossible without precisely identifying the system model.

Therefore, performing the system identification is the first step. This identification can also be carried out by use of optimal control theory. Using the basic Optimal Control Theory [4] system identification was carried out in our paper [2]. The required historical data was taken from U.S. energy Administration website [6] as presented in Table 1. All the fifty‐six unknown β‐ parameters were identified and given in Table 2.

Details of the mathematical procedure and the methodology for computation of optimum decision policy are given in [2]. Here we have considered three different scenarios for the electricity demand function D(t). In first scenario we use the electricity demand predicted by U.S. EIA [3] for twenty years plan period.

In the second and third scenario we assume 2% annual growth rate for the electricity demand for the twenty years plan period, and for the thirty years plan period.

In the computational scheme developed here the planner can easily choose any given trend for electricity demand and the duration of the plan period as necessary.

TABLE 1 GENERATION LEVEL OF ELECTRIC POWER SECTORS (TRILLION

KILOWATTHOURS)‐ HISTORICAL DATA [6]

Year 2010

Year 2015

Coal (x1)

1.831

1.562

Petroleum (x2)

0.034

0.026

Natural Gas (x3)

0.898

1.028

Conventional Hydropower (x4)

0.25532

0.29543

Wind (x5)

0.09449

0.15097

Solar (x6)

0.00128

0.00647

Geothermal (x7)

0.01567

0.01868

Wood and Biomass (x8)

0.01151

0.02128

TABLE 2 COMPUTED VALUES OF PARAMETERS

The level of electricity generation is given by xi, i = 1,…,8 and the decision policy is given by ui, i = 1,…,8 for eight energy sources of Coal, Petroleum, Natural Gas, Hydropower, Wind, Solar, Geothermal and Biomass, respectively. The weighting parameters qi, νi and Γ are given in Table 3.

TABLE 3 VALUES USED FOR THE WEIGHTS

AFTER 10,000

ITERATIONS βij

Value

βij

Value

βij

Value

βij

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Value

,

,

IN EXPRESSION (2)

qi

Value

qi

Value

νi

Value

Γ

Value

q1

10

q5

1

ν1

5

Γ

150

q2

5

q6

0.59

ν2

5*1.23

q3

3

q7

1.8

q4

3

q8

0.68

β12

0.00226

β23

0.00495

β35

0.00676

β48

0.00024

β21

0.00971

β32

0.00109

β53

0.08901

β84

0.02516

β13

0.02771

β24

0.00356

β36

3.55e‐6

β56

0.00010

Scenario 1

β31

0.02508

β42

0.00072

β63

0.03664

β65

0.00385

β14

0.00944

β25

0.00046

β37

4.35e‐5

β57

0.00027

β41

0.03931

β52

0.00176

β73

0.00750

β75

0.00078

β15

0.00460

β26

9.28e‐5

β38

3.20e‐5

β58

0.00093

β51

0.14887

β62

0.00238

β83

0.08850

β85

0.00931

β16

0.00089

β27

1.20e‐5

β45

0.00202

β67

0.00063

β61

0.07470

β72

0.00038

β54

0.02075

β76

1.07e‐5

β17

0.00073

β28

3.53e‐5

β46

2.75e‐5

β68

0.00046

In this scenario, we considered the electricity demand based on the U.S. prediction [3] as shown in Figure 1.

The value of Γ in Equation (2) is assumed as 150 and given in Table 3. This assumption was based on the high priority of meeting the electricity demand. The planner may easily reduce this value if, from his point of view, the priority of implementation and environmental costs is higher than the electricity demand.

β71

0.01529

β82

0.00735

β64

0.01041

β86

0.00012

β18

0.00407

β34

0.00070

β47

0.00033

β78

0.00009

β81

0.18045

β43

0.01928

β74

0.00213

β87

0.00054

The state trajectories representing the level of electricity generation from each of the energy sources are shown in Figure 1. As expected, the level of

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electricity production from polluted energy sources (Coal and Petroleum) is decreased by the end of the plan period. It is interesting to note that to meet the demand during early periods of the planning horizon the level of electricity production from Coal is increased. This is because the new sources are not yet developed and the demand is met by producing sufficient energy from the retiring sources. It is expected that the total energy produced from all the eight sources meet the demand over the plan period.

This is shown in Table 4.

decreasing as shown (Figure 1). Correspondingly the decision policy (u1) is initially positive and then negative as shown in Figure 2. The value of the cost function after 100,000 iterations is shown in Figure 6‐a.

Figure 6‐b shows the converging trend of this cost function from a closer view in the first 1000 iterations.

(a)

(b)

FIGURE 2. SCENARIO 1, 20 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) COAL AND (b) PETROLEUM ENERGY SOURCES

(a)

(b)

FIGURE 1. SCENARIO 1, 20 YEARS PLAN PERIOD, (a) DEMAND TREND AND STATE TRAJECTORIES FOR COAL, NATURAL GAS, HYDRO, WIND AND (b) STATE TRAJECTORIES FOR PETROLEUM, SOLAR, GEOTHERMAL AND BIOMASS TABLE 4 SCENARIO 1, 20 YEARS PLAN PERIOD, COMPARISON BETWEEN

THE ELECTRICITY DEMAND AND THE ELECTRICITY GENERATED FROM EACH ENERGY SOURCE DURING THE PLAN PERIOD (TRILLION KWH)

xi

Year 2015

Year 2020

Year 2025

Year 2030

Year 2035

x1

1.5620

0.7909

0.3833

0.2095

0.1211

x2

0.0260

0.0180

0.0127

0.0090

0.0065

x3

1.0280

2.7430

3.1580

3.0980

2.7992

x4

0.2954

0.4181

0.5171

0.6563

0.9110

x5

0.1510

0.1760

0.2264

0.3301

0.5632

x6

0.0065

0.0118

0.0235

0.0495

0.1098

x7

0.0187

0.0255

0.0348

0.0481

0.0669

x8

0.0213

0.0295

0.0512

0.1039

0.2461

3.1089

4.2128

4.4070

4.5044

4.8238

3.9000

4.2100

4.4050

4.5040

4.7000

Demand

(a)

(b)

FIGURE 3. SCENARIO 1, 20 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) NATURAL GAS AND (b) HYDROPOWER ENERGY SOURCES

(a)

(b)

FIGURE 4. SCENARIO 1, 20 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) WIND AND (b) SOLAR ENERGY SOURCES

Decision policies for generation levels of the eight energy sources during the plan period are shown in Figures 2‐5. For example, energy production from Coal given by the trajectory (x1) is increasing during the early periods of the plan period and then

4

(a)

(b)

FIGURE 5. SCENARIO 1, 20 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) GEOTHERMAL AND (b) WOOD AND BIOMASS ENERGY SOURCES

International Journal of Energy Science (IJES) Volume 4 Issue 1, February 2014

x5 x6 x7 x8

Demand

(a)

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0.1510 0.0065 0.0187 0.0213

0.1996 0.0162 0.0301 0.0394

0.2844 0.0448 0.0498 0.0874

0.4475 0.1287 0.0835 0.2118

0.9262 0.4239 0.1461 0.6746

3.1089

4.3016

4.7556

5.2045

5.8525

3.8580

4.2980

4.7500

5.2020

5.6660

Figures 8‐11 show the corresponding decision policies.

Comparing this with scenario 1, we observe that for some of the energy sources larger efforts (in terms of decision policies) are required to satisfy the greater demand (in scenario 2). This is seen in Figure 5(b) for scenario 1, where for the biomass energy source the growth rate is around 50% and this is changed to around 58% in scenario 2 as shown in Figure 11(b).

Similarly, for geothermal energy source, the growth rate in scenario 1 is around 10% while it is around 15% in scenario 2. The converging trend of the cost as a function of iteration is shown in Figure 12.

(b)

FIGURE 6. SCENARIO 1, 20 YEARS PLAN PERIOD, (a) COST FUNCTION AFTER 100,000 ITERATIONS AND (b) ZOOMED COST FUNCTION AFTER FIRST 1000 ITERATIONS

Scenario 2 Let us consider the case where the annual growth of electricity demand is 2%. In this case, we expect larger values (greater efforts) for the decision policies to satisfy this increased demand, compared to the previous scenario, where the mean annual growth rate of demand was only 0.958%. The values of the weights {qi ,i = 1,…,8; νi , i =1,2; Γ} for the cost function (Equation (2)) are given in Table 3.

Figure 7 shows the electricity demand and the state trajectories for all energy sources during the twenty years plan period. This demand is satisfied by increasing the level of electricity generated from the renewable energy sources due to lower implementation costs as shown in Table 3. Table 5 shows that during the plan period of twenty years, the sum of all the eight energy sources satisfies the total demand as recorded in the last two rows.

(a)

FIGURE 8. SCENARIO 2, 20 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) COAL AND (b) PETROLEUM ENERGY SOURCES

(a) (a)

(b)

(b)

(b)

FIGURE 9. SCENARIO 2, 20 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) NATURAL GAS AND (b) HYDROPOWER ENERGY SOURCES

FIGURE 7. SCENARIO 2, 20 YEARS PLAN PERIOD, (a) DEMAND TREND AND STATE TRAJECTORIES FOR COAL, NATURAL GAS, HYDRO, WIND AND (b) STATE TRAJECTORIES FOR PETROLEUM, SOLAR, GEOTHERMAL AND BIOMASS TABLE 5 SCENARIO 2, 20 YEARS PLAN PERIOD, COMPARISON BETWEEN

THE ELECTRICITY DEMAND AND THE ELECTRICITY GENERATED FROM EACH ENERGY SOURCE DURING THE PLAN PERIOD (TRILLION KWH) xi x1 x2 x3 x4

Year 2015 1.5620 0.0260 1.0280 0.2954

Year 2020 0.9266 0.0183 2.7020 0.3694

Year 2025 0.4772 0.0128 3.3700 0.4292

Year 2030 0.2738 0.0089 3.5240 0.5263

Year 2035 0.1443 0.0063 2.8355 0.6956

(a)

(b)

FIGURE 10. SCENARIO 2, 20 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) WIND AND (b) SOLAR ENERGY SOURCES

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(a)

(a)

(b)

FIGURE 11. SCENARIO 2, 20 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) GEOTHERMAL AND (b) WOOD AND BIOMASS ENERGY SOURCES

(b)

FIGURE 14. SCENARIO 3, 30 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) COAL AND (b) PETROLEUM ENERGY SOURCES TABLE 6 SCENARIO 3, 30 YEARS PLAN PERIOD, COMPARISON BETWEEN

THE ELECTRICITY DEMAND AND THE ELECTRICITY GENERATED FROM EACH ENERGY SOURCE DURING THE PLAN PERIOD (TRILLION KWH)

(a)

xi

Year 2015

Year 2020

Year 2025

Year 2030

Year 2035

Year 2040

Year 2045

x1

1.5620

1.0130

0.5352

0.3134

0.1847

0.1112

0.0579

x2

0.0260

0.0182

0.0125

0.0085

0.0058

0.0042

0.0029

x3

1.0280

2.7410

3.6200

4.1590

4.4530

4.3980

3.4192

x4

0.2954

0.2808

0.2583

0.2534

0.2792

0.3220

0.4300

x5

0.1510

0.1697

0.2017

0.2534

0.3321

0.4573

0.8745

x6

0.0065

0.0127

0.0270

0.0596

0.1325

0.2734

0.7289

x7

0.0187

0.0267

0.0393

0.0596

0.0921

0.1372

0.2250

x8

0.0213

0.0341

0.0626

0.1188

0.2229

0.4035

0.0555

3.1089

4.2962

4.7566

5.2257

5.7023

6.1068

6.7944

Deman d

3.8580

4.2930

4.7500

5.2250

5.6990

6.1060

6.5690

(b)

FIGURE 12. SCENARIO 2, 20 YEARS PLAN PERIOD, (a) COST FUNCTION AFTER 100,000 ITERATIONS AND (b) ZOOMED COST FUNCTION AFTER FIRST 1000 ITERATIONS

Scenario 3

In this scenario, we extend the plan period to thirty years. In this case, again we consider 2% annual growth rate of electricity demand. The demand trend and the state trajectories for all the eight energy sources are shown in Figure 13. It is interesting to observe that the demand is very well satisfied during the entire plan period as shown in Table 6. The corresponding optimum decision policies for all the eight energy sources are shown in Figures 14‐18 (including the convergence of the cost).

It may be interesting to note that by increasing the plan period from twenty years (scenario 2) to thirty years (scenario 3), the annual investment (or implementation) cost is reduced as shown in Table 7.

(a)

(b)

FIGURE 15. SCENARIO 3, 30 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) NATURAL GAS AND (b) HYDROPOWER ENERGY SOURCES

(a)

(b)

FIGURE 13. SCENARIO 3, 30 YEARS PLAN PERIOD, (a) DEMAND TREND AND STATE TRAJECTORIES FOR COAL, NATURAL GAS, HYDRO, WIND AND (b) STATE TRAJECTORIES FOR PETROLEUM, SOLAR, GEOTHERMAL AND BIOMASS

6

(a)

(b)

FIGURE 16. SCENARIO 3, 30 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) WIND AND (b) SOLAR ENERGY SOURCES

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resolved. The results presented are completely compatible with what is expected. The level of electricity generation from polluting energy sources such as Coal and Petroleum is reduced without compromising demand.

(a)

Different scenarios for electricity demand are presented. One of these is based on the official prediction for the electricity demand during 20 years plan period in the United States (around 0.95% annual growth). Another scenario is based on unexpected increase in demand, for example, 2% annual growth.

This is carried out for different plan periods such as 20 years and 30 years. The methodology developed here can be adapted to any given situation for any country.

The planner has to only change the basic data to suit the objectives of the country.

(b)

FIGURE 17. SCENARIO 3, 30 YEARS PLAN PERIOD, DECISION POLICIES FOR (a) GEOTHERMAL AND (b) WOOD AND BIOMASS ENERGY SOURCES

Future Work

(a)

(b)

The research presented in this article is based on the U.S. prediction during the plan period of 2015‐2035.

However, this prediction may change throughout this period due to uncertainties such as, economic growth, population increase, retirement of existing power plants, etc. Such uncertainties could affect the projection of electricity demand and in turn might change the generation policies. This point could be addressed as future work using Stochastic Optimal Control Theory.

FIGURE 18. SCENARIO 3, 30 YEARS PLAN PERIOD, (a) COST FUNCTION AFTER 100,000 ITERATIONS AND (b) ZOOMED COST FUNCTION AFTER FIRST 1000 ITERATIONS TABLE 7 COMPARISON BETWEEN THE CUMUATIVE AND ANNUAL

IMPLEMENTATION COSTS OF SCENARIO 2 AND SCENARIO 3

Scenario 2

Scenario 3

Plan Period

20 years

30 years

Cumulative Implementation Cost

70.0801

74.6538

Annual Implementation Cost

3.5040

2.4885

REFERENCES

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[1] M.

Miah,

NasirUddin

Ahmed,

Monjur

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The methodology for development of optimum decision policy for integration of renewable energy sources into the national power grid of any country is presented. As a case study we consider the United States since in this case complete historical data are available in the public domain. All eight main energy sources of Coal, Petroleum, Natural Gas, Hydropower, Wind, Solar, Geothermal and Biomass are considered in a single nonlinear dynamic model. Pontryagin Minimum Principle, which is well known to control theorists, is utilized to determine the optimum decision policies. All the costs of program implementation, environmental costs and demand are considered in a general objective function (cost function).

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