Optimal short-term operation of pumped-storage power plants with differential evolution algorithm

Energy 194 (2020) 116866

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Energy journal homepage: www.elsevier.com/locate/energy

Optimal short-term operation of pumped-storage power plants with differential evolution algorithm € o €n Serdar Ozy Electrical and Electronics Engineering Department, Kütahya Dumlupınar University, 43100, Turkey

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 August 2019 Received in revised form 24 October 2019 Accepted 26 December 2019 Available online 31 December 2019

In this study, the optimal short-term operation of the lossy electric power system which included the pumped-storage power generation unit (Ps) was carried out. The problem is quite complex in terms of its constraints and being a mathematically difficult problem. Therefore, in order to obtain optimal solutions of the problem, differential evolution algorithm (DE) was used. In this study, one day operation of a system, which includes thermal and Ps units, solved with different optimization algorithms in the literature was addressed. The addressed system was solved in two different ways. First of all, the solution is made assuming that the system consists only of thermal generation units. Then, a Ps unit was added to a bus in the system and it was solved again and the benefit obtained was discussed. In the solution of the sample system, transmission line losses are taken into consideration and Newton-Raphson power flow method was used to calculate these losses. The sample system was solved 30 times with DE and the optimal solution values were determined. All possible constraints in problem solving were provided through the punishment method. The results obtained from the solutions of the sample systems were compared and discussed. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Power generation systems Optimal short-term operation Pumped-storage hydraulic unit (Ps) AC load flow Differential evolution algorithm (DE)

1. Introduction In a world that is growing and developing economically, the need for electrical energy increases in parallel with these growth and developments. A large part of this increasing demand is provided by hydraulic and thermal power generation units. According to thermal generation units, electrical energy produced by hydraulic sources is an option which is both environment-friendly and free from fuel costs. In addition to these units, renewable energy generation units are other generation units that are widely used. Although the energy generated by these units is cost-efficient, the need for storage constitues a problem. At this point, pumpedstorage generation units gain great importance and their usage is becoming more common. The main purpose of the pumpedstorage generation units is to store the extra energy that can be produced in the system as hydraulic potential energy, while the cost of generation in the energy systems is low (low power demand). While the cost of this stored energy generation is high (high power demand), the energy demand of the system is met by converting it to electrical energy. The problem of operating and optimal

E-mail address: [email protected]. https://doi.org/10.1016/j.energy.2019.116866 0360-5442/© 2019 Elsevier Ltd. All rights reserved.

use of systems containing such units is of great importance in electrical engineering [1,2]. On the other hand, the optimal operation of pumped-storage hydroelectric power plants is important in the competitive electricity market because of their operational flexibility and their ability to react quickly to changes in spot price of electricity. The operation of pumped-storage generation facilities affects consumer and producer surplus in the individual market and accordingly leads to significant changes in energy prices [3,4]. Optimal operation of electricity generation systems over a period of time is defined as operation of the generation units in the system to supply an existing load with all the constraints and with the cheapest generation cost. The period mentioned here can be short, medium or long. The period addressed in this study is the optimal short-term operation, which covers one day (24 h) working time of the generation units [5]. When a comprehensive literature review is conducted, it is seen that there are many studies carried out on pumped-storage power systems. These studies have been precisely examined and evaluated in order to obtain faster, more efficient and better data for the problem. Mathematical modeling of pumped-storage generation units, the importance of their use, the benefits they provide to the system and various statistical data are given in Refs. [6e8]. In the literature, the problem of optimal operation of systems with

2

€ o €n / Energy 194 (2020) 116866 S. Ozy

pumped-storage generation units has been solved under different constraints by various approaches and optimization methods. These approaches can be listed as the lagrangian relaxation technique (LRT) [9], annealing neural network (ANN) [10], mixedinteger linear programming (MILP) [11,12], mixed-integer nonlinear programming (MINLP) [13e15], evolutionary particle swarm optimization (EPSO) [16], time varying acceleration coefficient particle swarm optimization with mutation strategies (TVAC-PSOMS) [17], the genetic algorithm (GA) [18e22], two stage stochastic programming (TS-SP) [23], pseudo spot price algorithm (PSPA) [24,25], modified subgradient algorithm based on feasible values (F-MSG) [26e29], binary artificial sheep algorithm (BASA) [30] and a new optimization methodology (NOM) [31]. Nowadays, optimization algorithms are frequently used in solving problems that are difficult or impossible to solve with complex and numerical methods. The reason for this is that the numerical methods of solving multidimensional problems with large search spaces, including many constraints, take a very long time. The problem of optimal short-term operation of pumpedstorage power plants which is solved in this study is also such a problem in terms of its dimensions and constraints. Numerous optimization algorithms have been developed and are being developed to solve such problems more quickly and decisively [5,24]. In this study, the problem of optimal operation of a lossy energy system consisting of thermal and pumped-storage generation units is solved by differential evolution algorithm (DE) which has a strong place in the literature. In the solution of the sample system, transmission line losses are taken into consideration and NewtonRaphson power flow method is used to calculate these losses. The sample system solved in this study is addressed in two separate ways such as the system with and without pumped-storage generation unit. The benefits of pumped-storage generation units to the system have been proved and the results have been discussed with the data obtained from the solutions.

2. Optimal short-term operation of pumped-storage power plants Thermal generation units, which cover a large portion of electrical power generation, cannot easily adapt to sudden load changes due to various constraints. Therefore, such generation units are generally operated at the base load. Hydraulic power plants are used to meet the peak load as they reach full capacity in a short time. It is important to meet the peak load demand for the system to operate it faultlessly. Another generation unit that is frequently used to meet the peak load demand is pumped-storage hydroelectric power plants. The purpose of these generation units is to store the water in the high reservoir in cases where the load is low, and to obtain energy with the stored water in case of peak load demand [24]. There are two types of operation modes including generation and pumping. Schematic representation of these generation units is given in Fig. 1. The solution of the Optimal Short-Term Pumped-Storage Power Plants problem is to find the active power generation values of the generation units where all possible thermal and hydraulic constraints are provided in the system, minimizing the total fuel cost during the expected operating time [24]. In this case, the total fuel cost of the objective function to be minimized by the determined optimization algorithm is given in equation (1). In the equations defined in this part n represents thermal generation units set (n2 NT ), m represents Ps hydraulic generation units set (m2 NPs ).

Upper Reservoir Electrical Generator

Lower Reservoir

Generating Turbine

Pumping

Fig. 1. Pumped-storage hydraulic unit.

TFC ¼ min

jmax X j¼1

tj

X

  Fn PGT;nj ; ð$Þ

(1)

n2NT

In general, the fuel cost function for each thermal generation unit is represented by a quadratic convex function. The fuel cost function of each generation unit is taken as in equation (2) [1e8].

  Fn PGT;n ¼ an þ bn :PGT;n þ cn :P 2GT;n ð$ = hÞ

(2)

In the equation, Fn (PGT,n), n. represents the fuel cost function of the generation unit, an, bn ve cn represents the coefficients, n. represents the fuel cost function coefficients and PGT,n represents the output power of the n. thermal generation unit and the unit is taken as MW. Active power balance constraints in a lossy system consisting of thermal and Ps hydraulic generation units are shown in equation (3) for the pumping state and equation (4) for the generation state. The reactive power balance constraints in the system are shown in equation (5) for the pumping state and in equation (6) for the generation state.

X

PGT;nj  PPPs;j  Pload;j  Ploss;j ¼ 0;

j2Jpomp

(3)

PGT;nj þ PGPs;j  Pload;j  Ploss;j ¼ 0;

j2Jgen

(4)

n2NT

X n2NT

X

QGT;nj  QPPs;j  Qload;j  Qloss;j ¼ 0;

j2Jpomp

(5)

QGT;nj þ QGPs;j  Qload;j  Qloss;j ¼ 0;

j2Jgen

(6)

n2NT

X n2NT

Active and reactive power working limit values of thermal generation units in the system are given in equations (7) and (8). max P min GT;n  PGT;nj  P GT;n ; max Q min GT;n  QGT;nj  Q GT;n ;

j ¼ 1; …; jmax j ¼ 1; …; jmax

(7) (8)

The electrical constraints of the Ps hydraulic generation units in the system are given in equations (9)e(14) and the hydraulic constraints of them are given in equations (15)e(22).

€ o €n / Energy 194 (2020) 116866 S. Ozy

max P min GPs;m  PGPs;mj  P GPs;m ; max Q min GPs;m  QGPs;mj  Q GPs;m ;

j2Jgen

(9)

j2Jgen

(10)

  max qmin m  qmj PGPs;mj  qm ;

j2Jgen

(11)

  max   P min PPs;m  PPPs;mj  P PPs;m ;

j2Jpomp

(12)

  max   Q min PPs;m  QPPs;mj  Q PPs;m ;

j2Jpomp

(13)

     qmax ; j2Jpomp qmin m  qmj PPPs;mj m

(14)

3

new individual is transformed to the next population, otherwise the old individual is transformed [3,32]. The basic algorithm parameters used in DE are the size of the population (NPmax), the number of the variables (D), maximum iteration number (gmax), crossover rate (CR) and scaling factor (F). The process steps of the algorithms can be defined as variable assignment and formation of the initial population, calculation of the fitness of the individuals, selection, mutation, crossover and stopping of the algorithm. Three chromosomes are required in order to generate a new chromosome in DE. Therefore, population size must be bigger than three (NP > 3). The generation of the initial population formed of NP numbered D dimensional chromosomes is done from the equation below.

h i  ðlÞ ðuÞ ðlÞ xj;i;g¼0 ¼ xj þ randj 0; 1 xj  xj

(23)

( )   dþePGPs;mj ðacreft=hÞ if 0


(

qPPs PPPs;mj ¼

qGPs;total ¼

X

)   if 0 < PPPs;mj   P max PPs   ; ðacre  ft=hÞ if PPPs;mj  ¼ 0

  f þ g PPPs;mj  ðacre  ft=hÞ 0

  qGPs PGPs;mj tj

(17)

j2Jgen

X

  qPPs PPPs;mj  tj

(18)

qGPs;total  qPPs;total ¼ qtotal ¼ 0

(19)

max V min m  Vmj  V m ;

(20)

qPPs;total ¼

j2Jpomp

 Vmj ¼

(15)

j ¼ 1; …; jmax

  Vmj1  qGPs PGPs;mj tj if   Vmj1 þ qPPs PPPs;mj tj if

j2Jgen j2Jpomp

¼ V end Vm0 ¼ Vmjmax ¼ V start m m

(21)

(22)

The solution of the Optimal Short-Term Operation of PumpedStorage Power Plants with DE is to determine the lowest fuel cost value that ensures all these equality and inequalities.

3. Differential evolution algorithm (DE) DE, which is a genetic algorithm-based and population grounded optimization technique, was first proposed by Price and Storn in 1995. At the basis of the algorithm lies obtaining a new solution by subjecting the chromosomes, each including a solution in itself, to the operators one by one. During these processes, mutation and crossover operators are used. If the fitness of the new individual, meanly its closeness to the solution is better than the old one, the

(16)

j2Jpomp

xj,i,g, taking place in the equation shows j parameter of i chromoðlÞ ðuÞ some in g population while ðxj ; xj Þ shows the lower and upper limits belonging to the variables [3,32]. After the formation of the population, mutation process takes place. Mutation is to make random changes on the genes of the selected chromosomes. Apart from the present chromosome three chromosomes different from each other are chosen for mutation process in the main structure of DE (r1, r2, r3). The difference of the first two is taken and multiplied with F. F is generally taken as a value changing between [0e2]. Weighted difference chromosome is summed up to the third chromosome. The mathematical expression of the mutation process applied in this study has been shown in equation (24). Different kinds of mutations are also present for DE in different studies [3,32].

  nj;i;gþ1 ¼ xj;r3 ;g þ F: xj;r1 ;g  xj;r2 ;g

(24)

In the equation nj,i,gþ1 shows the intermediate chromosome subjected to mutation and crossover, r1, r2, r3 show the randomly selected chromosomes that will be used in the generation of new chromosome r1;2;3 2f1; 2; 3; …; NPg, r1 sr2 sr3 si . New chromosome (ui,gþ1) is generated by using the difference chromosome obtained as a result of mutation and xi,g chromosome. The genes for the trial chromosome are selected from difference chromosome with CR probability and from the present chromosome with (1-CR) probability. j¼jrand condition, is used to guarantee the removal of at least one gene from the newly generated chromosome. A randomly selected gene at the point of j¼jrand, is selected from nj,i,gþ1 without looking at CR value [3,32].

 xj;u;gþ1 ¼

xj;n;gþ1 if rand½0; 1  CR or j ¼ jrand xj;i;g otherwise

 (25)

A new chromosome (trial chromosome) has been obtained by

€ o €n / Energy 194 (2020) 116866 S. Ozy

4

using three different chromosomes together with the objective chromosome and with the use of mutation and crossover operators. The chromosome, which will be transferred to the new population (g¼gþ1), is determined by looking at its fitness value. The fitness value of the objective chromosome is already known. The objective function value of the problem is calculated as fitness function. The chromosome whose fitness is high among others is transferred to the new population. The circle continues until it becomes (g¼gmax); when it becomes gmax, the best individual in the present population is taken as solution [3,32].

 xi;gþ1 ¼

    xu;gþ1 if f xu;gþ1  f xi;gþ1 xi;g otherwise

(26)

The purpose in algorithm is to obtain chromosomes with better fitness values continuously and to catch the optimum value or to approximate this value. The circle is carried on until it becomes g¼gmax. The stopping of the algorithm in the study depends on the defined iteration number. The flowchart belonging to the DE algorithm has been given in Fig. 2 [3,32].

It is necessary to provide all equalities and inequalities between equations (3)e(22) in order that the results obtained from the solution of optimal operation Ps hydraulic unit scheduling problem with DE algorithm can be suitable optimum solutions. Otherwise the obtained results are not the suitable solutions. When the algorithm starts the solution, first of all iteration number and the individual number in the population are determined and the solution process is started by reading the problem data from the data file. In the first step PGT,nj ve PPs,mj values are defined randomly in such a way that they can provide the constraints between equations (3)e(22) for all periods. Hence, PGT,nj and PPs,mj values are assigned by using equations (27) and (28). This process is maintained for the defined number of individuals. In this case each individual in the formed population becomes a potential solution of the problem.

Generate the starting population randomly with a specified number of (NPmax) individuals.

Calculate the fitness F(xi) of all individuals in the population.

Mutation

Crossover

Selection

g=g+1

j

¼ 1; …; jmax   max min PPs;mj ¼ P min Ps;m þ rand½0; 1  P Ps;m  P Ps;m ; m2NPs ; ¼ 1; …; jmax

Enter the DEparameters NPmax, CR, F, g, gmax

Randomly select three individuals different from the population.

3.1. The application of DE algorithm to the problem

PGT;nj ¼ P min GT;n þ rand½0; 1   min  P max GT;n  P GT;n ; nsslack; n2NT ;

START

(27)

Is maximum number of iterations reached? (g=gmax)

j

Yes

(28)

The generation values of the slack bus and total transmission line loss values are calculated for each of the suggested solutions by doing AC load flow with the generation values in these formed agents. A significant point to be considered before the load flow is the question of whether Ps hydraulic unit taking place in Equation (28) works as a pump or generator according to the random assignment. After defining the power generation values of all generation units in the system, it is controlled if the optimum solution is suitable to the constraints and the penalty functions defined between equations (29)e(31) for each constraint are calculated and added to the objective function. When any of the suggested solutions of the problem violates the foreseen constraints, it is punished by the help of these penalty functions. These foreseen constraints can be identified as slack bus, the minimum and maximum water volume of the reservoirs of the hydraulic units and initial and end volume values.

Take the best individual as a solution.

STOP Fig. 2. The flowchart of DE.

No

€ o €n / Energy 194 (2020) 116866 S. Ozy

PFs ¼

4. Numerical applications

8  2 min > > if PGT;sj < P min GT;s > CPFs PGT;sj  P GT;s > jmax < X   2

max ;j if PGT;sj > P max > CPFs P GT;s  PGT;sj GT;s j¼1 > > > : max 0 if P min GT;s  PGT;sj  P GT;s

¼ 1; …; jmax

(29)

8  2 > > > if Vmj < V min CPFVm Vmj  V min m m > jX < max X  max 2 max ;j PFVm ¼ > > CPFVm V m  Vmj if Vmj > V m j¼1 m2NPs > > min max : 0 if V m  Vmj  V m ¼ 1; …; jmax (30) 8  2 > > if V m CPFVend V m > end;jmax  Vend;m end;jmax < Vend;m > X <  2 PFVend ¼ m if V m > CPFVend Vend;m  V end;jmax end;jmax > Vend;m m2NPs > > > : 0 if Vend;m ¼ V m end;jmax (31) In the equations PFs shows slack bus penalty, PFVm shows the penalties belonging to the volumes of the water stored in the reservoirs of the hydraulic units, and PFVend shows the penalties belonging to the volumes of the water left in the reservoirs of the hydraulic units in the last time slice. CPFs, CPFVm and CPFVend values, which are constants belonging to the penalty functions, are defined by the user in accordance with the system that is solved in the program. After the first iteration completed as a result of all these processes, second iteration is started by calling DE algorithm in order to define the new power generation values that are required to obtain better solutions. The process is continued for the specified number of iterations and the algorithm is ended by taking the best individual in the last iteration as the solution.

V1 = 1.05 0o pu

5

The single line diagram of the selected system is given in Fig. 3 as an example of the optimal operation of energy units containing Ps units defined in the previous sections and whose constraints are determined. In this power system, there are 12 buss, 5 thermal generation generators, 1 hydraulic generation unit Ps, 25 transmission lines and 7 load bars [24]. All values used in the solution of the system and the results obtained were taken as pu. The base values used in the study were Sbase ¼ 100 MVA, Vbase ¼ 230 kV ve Zbase ¼ 529 Ohm. For this system, G1 generator connected to bus 1 has been selected as slack bus in all studies in the literature. The slack bus was fixed and the bus voltage value was taken as V1¼1:05:0o pu. The sample system was solved for two cases by taking in the account of the DE algorithm and transmission line losses. In the first case, the hydraulic unit Ps was neglected and the thermal fuel cost was minimized. In the second case, Ps hydraulic unit was included in the system and thermal fuel cost was calculated under the same load demand and Ps hydraulic unit’s benefit to the cost was determined. In this study, transmission line losses were found by making AC load flow with the Newton-Raphson method. 4.1. Case-I: Non Ps hydraulic unit In the first case, the solution was made assuming that the sample system determined consists only of thermal generation units. In the system, a one-day (short-term) operation period consisting of six equal time periods of 4 h (tj¼4h, j¼1, …,jmax) is considered. The line parameters R (resistance), X (reactance) and B (susceptance) values in pu of nominal p equivalent circuits of the transmission lines in the sample system are shown in Table 1, the active and reactive load values for each period are shown in Table 2, The initial reactive power values for each time period of the generation units are shown in Table 3, The fuel cost function coefficients and the active and reactive power generation limits of the

1 GPs

GT

6

10

7 2

5

11

GT

GT

8 4

3

GT

GT

9 GT

Thermal Unit

GPs Pumped-Storage Unit

Active Load Reactive Load

Fig. 3. One-line diagram of the example power system.

12

€ o €n / Energy 194 (2020) 116866 S. Ozy

6 Table 1 Transmission lines parameters. Line No

Table 4 Fuel cost function coefficients belonging to the thermal generation units.

Bus no (n)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

R

From

To

1 1 1 1 1 2 2 2 3 3 4 4 5 5 6 6 7 7 7 8 8 8 9 10 11

2 3 5 6 7 3 4 7 4 9 7 8 6 7 7 10 8 10 11 9 11 12 12 11 12

X

0.070 0.080 0.080 0.050 0.080 0.050 0.070 0.080 0.080 0.100 0.050 0.100 0.100 0.080 0.060 0.080 0.080 0.060 0.080 0.060 0.080 0.100 0.080 0.060 0.060

B

0.180 0.250 0.200 0.150 0.250 0.200 0.200 0.180 0.250 0.300 0.150 0.300 0.300 0.200 0.180 0.250 0.250 0.180 0.250 0.180 0.300 0.300 0.250 0.200 0.180

0.025 0.025 0.020 0.015 0.025 0.020 0.020 0.018 0.025 0.030 0.015 0.030 0.030 0.020 0.018 0.025 0.025 0.018 0.025 0.018 0.030 0.030 0.025 0.020 0.018

Table 2 Active and reactive loads. Bus no (n)

2 3 5 7 8 10 12

Period (j) 1

2

3

4

5

6

0.20 0.15 0.50 0.40 0.30 0.24 0.25 0.20 0.40 0.30 0.15 0.12 0.20 0.15

0.80 0.60 1.10 0.85 0.90 0.70 1.00 0.75 0.70 0.52 0.60 0.45 0.90 0.70

1.00 0.75 1.20 0.90 0.80 0.60 1.10 0.85 0.90 0.70 1.05 0.80 0.95 0.75

0.80 0.60 1.10 0.85 0.90 0.70 1.00 0.75 0.70 0.52 0.60 0.45 0.90 0.70

0.40 0.30 0.60 0.45 0.25 0.20 0.50 0.40 0.30 0.24 0.45 0.35 0.50 0.40

0.20 0.15 0.50 0.40 0.30 0.24 0.25 0.20 0.40 0.30 0.15 0.12 0.20 0.15

2.00 1.56

6.00 4.57

7.00 5.35

6.00 4.57

3.00 2.34

2.00 1.56

1

2

3

4

5

6

0.080 0.080 0.080 0.080

0.700 0.700 0.700 0.700

1.000 1.000 1.000 1.000

0.700 0.700 0.700 0.700

0.200 0.200 0.200 0.200

0.080 0.080 0.080 0.080

P Q P Q P Q P Q P Q P Q P Q

SP SQ

Table 3 Starting reactive power values. Bus no (n)

4 7 9 11

Bus no (n)

1

4

7

9

11

an bn cn P min GT;n (MW)

527.0 7.48 0.001495 50.0

561.0 7.92 0.001562 45.0

310.0 7.85 0.001940 40.0

476.0 9.52 0.004360 5.0

460.0 9.40 0.003970 3.0

P max GT;n (MW)

350.0

180.0

175.0

100.0

100.0

Table 5 DE parameters. Iteration Individual number (gmax) number (NPmax)

Function Call (FCall)

Crossover Rate (CR)

Scaling Factor (F)

Run

300

15000

0.8

0.4

30

50

Table 6 The values obtained from 30 times solutions (Case-I: Non Ps). DE Worst Mean Best Std Total time (s) Mean time (s)

127918.885753 (Run: 16) 125873.811997 124246.594531 (Run: 13) 1357.761599 2943.454928 98.115164

2637 v4 3.50 GHz processor and 128 GB RAM memory. The machine features on which the algorithm is run are important in terms of time comparisons. The parameter values given in Table 5 and the statistical values of the problem solved 30 times are given in Table 6. The optimal power generation values, transmission line loss and solution time are given in Table 7. The variation of the TFC according to iterations of the optimal solution obtained from the solution of the sample system 30 times with the DE algorithm is given in Fig. 4, the change of transmission line loss according to iterations is shown in Fig. 5 and the active powers generated in each period are given in Fig. 6. The plot of all TFCs obtained from 30 times solutions conducted for the Case-I is given in Fig. 7 and the boxplot of these solutions is given in Fig. 8. When Figs. 7 and 8 are analyzed, it is seen that the worst cost value is obtained as 127918.885753 $ in the 16th solution and 124246.594531 $ in the 13th solution for the 30 times solution of the sample prompt with DE in the case-I. The DE algorithm searched in the range of 3672.2912 $, which provided all constraints in 30 solutions.

Period (j)

thermal generation units are shown in Table 4 [24]. The DE parameter values used for both cases are given in Table 5. In order to solve the problems, the program parts developed in MATLAB R2015b were run on a workstation with Intel Xeon E5-

4.2. Case-II: With Ps hydraulic unit In the second case, in order to determine the benefit provided by Ps hydraulic unit to the system, a solution was made by connecting a Ps hydraulic unit, of which features are given in Table 8, to the sixth bus in the sample system. In order to make the comparisons more accurate, the reactive generation and reactive pumping load of the Ps hydraulic unit was taken as 0 MVAr in both cases. The statistical values of the problem solved 30 times for the Case-II are given in Table 9. The optimal power generation values,

€ o €n / Energy 194 (2020) 116866 S. Ozy

7

Table 7 The values regarding the optimal solution (Case-I: Non Ps) (Run: 13). Bus no (n)

1 4 7 9 11

Generation (pu)

Period (j)

PGT,1 QGT,1 PGT,4 PGT,7 PGT,9 PGT,11

Total Fuel Cost ($) Ploss (pu) Time (s)

1

2

3

4

5

6

1.114921 0.807846 0.450000 0.480705 0.000000 0.000000

2.898354 2.615450 1.800000 1.750000 0.000000 0.000000

3.123533 2.268483 1.800000 1.750000 0.000000 0.804240

2.898354 2.615450 1.800000 1.750000 0.000000 0.000000

1.438687 1.319420 0.693893 0.973322 0.000000 0.000000

1.114923 0.807847 0.450000 0.480714 0.000000 0.000000

124246.594531 1.571657 108.102040

105

1.5

Total fuel co

($)

1.45

1.4

1.35

1.3 X: 111 Y: 1.242e+05

1.25

1.2

0

50

100

150

200

250

300

250

300

Iteration number Fig. 4. TFC’s variation according to the iterations (Case-I: Non Ps).

1.7 X: 147 Y: 1.571

1.6

Ploss (pu)

1.5

1.4

1.3

1.2

1.1

0

50

100

150

200

Iteration number Fig. 5. Ploss’s variation according to the iterations (Case-I: Non Ps).

3.5 Psal P4 P7 P9 P11

3

Generation (pu)

2.5 2 1.5 1 0.5 0

1

2

3

Period

4

5

6

Fig. 6. Generated power’s variation according to the iterations (Case-I: Non Ps).

130000,00 Worst Solution 127918.885753 $

129000,00

TFC

Best values

128000,00 127000,00 126000,00 125000,00 124000,00 123000,00

Best Solution 124246.594531 $ 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Run Fig. 7. TFC values obtained in 30 times solutions (Case-I: Non Ps).

105 1.28 1.275 1.27 1.265 1.26 1.255 1.25 1.245

DE Fig. 8. Boxplot (Case-I: Non Ps).

€ o €n / Energy 194 (2020) 116866 S. Ozy

9

Table 8 Ps hydraulic unit parameters. Bus no 6

P min GPs P min PPs

P max GPs

(MW)

0 V

(MW)

P max PPs 130

initial

(acre-ft)

V

10000

end

(acre-ft)

e

200

2.0

V

10000

min

(acre-ft)

V

5000

Table 9 Values obtained from 30 times solutions (Case-II: With Ps). DE Worst Mean Best Std Total time (s) Mean time (s)

d, f

125045.751736 (Run: 20) 124172.526166 123685.907144 (Run: 15) 348.761546 3316.775643 110.559188

transmission line loss and solution time are given in Table 10. When Table 9 is examined, it is seen that the worst cost value is obtained as 125045.751736 $ in the 20th solution and 123685.907144 $ in the 15th solution for the 30 times solution of the sample prompt with DE in the case-II. The DE algorithm searched in the range of 1359.8446 $, which provided all constraints in 30

g

max

1.33 Efficiency (g/e ¼ m)

(acre-ft)

15000

0.67

solutions. In the second case, the variation of the TFC related to the optimal solution obtained from the solution of the sample system 30 times with DE algorithm is given in Fig. 9, the variation of transmission line loss according to iterations is given in Fig. 10, the amount of water in the upper reservoir at the end of each iteration of the Ps hydraulic unit is given in Fig. 11, the active powers generated in each period are given in Fig. 12, amount of drained-pumped water in Ps units is given in Fig. 13, the amount of water remaining in the upper reservoir at the end of the each period during the operating time of the hydraulic unit Ps is given in Fig. 14. When the TFC value of the optimal solution in Fig. 9 changes according to iterations, it is seen that the DE algorithm approximately captures the optimal TFC value at the 237th iteration. However, the loss of power in the transmission lines of the sample system has reached its lowest value at the 203rd iteration. The iteration differences between them are that there are different

Table 10 The values regarding the optimal solution (Case-II: With Ps) (Run: 15). Bus no (n)

Generation (pu)

1

PGT,1 QGT,1 PGT,4 PPs,6 PGT,7 PGT,9 PGT,11

4 6 7 9 11

Period (j)

Vend Total Fuel Cost ($) Ploss (pu) Time (s)

1

2

3

4

5

6

1.634875 0.888340 0.601433 ¡0.838274 0.672982 0.000000 0.000000

2.869104 2.608977 1.790055 0.067480 1.719527 0.000000 0.000000

3.489639 2.844314 1.784221 0.635387 1.747963 0.000000 0.000000

2.946704 2.623064 1.755962 0.001083 1.746975 0.000000 0.000000

1.501397 1.325340 0.679166 ¡0.113470 1.040650 0.000000 0.000000

1.234613 0.818223 0.450000 ¡0.104257 0.468294 0.000000 0.000000

10000.040026 123685.907144 1.781514 106.027370

105

1.3 1.28

Total fuel co

($)

1.26

X: 237 Y: 1.238e+05

1.24 1.22 1.2 1.18 1.16 1.14

0

50

100

150

200

Iteration number Fig. 9. TFC’s variation according to the iterations (Case-II: With Ps).

250

300

€ o €n / Energy 194 (2020) 116866 S. Ozy

10

450 400 350

Ploss (pu)

300 250 200 150 100 X: 203 Y: 1.778

50 0

0

50

100

150

200

250

300

Iteration number Fig. 10. Ploss’s variation according to the iterations (Case-II: With Ps).

10300 10200

Vson (acre-ft)

10100 10000 9900 9800 9700 9600

0

50

100

150

200

250

300

Iteration number Fig. 11. Variation of the amount of water in the upper reservoir of the Ps unit.

3.5 Psal P4 P6 P7 P9 P11

3

Generation (pu)

2.5 2 1.5 1 0.5 0 -0.5 -1

1

1.5

2

2.5

3

3.5

4

4.5

5

Period Fig. 12. Generated power’s variation according to the iterations (Case-II: With Ps).

5.5

6

€ o €n / Energy 194 (2020) 116866 S. Ozy

11

1500

1000

q (acre-ft)

500

0

-500

-1000

-1500

1

2

3

4

5

6

5

6

Period Fig. 13. Amount of discharge-pumped water in Ps units (Case-II: With Ps).

11500 11000

V (acre-ft)

10500 10000 9500 9000 8500 8000

1

2

3

4

Period Fig. 14. Amount of water remaining in the upper reservoir of Ps unit (Case-II: With Ps).

constraints that must be met in the problem. The plot of all TFCs obtained from 30 times solutions conducted for the Case-II is given in Fig. 15 and the boxplot of these solutions is given in Fig. 16. The comparison of the optimal TFC of the sample power system using DE for both cases is given in Table 11. When the table is examined, it is seen that TFC values obtained with DE in both cases of the same problem previously solved are better than PSPA, GA and F-MSG approaches. 5. Conclusions In this study, optimal operation of Ps hydraulic units, which has a great advantage in storage of generated electrical energy, is addressed. The economic benefit provided by the use of thermally fueled power generation units on their own and in combination with these units has been calculated approximately. With these units, water was stored in the upper reservoirs by pumping with extra energy generated by the thermal generation units for low load

demand, and generation was contributed to the continuity of the system with high power demand. The problem of optimal operation of Ps hydraulic units was discussed in two different approaches and solved by DE algorithm. In the first approach, the sample system was solved 30 times with the DE algorithm under the current constraints, assuming that only the thermal generation units were in use. The optimal thermal fuel cost value obtained from these solutions was 124246.594531 $. The optimal TFC with the best solution value is calculated from the 157.1657 MW lost power load flow on the transmission lines of the system. In the second approach, the same system was solved by adding one Ps hydraulic unit and the optimal thermal fuel cost value was found as 123685.907144 $. In the sample system, 560.6874 $ cost benefit was achieved with the same Ps hydraulic unit at the same operating time and same load demand values. While obtaining this benefit, the benefit of Ps hydraulic unit was taken as 0.67. In optimal TFC, where the best solution value is found, a loss of 178.1514 MW of power was occurred on the transmission lines of the system.

€ o €n / Energy 194 (2020) 116866 S. Ozy

12

126000,00

TFC

Best values

Worst Solution 125045.751736 $

125000,00

124000,00

Best Solution 123685.907144 $

123000,00

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Run Fig. 15. The TFC values obtained from 30 times solutions (Case-II: With Ps).

105 1.25 1.248 1.246 1.244 1.242 1.24 1.238

DE Fig. 16. Boxplot (Case-II: With Ps).

DE

In the future studies, it is considered to integrate renewable energy sources together with Ps hydraulic units to the sample system and to provide sample prompt solutions to the literature.

124246.594531 123685.907144

References

Table 11 Literature comparison.

Non Ps With Ps a

a

PSPA [24]

GA [24]

FMSG [27]

125268.540 124604.982

e 124916.000

124987.1462 124305.989

With reactive power optimization.

The aim of this study is to show that the optimal operation of Ps hydraulic generation units, which is of importance in electrical engineering and one of the optimization problems which are difficult to solve by mathematical methods, which is complex, nonlinear and has many constraints, is solvable by DE. In addition, another objective is to determine the cost benefit to the manufacturer by using the Ps hydraulic generation unit, together with the thermal generation units, in an electric power generation system.

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