Computational Intelligence and Neuroscience
Volume 2019 (2019), Article ID 6192980, 12 pages
https://doi.org/10.1155/2019/6192980

Research Article

Optimal Operation of the Hybrid Electricity Generation System Using Multiverse Optimization Algorithm

Muhammad Sulaiman,¹ Sohail Ahmad,¹ Javed Iqbal,¹ Asfandyar Khan,¹ and Rahim Khan²

¹Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, KP, Pakistan
²Department of Computer Science, Abdul Wali Khan University Mardan, Mardan, KP, Pakistan

Correspondence should be addressed to Muhammad Sulaiman; sulaiman513@yahoo.co.uk

Received 12 September 2018; Revised 31 January 2019; Accepted 11 February 2019; Published 11 March 2019

Academic Editor: Antonino Laudani

Copyright © 2019 Muhammad Sulaiman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The ongoing load-shedding and energy crises due to mismanagement of energy produced by different sources in Pakistan and increasing dependency on those sources which produce energy using expensive fuels have contributed to rise in load shedding and price of energy per kilo watt hour. In this paper, we have presented the linear programming model of 95 energy production systems in Pakistan. An improved multiverse optimizer is implemented to generate a dataset of 100000 different solutions, which are suggesting to fulfill the overall demand of energy in the country ranging from 9587 MW to 27208 MW. We found that, if some of the power-generating systems are down due to some technical problems, still we can get our demand by following another solution from the dataset, which is partially utilizing the particular faulty power system. According to different case studies, taken in the present study, based on the reports about the electricity short falls been published in news from time to time, we have presented our solutions, respectively, for each case. It is interesting to note that it is easy to reduce the load shedding in the country, by following the solutions presented in our dataset. Graphical analysis is presented to further elaborate our findings. By comparing our results with state-of-the-art algorithms, it is interesting to note that an improved multiverse optimizer is better in getting solutions with lower power generation costs.

1. Introduction

Proper management and finding out optimal solutions to utilize the available power production sources is an important optimization problem in Pakistan, which is the main theme of this paper [1–11]. According to the reports in [12, 13], the demand of the power is increasing exponentially and which is causing an increase in load shedding each year [14]. The cause of this shortage is mismanagement of power production sources.

According to a report published by National Electric Power Regulatory Authority (NEPRA) in 2016 [15], the total capacity of 27240 MW is installed in the country, whereas the demand rises to 22000 MW in summer [16]. These production units can produce enough energy if all are managed optimally. In [17], these production units were normally producing up to 13240 MW with 4760 MW of shortfall. In nomenclature, we have indicated all the 95 installed capacities of electricity production with majority of them depending on thermal sources [15]. In [14], it is reported that the gap between demand and production is expected to increase in the coming years. It is further reported in the literature [14, 18] that urban areas are facing a load shading in summer season as of 10–12 hours, and in rural areas, it is even worse as 16–18 hours a day. Urbanization and rapid increase in population have caused an increase in the industrial zones [19], and it is worth noting that the demand of electricity was 16000 (MW) during summer of 2012 while the short fall was 8500 (MW) [20].

In this study, we have explored a scientific decision-making approach in terms of mathematical programming for providing a strategy to utilize the available production sources of Pakistan to meet the demand of energy in different seasons. However, in [21], a general mechanism was presented in terms of 5 major energy production types as to . They have fixed the per unit price to estimate the overall mathematical model. We have extended the model and included 95 production sources, which generates electricity using different means (oil, gas, wind, hydropower, and coal). We have implemented per unit price according to the actual price of fuel used to run these sources. The production capacities of each source are given in Table 1 [22]. Our model will help to reduce the shortfall by utilizing the power production sources properly.

Table 1: Electricity production constraints on each source.

Power unit
Lower limit
Upper limit

0
3478

0
1000

0
1450

0
243

0
184

0
130

0
121

0
72

0
30

0
22

0
22

0
20

0
17

0
14

0
13.5

0
4

0
1

0
1

0
1655

0
1350

0
850

0
244

0
195

0
174

0
150

0
132

0
59

0
39

0
35

0
17

0
247

0
100

0
100

0
1260

0
560

0
900

0
362

0
365

0
29

0
225

0
165

0
330

0
94

0
152.5

0
226.5

0
157

0
110

0
179

0
165

0
188

0
201.5

0
136

0
140

0
225

0
225

0
1292

0
165

0
120

0
131

0
1638

0
232

0
202

0
660

0
200

0
200

0
225

0
164

0
412

0
114

0
225

0
235

0
350

0
52

0
80

0
110

0
133.5

0
126

0
990

0
137

0
325

0
325

0
340

0
425

0
150

0
50

0
50

0
49

0
50

0
52

0
50

0
56

0
30

0
49

0
50

0
50

In the last few years, evolutionary algorithms (EAs) are implemented to solve different real-world optimization problems. EAs simulate social behavior or natural procedures in order to handle optimization problems with best solution. Among the popular EAs are the Bat algorithm [23, 24], grasshopper optimization algorithm (GOA) [25], and firefly algorithm (FA) [26, 27]. The efficiency of metaheuristics is better as compared to classical optimization techniques in solving optimization problems with iterations and random search behavior. The main idea behind all EAs is the survival of the fittest, which in return increases the fitness of individuals in population. EAs are implemented with different frameworks, which includes population structure and search equations [28]. EAs with different frameworks generate random populations. Then, all solutions are evaluated according to a given objective function. The search space is explored and exploited in two phases. EAs get stuck in local optima due to their random search. Several papers are published in the literature to overcome this problem in EAs [29, 30].

The multiverse optimization algorithm (MVO) is a nature-inspired optimization technique given in [31]. The key idea of MVO is inspired from the theory of multiverses. Since its invention, it is applied to solve several real-world problems. In [31], it is shown that MVO obtained very competitive results compared with other metaheuristic optimization algorithms. Also, Faris et al. [32] employed the MVO for training the multilayer perceptions in neural network. The efficiency of their techniques was evaluated on different medical datasets from UCI repository. The outcome of their experiments is then compared to different metaheuristics, namely, Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Differential Evolution (DE) Algorithm, and Cuckoo Search (CS). It was observed that MVO was improved in terms of convergence and avoidance of local optima. MVO still lacks behind in accuracy of results and convergence speed. Fewer studies, like Levy-flight-based MVO [33] and a quantum version of MVO [34], are recently proposed to further enhance the capabilities of MVO. Experimental results show that they have improved the solutions to some extent.

Currently, as can be seen in the former review, there are limited works on improving the performance of MVO. This work presents a new method of initialization with the MVO algorithm called improved multiverse optimization algorithm (IMVO). The experimental results on mathematical model of economic dispatch problem of electricity generation system in Pakistan and its three case studies show that IMVO is very competitive over FA, Bat algorithm, and GOA.

Furthermore, we have presented solutions to five case studies for proper management and production by all the 95 energy producers in the country. Our model can be implemented to reduce the load shedding and distribute the burden of production on respective power production units.

This work is organized as follows: Section 2 describes the concept of multiverse optimizer and improvements in it. The mathematical model for the problem is described in Section 3. Experimental settings, results, and discussions based on numerical simulations are given in Section 4. Conclusion of our findings is presented in Section 5.

2. An Improved Multiverse Optimizer

The MVO algorithm simulates the theory of multiverse, where three verses play main part in the whole algorithmic procedure: these are white holes, black holes, and wormholes. This is a population-based algorithm, where the population is searched in two phases: exploration in which the search space is visited very well and exploitation performs the search in neighborhood of a solution extensively. White hole and black hole, in the theory of multiverses, are used as exploration agents, and wormhole is acting as exploitation agent in this algorithm. In Figure 1, black, yellow, green, and pink are the objects which are moving through the white/black hole, and white points represent objects which move through the wormhole. There are 5 assumptions that are applied in MVO to the population/universe [31]:

(a)The probability of having the white holes is directly proportional to the inflation rate.(b)Individuals (universes) having higher inflation rate send objects through white holes with higher probability.(c)Individuals (universes) having higher inflation rate send objects through white holes with higher probability.(d)Individuals (universes) having lower rate of inflation send objects through black holes with higher probability.(e)The objects present in all universes are moved randomly towards the current best individual (universe) through wormholes. This random process occurs without taking into account the inflation rate. A sketch of this algorithm is shown in Figure 1.

Figure 1: A sketch of MVO [31].

The initialization phase of MVO can be written as follows:

where represents the matrix of candidate solutions, is the dimensions of the problem, and is the number of universes (population of solutions) which can be represented as follows:

where is the parameter of universe, denotes the universe, NI () is the scaled inflation rate of the universe, , and is the parameter of universe selected by a roulette wheel selection mechanism. The objects are changed among the universes without perturbation. To maintain the balance between exploration and exploitation during the searching process in MVO, each universe is randomly treated to have wormholes to transport its objects through space. In order to ensure the exploitation around the current best solutions, particular wormhole tunnels are always established between a universe and the best universe formed so far, which can be represented mathematically as follows:

where is the parameter of best universe formed so far, (travelling distance rate) and (wormhole existence probability) are coefficients, and are the lower bound and upper bound of variable, respectively, is the parameter of universe and r₂, r₃, and r₄ are random numbers between 0 and 1. and are treated as adaptive formula as follows:

where is the minimum (in this paper, it is set to 0.2), is the maximum (in this paper, it is set to 1), is the current iteration, is the maximum iterations, and is the exploitation accuracy over the iterations (in this paper, it is set to 6). Higher the value of , the sooner and more accurate exploitation/local search is performed [31]. It is worth to highlight that the MVO algorithm depends on number of iterations, number of universes, roulette wheel mechanism, and universe sorting mechanism. The quick sort algorithm is used to sort universe at each iteration, and roulette wheel selection is run for each variable in every universe in all iterations. Detailed description of MVO can be seen in [31].

3. Our Proposed Initialization Strategy

According to the trend in stochastic techniques, random initialization in each generation is the frequently used method to initialize algorithms without knowing any priory information about the problem in hand. The idea of random initialization is often pushing the algorithm to explore the given domain very well, while the exploitation around best solutions is not carried out by this type of initialization. We have used an initialization strategy in which IMVO is balanced in terms of exploration and exploitation. We have designed a two-stage initialization strategy: in the first stage, the algorithm is initialized randomly for a certain number of scattered solutions in the whole search domain. In stage two, after the algorithm completes a certain number of iterations and collects best points in the search domain, we initialize the IMVO with population of these best solutions to concentrate the algorithm on exploitation around the best solutions in the space. Hence, in current studies, we have assigned the function evaluations, fixed to solve a problem, in two parts. The first part initializes with random populations and collects the best solutions of each iteration by using the following equation:

In the second phase, the algorithm is directed to search in the neighborhood of the best solutions obtained in the first part and exploits the search space around the population of these best solutions.

4. Modeling of Electricity Production Cost as a Single Objective Linear Programming Problem

In this paper, we have presented an optimal mixture of energy production system as a linear programming (LP) model consisting of different decision variables, representing different electricity production sources of Pakistan, see Nomenclature [21]. Each variable is bounded and generates electricity through different fuel types like hydro, thermal, nuclear, wind, and solar energies. The suggested LP model includes different production sources as and represent different costs per for each source. The objective function represents the total cost () incurred to meet the demand of energy in different situations. Mathematically,

where represents the cost per and are the different sources given in nomenclature. The objective of this study is to produce a data set of solutions so that we can select the required optimal solution fulfilling the given demand with lowest cost per as compared to other solutions in the data set. For more details, about the data set generated, the reader may refer to [35]. The problem is to search for best solutions among several candidate solutions and to meet the required power demand and satisfy constraints on each variable. These constraints are given in Equation (7) and Table 1 as follows.

Table 2: Case Study A: demand for 15000 megawatt, where represent the collection of sources operating on 8 fuel types.

Firefly
BAT
GOA
IMVO

Price in PKR
390.831986
252.289853
201.3324084
111.5048004

Variable (MW)
15017.4015
15006.4398
15002.26667
15067.46055

4035.19978
2893.04991
3513.562502
4785.656

1407.73428
1873.28453
1072.160605
1338.551

981.013264
158.209497
62.4800805
431.9034

1137.47918
567.079361
395.4767659
1476.913

5684.88097
8502.04227
8812.308041
5366.239

921.438836
523.099019
684.1945259
1180.03

393.294457
21.8122192
442.1898083
150.7902

456.360696
467.863004
19.8943367
337.3776

Table 3: Case Study B: demand for 19000 megawatt, where represent collection of sources operating on 8 fuel types.

Firefly
BAT
GOA
IMVO

Price in PKR
327.227212
236.951036
233.3069629
155.6167319

Variable (MW)
19076.3946
19017.236
19023.02749
19044.11301

6761.41122
4928.40531
5283.899334
3905.454

2395.21169
2006.62552
1695.397916
1655

1469.31172
204.258471
167.1485397
1674.898

1701.02742
1102.9362
1720.041194
2433.113

5583.69119
10116.7438
9981.020494
8160.88

517.009725
153.65083
28.65642935
480.2648

489.383474
93.3714342
137.4132769
394.7306

159.348169
411.244455
9.450307236
339.7717

Total demand constraint:

5. Experimental Settings, Results, and Discussion

We have implemented IMVO which is coded in Matlab R2016 [31]. The main features of our computer included 8 GB RAM with Intel® Core™ i7-7500U CPU @ 2.7 GHz 2.9 GHz with a 64-bit operating system Windows 10. For a fair comparison, the population size was taken as 100 and the number of iterations was 100. We have repeated our simulations 10 times. To elaborate the efficiency of IMVO, we have considered three special cases. The experimental outcome is compared with state-of-the-art algorithms: firefly algorithm, bat algorithm, and grasshopper optimization algorithm, as in Tables 2, 3, and 4. Also, we have picked 5 solutions from overall 100000 solutions and presented these solutions as different case studies with their respective costs per MWh (Table 5), which are based on the power demand in summers as provided in [20, 36–38]. The power demands and decision variables, which represent the 95 production sources in Pakistan and cost incurred along with the total power produced, are given in Table 5. The individual variations of 20 best variables required to produce a particular solution in all case studies are furnished in Table 6, and a weighted comparison of the 95 energy sources is given in Figure 2. Looking at these graphs, the reader can deduce the importance of each production source of electricity in national grid of the country. One can deduce which solution to choose if a particular source is down due to some fault or which sources are very impotent and acting as a backbone in generating the required electricity. We have depicted five case studies graphically as shown in Figure 3. These graphs are collective representation of 95 production sources acting to produce a certain demand. It is interesting to note that the production of electricity from available sources is manageable, if proper techniques, like IMVO algorithm, are implemented to solve the problem. It is worth to note that we have pointed out the top 20 production sources of electricity in Pakistan according to the five case studies listed as in Table 5. These sources play vital part in the energy production as shown in Table 6. It is obvious that certain sources like are producing higher demand which are overburdened and could be relaxed by installing or expanding alternate sources. In Table 7, prices depending on different fuel types are listed for a megawatt of electricity. The convergence plots for these three case studies are depicted in Figures 4–6. It is interesting to note that, in all cases with demands of 15000, 19000, and 23000 megawatts, IMVO produced better solutions with lowest costs in millions (Table 2–4).

Table 4: Case Study C: demand for 23000 megawatt, where represent collection of sources operating on 8 fuel types.

Firefly
BAT
GOA
IMVO

Price in PKR
350.831986
263.667565
276.6948704
187.660591

Variable (MW)
23061.0476
23022.7254
23044.31656
23064.1461

7331.80841
6292
7333.779792
5069.58882

2493.78785
2618
1725.823048
1670.682

1871.00046
219
56.0182213
2517.06661

1756
1756
533.408886
2467.49605

7685.67332
11612
11553.22811
9782.75386

1104.62969
0.02326828
1089.543135
696.3726

436.442721
525
462.5011979
448.716537

381.705102
0.70208538
290.0141633
411.469586

Table 5: Electricity production of all 95 sources for five case studies.

Case Study 1
Case Study 2
Case Study 3
Case Study 4
Case Study 5

Price in PKR
1.12E + 08
1.51E + 08
1.84E + 08
1.97E + 08
2.16E + 08

Variables (megawatts)
15003.16
19003
23000.08
25000.3
27208.36

TD ()
2379.137
1566.35
2938.164
3478
3476.16

MD ()
818.677
1000
905.7829
763.4957
983.48

GBHPP ()
516.5337
1450
1023.352
1347.306
1421.054

WD ()
130.7975
210.4076
177.7411
106.4834
181.5488

CB ()
38.74718
184
75.42442
174.4151
163.8171

DKD ()
42.56789
116.3349
116.0204
60.36495
127.4962

AKHPP ()
25.02909
0.415286
119.0034
33.87101
47.99439

KKHP ()
6.04886
15.92544
41.72064
72
19.27703

JagHP ()
14.17929
30
12.85605
8.998066
21.38508

JabHP ()
8.375935
10.57083
18.06
9.438497
0.704657

RG ()
6.866051
3.330527
0.380166
4.987555
9.745063

DHP ()
1.428187
13.53056
12.65834
20
2.044366

GHD ()
6.05225
0.280779
2.219217
9.40119
15.5921

NHP ()
6.647023
11.40453
7.595507
6.010873
13.86329

SHP ()
11.55404
11.2018
8.391618
11.16708
13.093

KHP ()
1.869716
1.137132
1.534735
2.692804
1.023365

RKHP ()
0.905093
1
1
0.651626
0.854247

CHPP ()
0.864139
0.815802
0.815802
0.965926
0.830172

GTPP ()
1305.331
1655
1655
1655
1645.54

MTPP ()
640.0394
546.27
1350
1350
1350

JTPP ()
149.5429
850
850
850
849.0554

FGTPP ()
148.9675
181.2005
188.0609
103.1949
217.891

MGTPP ()
4.385999
74.8375
155.4596
145.8149
168.1718

MGTPP ()
127.056
65.27735
166.15
34.1201
110.9528

KGTPP ()
112.8329
116.059
30.98832
0.567566
149.5035

LTPP ()
40.37149
0.883692
24.3518
120.5104
117.4427

FSPP ()
39.19613
54.27258
59
56.63659
55.09061

SGTPP ()
0.914076
21.86785
14.66417
22.23108
31.08445

PGTPP ()
27.30059
14.45997
0.589928
8.484009
23.07876

PTPP ()
12.70353
17
6.695689
16.85053
11.95603

KGTPS ()
91.79519
228.2708
247
127.6751
246.8065

KPC ()
91.12861
100
0.076503
16.21392
1.324039

GTPS ()
21.39859
30.86809
0.608317
41.91513
1.135833

TPS ()
1019.436
852.5889
921.7966
1132.252
1260

CCPP 1 ()
363.8285
375.9421
485.8476
538.1132
552.4153

CCPP 2 ()
67.16314
359.9164
551.1955
798.5884
899.1415

AES LL ()
12.48976
362
330.1038
341.4665
346.4409

AES PG ()
87.79272
289.7815
365
244.1401
364.9033

AEL ()
3.047575
29
29
12.08485
19.5939

AP ()
146.5024
194.7207
214.9626
225
225

AGL ()
153.9627
107.4664
152.7011
114.3378
113.5217

CHMC PL ()
75.13341
265.198
273.3111
255.6046
322.7874

DHA C L ()
68.73455
0.476099
6.545028
62.96757
70.30826

EPC ()
23.40477
98.656
37.10004
141.3346
152.0636

EPQL ()
130.7633
214.0389
222.3562
226.5
226.2002

FKPCL ()
95.2459
157
151.2643
27.8556
155.6898

FTSM ()
56.16674
104.4668
100.8249
76.57812
107.0448

FPCDL ()
10.1152
164.3581
127.2432
106.039
177.7063

GHLLP ()
101.702
56.26072
109.8266
140.2346
157.6599

GEPtvL ()
162.3922
188
144.4615
179.5249
131.9891

GEL ()
8.193736
89.19682
201.5
84.89962
201.5

GAEL ()
100.5592
92.35572
57.18626
110.4825
82.02294

HCPC ()
111.3358
58.26757
110.9772
43.47083
111.5484

HPGC ()
169.7313
0.807693
192.3602
125.0309
215.355

HUBCO NPP ()
147.8875
225
212.279
209.5134
223.5541

HUBCOHPP ()
148.0634
1292
1167.515
1292
1292

Iptvl ()
21.53738
158.6019
141.6202
164.2585
164.9125

JPG ()
84.13137
120
120
65.22738
119.424

KEL ()
43.50688
47.52342
103.8332
30.58684
130.8207

KAPCL ()
811.4451
1223.218
1185.71
1638
1638

LPL ()
153.0046
0.52402
94.67205
196.4468
182.2104

LPTL ()
124.1896
68.1917
50.95543
138.8238
198.6912

LEPCL ()
255.6012
589.7144
578.5459
660
658.6634

NCP ()
158.2538
200
95.85135
200
151.3611

NPL ()
32.48763
200
168.4331
168.8323
199.1059

OPCL ()
205.2865
0.89022
132.7586
65.89102
220.0356

REPGCL ()
30.20849
100.2075
52.99079
164
163.4477

RP ()
327.1605
276.0325
302.1523
412
411.8322

SPC ()
72.08175
75.02883
91.21424
109.1339
112.6283

SPPQ ()
101.0073
225
124.0716
211.05
221.962

SESL ()
208.1023
26.5598
66.00281
215.5041
200.3566

SSEL ()
220.5401
14.1361
321.1493
310.6587
343.2993

SNPCL ()
39.3573
44.14536
0.398791
20.11775
51.89066

SE ()
77.28166
11.07506
66.37944
23.84816
16.50386

SEPC ()
12.01473
71.11238
40.1338
47.43561
83.28035

SPGL ()
71.382
35.85756
10.86969
88.37954
113.884

TEL ()
110.2271
25.38428
113.5676
26.5028
94.01525

UchPL ()
735.2781
28.16507
990
990
989.7866

KANUPP ()
50.4151
92.10906
137
105.7683
134.7101

CHASNUPP-1 ()
282.9386
86.3647
275.8089
325
268.7866

CHASNUPP-2 ()
200.2746
243.2484
149.4219
325
320.8773

CHASNUPP-3 ()
191.8579
340
319.874
301.6324
214.0955

NPP ()
18.16426
173.693
374.8168
336.2965
420.7756

QezSP ()
49.71705
106.9033
70.03342
147.3463
126.7687

YEL ()
15.58682
2.334511
46.23074
24.48091
12.86986

MWPCL ()
41.01434
21.35616
27.94642
36.45242
45.21504

TGL ()
9.466357
25.20002
49
47.79046
47.90783

GAWPL ()
15.35919
18.91722
45.14516
15.24926
27.94244

MWEL ()
14.96827
21.99775
52
47.87871
46.00231

FFCEL ()
30.26861
42.68928
50
50
15.58795

ZEP ()
20.88768
14.37346
44.67045
30.45881
40.17728

TWEL ()
15.28468
12.1674
30
29.52479
28.65781

HCDPL ()
9.876255
20.00362
47.25205
31.9031
41.52962

FWEL 1 ()
43.69201
29.80095
45.98028
29.68556
22.61687

FWEPL 2 ()
6.404915
50
6.872093
21.65334
40.28912

Figure 2: A weighted comparison of the 95 energy sources used in five case studies. Series 1–5 represent case studies 1–5, respectively.

(a)

(b)

(c)

(d)

(e)

Figure 3: Comprehensive plots of five case studies discussed in Table 4. All production sources are plotted against the electricity production in megawatts. (a) Case Study 1: demand of 15003 MW. (b) Case Study 2: demand of 19003 MW. (c) Case Study 3: demand of 23000 MW. (d) Case Study 4: demand of 25000.3 MW. (e) Case Study 5: demand of 27208.3 MW.

Table 6: Top 20 electricity production sources according to the five case studies.

No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
No. 7
No. 8
No. 9
No. 10
No. 11
No. 12
No. 13
No. 14
No. 15
No. 16
No. 17
No. 18
No. 19
No. 20

Figure 4: IMVO convergence graph demand for 15000 megawatt.

Figure 5: IMVO convergence graph demand for 19000 megawatt.

Figure 6: IMVO convergence graph demand for 23000 megawatt.

Table 7: Per megawatt cost of electricity for different fuel types.

Fuel type
Price in PKR per megawatt

Hydel
2500

LNG
9070

Furnace oil
11050

High-speed diesel
17960

Coal
12080

Solar
16950

Wind
16630

Nuclear
6860

Regasified liquefied natural gas
11270

6. Conclusions

In this paper, an improved multiverse optimization algorithm is implemented for solving a linear programming model for proper management of electricity production sources in Pakistan. To address the proper management of power production sources, several bound constraints, such as constraint for total demand, and bounds on all variables are considered. The objective function is linear with 95 decision variables and different coefficients. The IMVO algorithm is easy to implement with few parameters. We have found solutions for five different case studies and analyzed all solutions graphically. A link to the data set of 100000 different solutions is provided in this paper. In our results, we have obtained better solutions to meet the power demand in summer. It is recommended that current resources can produce the required power demand but some of the resources are required to be expanded in capacity along with installation of new sources. In future, we intend to include other factors, like power loss, in our model.

Nomenclature

TD: | Tarbela Dam ()
MD: | Mangla Dam ()
GBHPP: | Ghazi Barota Hydropower Project ()
WD: | Warsak Dam ()
CB: | Chashma Barrage ()
DKD: | Duber Khwar Dam ()
AKPH: | Allai Khwar Hydropower Plant ()
KKPL: | Khan Khwar Hydropower Plant ()
JHP: | Jagran Hydropower Plant ()
JabHP: | Jabban Hydropower Plant ()
RB: | Rasal Barrage ()
DHP: | Dargai Power Plant ()
GZD: | Gomal Zam Dam ()
NHP: | Nandipur Hydropower Plant ()
SHP: | Shadiwal Hydropower Plant ()
KHP: | Khuram Hydropower Plant ()
RKHP: | Renela Khuard Hydropower Plant ()
CHPP: | Chatral Hydropower Plant ()
GTPP: | Guduu Thermal Power Plant ()
MTPP: | Muzaffargarh Thermal Power Plant ()
JTPP: | Jamshoro Thermal Power Plant ()
FGTPP: | Faisalabad Gas Turbine Power Plant ()
MGTPP: | Multan Gas Turbine Power Plant ()
KGTPP: | Kotri Gas Turbine Power Plant ()
LTPP: | Larkana Thermal Power Plant ()
FSPP: | Faisalabad Steam Power Plant ()
SGTPP: | Shahdra Gas Turbine Power Plant ()
PGTPP: | Panjgur Gas Turbine Power Plant ()
QTPP: | Quetta Thermal Power Plant ()
PTPP: | Pasni Thermal Power Plant ()
KPC: | Korangi Power Complex ()
KGTPS: | Korang Gas Turbine Power Plant Station ()
GTPS: | Gas Turbine Power Station ()
KPS: | Thermal Power Plant ()
CCPP1: | Combined Cycle Power Plant 1 ()
CCPP2: | Combined Cycle Power Plant 2 ()
AESLL: | AES Lalpir Limited ()
AESPG: | AES Pak Gen ()
AEL: | Altern Energy Limited ()
AP: | Altes Power ()
AGL: | Attock Gen Limited ()
CMECP: | CMEC Power Pvt. ()
DHACL: | DHA Cogen Limited ()
EPC: | Eastern Power Company ()
EPQL: | Engro Powergen Qaidpur Limited ()
FKPP: | Fauji Kabirwala Power Plant ()
FTSM: | First-Star Modaraba ()
FPCDL: | Foundation power company Daharki Limited ()
GHLLP: | Grang Holding Limited Power Plant ()
GEPtvl: | Green Electric Pvt. Limited ()
GEL: | Gujranwala Energy Limited ()
DAEL: | Gul Ahmad Energy Limited ()
HCPC: | Habibullah Coastal Power Company ()
HPGC: | Halmore Power Generation Company ()
HUBCONPP: | HUBCO Narowal Power Plant ()
HUBCOHPP: | HUBCO Hub Power Plant ()
IPtvL: | Intergen PTV Limited ()
JPG: | Japan Power Plant ()
KEL: | Kohinor Energy Limited ()
KAPCL: | Kot Addu Power Company Limited ()
LPL: | Liberty Power Limited ()
LPTL: | Liberty Power Tech. Limited ()
LEPCL: | Lucky Electric Power Company Limited ()
NCP: | Nishat Chunian Power ()
NPL: | Nishat Power Limited ()
OPCPrvL: | Orient Power Company (Pvt.) Limited ()
REPGC: | Radian Energy Power Generation Company ()
RPK: | Rousch Power, Khanewal ()
SPC: | Saba Power Company ()
SPPQ: | Saif Power Plant, Qadirabad ()
SECL: | Sapphire Electric Company Limited ()
SSEL: | Siddiqsons Energy Limited ()
SNPCL: | Sindh Nooriabad Power Company (Pvt.) Limited ()
SE: | Sitara Energy ()
SEPC: | Southern Electric Power Company Limited ()
SPGL: | Star Power Generation Limited ()
TEL: | Tapal Energy Limited ()
UchPL: | Uch (Uch-I and Uch-II) Power Limited ()
KANUPP: | KANUPP ()
CHASNUPP-1: | CHASNUPP-1 ()
CHASNUPP-2: | CHASNUPP-2 ()
CHASNUPP-3: | CHASNUPP-3 ()
NPP: | Nandipur Power Project ()
QeaSP: | Quaid-e-Azam Solar Park ()
YEL: | Yunus Energy Limited ()
MWPCL: | Metro Wind Power Co Limited ()
TGL: | Tenaga Generai Limited ()
GAWPL: | Gul Ahmed Wind Power Limited ()
MWEL: | Master Wind Energy Limited ()
FFCEL: | FFC Energy Limited, Jhimpir ()
ZEPJ: | Zorlu Enerji Pakistan, Jhimpir ()
TWEL: | Tapal Wind Energy Limited ()
HCDPL: | HydroChina Dawood Power Limited ()
FEW-1L: | Foundation Wind Energy-I Limited ()
FEW-2PT: | Foundation Wind Energy-II Private Limited ().

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the writing of this paper. And all authors read and approved the final manuscript.

Acknowledgments

This paper was supported by the Department of Mathematics and the Office of Research, Innovation, and Commercialization of Abdul Wali Khan University Mardan, Pakistan.

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