Economic Dispatch and Losses Minimization using Multi-Verse Optimizer on 150 kV Mahakam Transmission System

Economic Dispatch and Losses Minimization using Multi-Verse Optimizer on 150 kV Mahakam Transmission System

Yun Tonce Kusuma Priyanto Electrical Engineering Department Kalimantan Institute of Technology Balikpapan, Indonesia

Muhammad Robith Electrical Engineering Department Kalimantan Institute of Technology

Balikpapan, Indonesia

Abstract-On this paper, Multi-Verse Optimizer (MVO) is proposed to solve multiobjective optimal power flow. This algorithm inspired from interaction of universes using black holes, white holes, and wormholes. This algorithm is used to solve multiobjective optimal power flow on 150 kV Mahakam transmission system on East Kalimantan. As comparison, PSO and FA would be used to solve the same problem. As seen on discussion section, each algorithm provide really competitive result at economic dispatch, losses minimization, and both. On the first case, MVO successfully solve the problem with most plausible result, reaching 442552.19 point. MVO also succeed solve the latter case and overcome another algorithms with 4.42 MW. On last case, MVO still solve the problem with the best result with 279499.774 point. From this results, MVO can be used to solve multiobjective optimal power flow.

Keywords-Economic Dispatch, Losses Minimization; Multiobjective Optimization; Optimal Power Flow; Multi Verse Algorithm; Power Generation, Capacitors

1. INTRODUCTION

   Population growth and technological advances are some

2. MULTIOBJECTIVE OPTIMAL POWER FLOW FORMULATION

   Generally, every optimization problem can be represented using this following model:

   minimize/maximize ()

   subject to () = 0 (1)

   () 0

   On (1), f(x) is the objective function, where x is a vector containing all variables that can be controlled. g(x) and b(x) are constraints in equality or inequality forms, respectively. On this paper, cost and losses function are used as objective function. Objective function and constraints used here will explained on next section [8].

   1. Power Losses Function

      Power losses represented on this equation:

      P Nl g t V 2 V2 2t V V cos

      causes that increases demand of electrical energy [13]. This demand increases faster than number of electrical energy resource discovered. To solve this problem, electrical energy

      loss

      k k i

      k1

      j k i j i j

      (2)

      must be managed optimally. Optimality of this management can be seen on many factors, two of them are their cost and power losses that happens on system. This problem categorized as Optimal Power Flow (OPF), where the aiming for the best combination of some variables like generated power and Static VAR Compensators (SVC). OPF problems are really flexible and complex problems [1]. This means that

      OPF problems may have many objectives to deal with, and these objectives may conflict each other. Until now, there are

      On (2), Vi and Vj are voltage magnitude on bus i and bus

      j respectively; Nl is total branches; gk is conductance of branch k; tk is transformer tap ratio installed on branch k; i and j are voltage angle on bus i and bus j respectively.

   2. Cost Function

      Operational cost of a thermal generators modeled as a cost function based on real power generated by that generator. Mathematical model used is quadratic function as

      some methods proposed to solve this problem, from classical differentiation-based methods like Newton-Raphson method [2,3,4,6,8] to metaheuristic methods like Particle Swarm Optimization (PSO) [5,7]. However, first mentioned methods

      below:

      Ng

      Fe Pg i

      i1

      • P2

      (3)

      i i gi

      sometimes trapped on local optima. On the other hand, metaheuristic methods successfully overcome this problem. On this paper, new algorithm is proposed to OPF problems, called Multi-Verse Optimizer (MVO) [14,15]. This algorithm inspired from universes interaction mechanism. This algorithm will be tested on 150 kV Mahakam transmission system on East Kalimantan, and will be compared with two well-known algorithms, PSO and Firefly Algorithm (FA) [9- 12].

      On (3), , , and are cost characteristic coefficients. Pgi

      is real power supplied by generator i and Ng is total generators.

   3. Real and Reactive Power Balance

      On power flow, (4) and (5) must hold, where P and Q are real and reactive power respectively. Gi, load, and losses indexes are tags to mark any variables above to generator i, load, and losses respectively.

      M

      PGi Pload Plosses i1

      M

      QGi Qload Qlosses i1

   4. Constraints

   (4)

   (5)

   1. Multi-Verse Optimizer

      Multi-Verse Optimizer (MVO) founded by Seyedali Mirjalili on 2015 [14]. MVO is a new algorithm that inspired by interaction between universes with a mechanism known as black holes, white holes, and wormholes. There are some theories that explain universes origin, one of them is Multi- Verse Theory. This theory states that there are other universes outside the universe that mankind live, where each universes interact each other. When interaction occurs, they interact using some mechanism known as black holes and white

      These following inequalities are constraints used in this

      paper:

      holes. These holes connect two different universes where an object enters black hole and come out through white hole. In addition to these holes, there are wormholes that connects

      P P P

      min max

      Gi Gi Gi

      Q Q Q

      min max

      Gi Gi Gi

      Qmin Q Qmax

      (6)

      (7)

      (8)

      two point on the same universe. MVO is created using these interaction described above. To convert this to a mathematical model, we apply these approachs:

      • A galaxy is assumed as a combination of some objects

   shunt i

   shunt i

   shunt i

   (or variables) to be optimized. This algorithm search

   t min t t max

   (9)

   for a galaxy with the best objective value through

   i i i

   On (6) to (9), Pgi is real power supplied by generator i, Qgi is real power supplied by generator i, Qshunt-i is capacity of capacitor banks installed on bus i, and ti is transformer tap ratio installed on branch i. min and max indexes are tags to mark any variables above to maximum and minimum values respectively. Inequality (6) represent real power constraint; inequality (7) represent reactive power constraint; inequality

   (8) represent capacity constraint on installed capacitor; and inequality (9) is transformer tap ratio constraint.

3. METHODOLOGY

1. Weighted Sum Method

   Optimal solutions of multiobjective function are solutions from some objective functions simultaneously. To simplified those functions, weighted sum method is proposed. This methods combine all objective functions into a single objective function. For multiobjective optimal power flow in this paper, this method formulated as follow:

   some mechanisms.

   • Probability of black holes or white holes existence on a galaxy determined from its objective value. White holes probability is higher whenever its objective value is far from optimum, and vice versa.

   • Every objects has chances to moving randomly in the same galaxy.

Flowchart of this algorithm for multiobjective power flow is given in fig. 1. First operation executed is black and white holes mechanism. First, each galaxy are sorted based on their objective values, then normalied them. For each variable, we assign a variable form a galaxy randomly (not nessecary different). Randomly selected galaxy are chosen by roulette wheel method. This method chosen for provide variables from the best galaxy to others. This mechanism works like GAs crossover, but GA exchange their gen with others. Pseudocode of this mechanism is given at fig. 2.

Second operation executed is wormholes mechanism. On this mechanism, each variable may move randomly. There are two parameters used for this mechanism,

F w1f1 w2f2

Nl

fref max

PLosses Max k1

(10)

(11)

Wormhole Existence Probability (WEP) and Travelling Distance Rate (TDR). WEP determine each variable move or not, and TDR determine how far they move. This movement following one of these equations:

, = , + TDR × (3 × ( ) + ) (14)

, = , TDR × (3 × ( ) + ) (15)

Ng

fref max

• P

(12)

Where Xi,j is the best variable reached so far, ub and lb

i i i1

2

i gi

are upper and lower bound of that variable respectively. Constant value may be assigned for WEP and TDR, but these

w1 w2 1

(13)

values may be vary following these equation:

On (10), f1(x) and f2(x) will be subtituted with (2) and (3). and are penalty factor. w1 and w2 are weighting factors,

WEP = min + × max min

1/

(16)

where |w| 1 and satisfy (13) [10].

TDR = 1 ( )

(17)

Fig. 1. Multi-verse optimizer flowchart for this paper

SG = sorted_galaxy

NF = normalized_fitness for each galaxy indexed by i

black_hole_index = i

for each variable indexed by j r1 = random(0,1)

if r1 < NF(xi)

white_hole_index = roulette_wheel(NF) G(i,j) = SG(white_hole_index,j)

end if end for

end for

1. SIMULATION AND DISCUSSION

   Input data, test cases, and algorithms used as comparison algorithm will be summarized before simulations started. Transmission system that would be used on this paper is 150 kV Mahakam transmission system on East Kalimantan. Single line diagram of this system given in fig.3. There are three test cases to be examined. They are economic dispatch, losses minimization, and both. Algorithms used as comparison algorithm on this paper are PSO and Firefly Algorithm (FA). On this paper, each algorithm using 20 search agents that searching for best combination for 10000 iterations, performed 10 times on 64-bit Intel Core i7-6700 computer with 16 GB RAM. Parameters used by all algorithms and cost characteristic functions of all generators shown in tables below.

   apply (16) and (17)

   for each galaxy indexed by i

   for each variable indexed by j r2 = random(0,1)

   if r2 < WEP

   r3 = random(0,1) r4 = random(0,1) if r4 < 0.5

   apply (14)

   else apply (15) end

   end if end for

   end for

   Fig. 3. Wormhole mechanism

   Fig. 2. Black and white hole mechanism

   Generator	Cost characteristic function	Minimum Power	Maximum Power
   Generator 1	2 + 2288,5P C1 = -16,873 P1 1 1524,5	41	100
   Generator 2	C2 = 1658,7P2	20	80
   Generator 3	C3 = 2213,2P3	11	190
   Generator 4	C4 = 2628,8P4	1.74	50

   On (16) and (17), min and max are manimum and maximum values assigned for WEP. On this paper min = 0.2 and max = 1 are assigned. p describe algorithms exploitation ability, where 6 is assigned on this paper. l and L are on-going and maximum iteration respectively. Pseudocode of this mechanism given on fig. 3.

   Fig. 4. 150 kV Mahakam transmission system single line diagram TABLE 1. GENERATORS COST CHARACTERISTIC FUNCTION

   TABLE 2. LOAD DATA FOR EACH BUS

   Bus Number	Bus Code	Real Power Load (MW)	Reactive Power Load (MW)
   1	0	56.164	18.396
   2	0	60.925	19.374
   3	2	0	0
   4	0	49.622	11.519
   5	2	23.264	4.385
   6	0	77.442	32.866
   7	1	57.116	8.033
   8	0	18.747	4.790
   9	0	18.331	6.368
   10	0	11.179	2.785
   11	0	23.439	9.759
   12	2	15.452	4.541

   TABLE 3. LINE IMPEDANCE DATA FOR EACH BUS

   Bus Number		Resistance (p.u.)	Impedance (p.u.)	Supceptance (p.u.)
   From	To
   1	2	0.058135	0.167716	0.002392
   2	3	0.016497	0.048836	0.000825
   3	4	0.016497	0.048836	0.000825
   4	5	0.185652	0.549582	0.009285
   5	6	0.020436	0.060498	0.001022
   5	9	0.032907	0.094933	0.001354
   6	7	0.038903	0.115164	0.001946
   7	8	0.056216	0.162178	0.002313
   9	10	0.017728	0.052480	0.000887
   10	11	0.110800	0.328000	0.005541
   10	12	0.221600	0.656000	0.011083

   TABLE 4. MVO PARAMETERS

   Parameter	Value
   min	0.2
   max	1
   p	6

   TABLE 5. PSO PARAMETERS

   Parameter	Value
   w	0.9
   c1	0.1
   c2	0.1

   TABLE 6. FA PARAMETERS

   Parameter	Value
   beta	1
   gamma	0.5

   1. Test Case 1: Economic Dispatch

      On the first case, each algorithms perform economic dispatch on 150 kV Mahakam transmission system. Table 5 and 6 shows simulation result of this case. From table 9, MVO and PSO provide adjacent power generation, while FA provide a little different result power generation 3 and 4. However, even each algorithms provide variative capacitor result, these algorithms provides competitive itness values. MVO successfully overcome other algorithms, as seen on table 6. To obtain each algorithms characteristic, fig. 5 provide the best fitness value reached each iterations from any randomly selected data, where MVO represented on blue curve, PSO on red curve, and FA on yellow curve. It can be seen that MVO actually can overcome all other algorithms near 1000th iterations and converge even at the start at process.

      TABLE 5. OBTAINED VARIABLES FROM EACH ALGORITHM FROM TEST CASE 1

      Variable	MVO	PSO	FA
      Generation Power 1 (MW)	99.999	99.973	99.977
      Generation Power 2 (MW)	79.910	78.672	79.428
      Generation Power 3 (MW)	110.92	112.570	101.19
      Generation Power 4 (MW)	2.2490	1.996	11.847
      SVC 1 (MVAR)	28.302	19.858	38.092
      SVC 2 (MVAR)	18.310	48.716	35.159
      SVC 3 (MVAR)	19.266	10.283	27.068
      SVC 4 (MVAR)	39.385	21.040	42.041
      SVC 5 (MVAR)	27.372	27.964	20.575

      TABLE 6. TEST CASE 1 SIMULATION RESULT

      Method used	Best Fitness
      MVO	442552.19
      PSO	443506,526
      FA	445471.413

      Fig. 5. Total cost convergence curve

   2. Test Case 2: Losses Minimization

      On the second case, each algorithms perform losses minimization on 150 kV Mahakam transmission system. Table 7 and 8 shows simulation result of this case. From these tables, each algorithm give adjacent result on all generation power, but slightly different SVC results. However, each algortihm provides really competitive fitness value. MVO successfully overcome other algorithms, as seen on table 8. To obtain each algorithms characteristic, fig. 6 provide the best fitness value reached each iterations from any randomly selected data, where MVO represented on blue curve, PSO on red curve, and FA on yellow curve. It can be seen that MVO sometimes get another best fitness value, different than other algorithms that can converge at the start of iteration.

      Variable	MVO	PSO	FA
      Generation Power 1 (MW)	99,999	99,828	99,883
      Generation Power 2 (MW)	79,985	79,565	79,260
      Generation Power 3 (MW)	93,196	89,957	91,512
      Generation Power 4 (MW)	18,839	22,809	21,457
      SVC 1 (MVAR)	24,090	27,151	29,034
      SVC 2 (MVAR)	23,013	32,392	20,631
      SVC 3 (MVAR)	10,209	22,313	13,298
      SVC 4 (MVAR)	35,921	48,418	49,444
      SVC 5 (MVAR)	24,295	13,703	47,066

      TABLE 7. OBTAINED VARIABLES FROM EACH ALGORITHM FROM TEST CASE 2

      TABLE 8: TEST CASE 1 SIMULATION RESULT

      Method used	Best Fitness
      MVO	4.42
      PSO	4.513
      FA	4.56

   3. Test Case 3: Economic Dispatch and Losses Minimization

      As the last test case, all algorithms would be used to solve multiobjective optimal power flow on same transmission system, where both Economic Dispatch and Losses Minimization melted into a single objective function by weight sum method. Table 9 and 10 shows simulation result of this case. From table 9, one can see that these results has almost the same characteristic with results on test case 1. MVO gives the minimum fitness value than others, as seen on table 10. To obtain each algorithms characteristic, fig. 7 provide the best fitness value reached each iterations from any randomly selected data, where MVO represented on blue curve, PSO on red curve, and FA on yellow curve. It can be seen that result of this case is almost the same with previous case.

      Fig. 6. Total losses convergence curve

      TABLE 9: OBTAINED VARIABLES FROM EACH ALGORITHM FROM TEST CASE 3

      Variable	MVO	PSO	FA
      Generation Power 1 (MW)	100	99,846	99.978
      Generation Power 2 (MW)	80	79,319	79.622
      Generation Power 3 (MW)	95.689	96,224	94.283
      Generation Power 4 (MW)	22.335	16,683	18.225
      SVC 1 (MVAR)	2.869	23,956	22.203
      SVC 2 (MVAR)	32.728	21,581	40.113
      SVC 3 (MVAR)	8.948	4,767	29.834
      SVC 4 (MVAR)	38.725	23,425	45.350
      SVC 5 (MVAR)	31.029	12,323	26.731

      TABLE 10: TEST CASE 3 SIMULATION RESULT

      Method used	Best Fitness	Total Cost	Power Losses (MW)
      MVO	279499.774	448513.690	4.470
      PSO	280474.940	447149.492	4.472
      FA	280982.273	447265.95	4.507

      Fig. 6. Test case 3 convergence curve

2. CONCLUSION

On this paper, a new algorithm is proposed to solve multiobjective optimal power flow. This algorithm called Multi-Verse Optimizer (MVO). This algorithm inspired from interaction of universes using black holes, white holes, and wormholes. This algorithm is used to solve multiobjective optimal power flow on 150 kV Mahakam transmission system. As comparison, PSO and FA would be used to solve the same problem. As seen on discussion section, each algorithm provide really competitive result at economic dispatch, losses minimization, and both. Each algorithms provide almost the same result on generation power on each generator, but really different SVC values, that proves non- linearity of each cases. MVO successfully overcome other algorithms on each cases. This makes MVO as a option to solve any multiobjective optimal power flow.

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