Simulated Annealing Based Optimization for Solving Large Scale Economic Load Dispatch Problems

Simulated Annealing Based Optimization for Solving Large Scale Economic Load Dispatch Problems

Kamlesh Kumar Vishwakarma Department of Electrical Engineering Madhav Institute of Technology & Science Gwalior (M.P), India-474005

Hari Mohan Dubey

Department of Electrical Engineering Madhav Institute of Technology & Science Gwalior (M.P), India-474005

Abstract

This paper presents a Simulated Annealing (SA) approach for solving Economic Load Dispatch (ELD) problems in electrical power system. The objectives of ELD problems in electric power generation is to programmed the devoted generating unit outputs so as to meet the mandatory load demand at lowest amount operating cost while satisfying all units and system equality and inequality constraints .Global optimization approaches is inspired by annealing process of thermodynamics. The proposed method work s very fast, this aspect of algorithm is strik ing when applied for a large ELD system. Simulation has been performed over two different cases. Case study-I consist 38 generating units and Case study-II consist 110 generating units, both cases having convex fuel cost characteristics. The proposed method results have been compared with other relative existing approaches and finally SA proves luminous feasibility, robustness and fast convergence for optimization of ELD problems.

1. Introduction

   The main aim o f the economic dispatch is to include variables that affect operational cost, su ch as the generator distance from the load, type of fuel, load capacity and transmission line losses. By including these variables one will be ab le to perform economic dispatch and interconnected generators to minimize operating cost.

   Economic dispatch is the process of allocating the required load demand between the available generating units such that the cost of operation is minimu m as passable. The process of solving such a problem is referred to as optimization. ELD is a constrained nonlinear optimizat ion proble m.

   Simu lated Annealing (SA) has been proved to be effective and quite robust in solving the optimizat ion problems. SA can provide near g lobal solutions and can

   also handle effectively the discrete control variables. SA does not stick into local optima because SA begins with many init ial points and search for the most optimu m in para lle l. SA considers only the pay -off informat ion of objective function regardless whether it is differentiable or continuous. Consequently, the most realistic cost characteristic of power plants can be formulated.

   In recent times, different heuristic approaches have been proved to be effective with promising performance. These include evolutionary progra mming (EP) [1], genetic algorith m (GA) [2], differentia l evolution (DE) [3], part icle swarm optimizat ion (PSO) [4], etc. Improved fast Evolutionary programming algorith m has been successfully applied for solving the ELD proble m [5, 6]. Other a lgorith ms like improved coordination aggregated based PSO [7], SOH-PSO [8] and BFONM [9] are some of the, those which have been successfully applied to solve the ELD proble m.

   Th is paper present SA approach for optimization has been used to get to the bottom of economic load dispatch problems. Simulated Annealing (SA ) is a stochastic optimization technique which is based on the process of annealing in Thermodynamics proposed by Kirkpatrick [10].

   Mathematical model of simulated annealing describes how the molecules of liquidated metal move freely with respect to each other and by gradually cooling (thermodynamic process of annealing) therma l mobility are lost. The atoms start to get arranged and finally form crystals, having the minimu m energy which depends on the cooling rate. The proposed method is found to give optima l results while working with constraints in the ELD.

   Th is paper provides a brief e xp lanation and

   mathe matica l formu lation of ELD proble ms in Section

2. The concept of Simu lated Annealing (SA) is discussed in Section 3. Section 4 p rovides the imple mentation process of the algorith m used in the test system. The parameter settings for the test system to evaluate the performance of SA and the simulat ion

studies are discussed in Section 5. Finally, Section 6 presents the conclusions.

1. Problem Formulation

   2.2 The Generator Constraints: The power generated by each generator shall be within their lo wer limit Pimin and upper limit Pimax so that

   P

   In a power sys tem, the unit co mmit ment proble m has various sub-problems varying fro m linear programming proble ms to co mple x non-linear

   min

   P

   P

   i i

   max i

   (5)

   problems. The concerned problem, i.e., Economic Load Dispatch (ELD) proble m is one of the different non – linear progra mming sub-problems of unit commit ment. The ELD proble m is about minimizing the fuel cost of generating units for a specific period of operation so as to accomplish optima l generation dispatch among operating units and in return satisfying the system load demand considering generator operational constraints.

   The objective function corresponding to the production cost can be approximated to be a quadratic function of the active power outputs from the generating units. Symbolica lly, it is represented as

2. Simulated Annealing Method

   Simu lated Annealing (SA) a lgorith m is a nature- inspired method which is adapted from process of gradual cooling of metal in nature. In the metallurgica l annealing process, a solid is melted at high temperature until all mo lecules can move about freely and then a cooling process is performed until therma l mobility is lost. The perfect crystal is the one in which all ato ms are arranged in a lo w level pattern, so crystal reaches the minimu m energy.

   It is basically a stochastic optimizat ion technique

   F

   which is based on the principles of statistical engineering. The search for global minima of a

   Minimize

   cos t t

   NG

   fi (Pi )

   (1)

   mu ltid imensional function is quite a comple x p roble m especially when a big number of local min ima

   Where

   f i (Pi )

   a i P 2

   i

   bi Pi

   1

   ci ,

   i 1,2,3, …, N G

   i

   (2)

   correspond to the respective function. The main

   purpose of the optimization is to prevent hemming about to local minima . The originality of the SA method lies in the application of a mechanis m that

   is the expression for cost function corresponding to ith generating unit and ai, bi and ci are its cost coefficients. Pi is the real power output (MW) of ith generator corresponding to time period t. NG is the number of online generating units to be dispatched.

   Th is constrained ELD proble m is subjected to a variety of constraints depending upon assumptions and practical imp lications. These include power balance constraints to take into account; these constraints are discussed as under

   2.1 Power Balance Constraints or Demand

   Constraints: This constraint is based on the principle of

   NG

   equilibriu m between total system generation ( Pi)

   i 1

   and total system loads (PD) and losses (PL). That is,

   NG

   guarantees the avoidance of local min ima.

   Following its introduction fro m [10], simulated annealing is main ly applied to large-scale combinatoria l optimizat ion proble ms.

   1. The Process of Annealing in Thermodynamics :

      At high temperature, the metal is in liquid stage. The molecules of liquidated metal move freely with respect to each other, via gradual cooling (thermodynamic process of annealing) thermal mobility is lost. The atoms start to get arranged and finally fo rm crystals, having the min imu m energy which depends on the cooling rate. If the temperature is reduced at a very fast rate, the crystalline state transforms to an amorphous structure, a meta-stable state that corresponds to a local min imu m of energy [11]. Annealing process of metal influences SA algorith m.

      If the system is at a therma l balance for g iven

      Pi PD PL

      i 1

      (3)

      temperature T, then the probability PT(s) that it has a configuration s depends on the energy of the corresponding configuration E(s), and is subject to the

      Where the transmission loss PL is e xpressed using B- coeffic ients [20] given by

      Bolt zmann distribution

      e E (s) / kT

      NG NG NG

      PL Pi Bij Pj

      B0i Pi

      B00

      PT (s)

      e E (w) / kT

      W

      (6)

      i 1 j 1 i 1

      (4)

      Where, k is the Boltzmann constant and the sum W

      includes all possible states W.

      Metropolises [12] we re the first to suggest a method for calcu lating a distribution of a system of ele mentary particles (mo lecules) at the thermal ba lance state.

      Let the system has a configuration g, which corresponds to energy E(g). When one of the molecules of the system is displaced from its starting position, a new state occurs which corresponds to energy E(). The new configuration is compared with the old one. If E() E(g) , then the new state is accepted. If E()>E(g), then the new state is accepted with probability :

      1. Starting Te mperature

      2. Final Te mperature

      3. Te mpe rature Decre ment

      4. Iterations at each Temperature

         1. Starting Te mperature

            The starting temperature must be set to a big enough value, in order to make possible a big probability of acceptance for non optimized solutions during the first stages of the algorithms application. However, if the value of the starting temperature gets too big, SA algorith m beco mes non-effective because of its slow convergence and in general, the optimizat ion process

            e ( E (

            ) E ( g ))

            kT

            (7)

            degenerates to a random wa lk.

            On the contrary, if the starting temperature is lo w

            Where, k is the Boltzmann constant.

            Table 1: Connection between Thermodynamic and Co mbinatoria l Optimization

            Ther modynamics simulati on	Combinatorial Opti mization
            System state	Feasible So lutions
            Energy	Cost
            Change of state	Neighboring So lutions
            Temperature	Control Para meter
            Frozen state	Heuristic Solution

            The basic step of the simulated annealing algorith m is presented with the following Pseudo-code.

            • Get the initial solution S.

            • Get the initial Temp T>0

            • While not yet frozen

            (a)perform the following loop L times

            *peak the random neighbor, S of S

            *let =cost(S)-cost(S)

            *If 0 ,set S=S

            / T

            then there is a greater probability of achieving local minima. There is no particular method for finding the proper starting temperature that deals with the entire range of proble ms.

            Various methods for finding the appropriate starting temperature have been developed [13]. suggests to quickly raise the temperature o f the system initia lly up to the point where a certain percentage of the worst solutions is acceptable and after that point, a gradual decrement of te mperature is proposed.

         2. Final Te mper ature

            During the application of the SA algorith m it is common to let the temperature fall to zero degrees. However, if the decre ment of the temperature becomes e xponential, SA a lgorithm can be e xecuted for much longer time . Finally, the stopping criteria can either be a suitable low temperature or the point when the system is fro zen at current te mperature.

         3. Te mper ature Decre ment

            Since the starting and final te mperatures have been defined, it is necessary to find the way of transition fro m the starting to the final te mperature.

            The way of the temperature decrement is very important for the success of the algorithm [ 14] suggested the following way to decre ment the temperature:

   2. Control parameters of SA algorithm:

      T (t)

      d

      log(t)

      *If >0, set S=S with probability e

      (b)set T= T (reduce Temperature) Return S

      (8)

      For the successful application of the SA algorithm is the annealing schedule is vital, which refers to four control parameters that directly influence its

      Where d is a positive constant.

      An alternative is the geometric re lation:

      convergence (to an optimized solution) and consequently its efficiency [11]. The parameters are the following:

      T (t)

      a.t

      (9)

      Where parameter a, is a constant near 1. In effect, its typical values range between 0.8 and 0.99.

      1. Iterations at each Te mper ature

      For inc reased effic iency of the algorith m, the number of ite rations is very important. Using a certain number of iterations for each te mperature is the proper solution. [15] suggests the realizat ion of only one, iteration for each temperature, wh ile the temperature decrement should take place at a really slow pace that can be expressed as:

      The key para meters of algorith m a re In itia l temperature, Fina l te mperature, Cool Sched () and ma ximu m nu mber of generations which is used here as a stopping criteria to choose the best suitable values of key parameters. The setup of SA approach was the following: In itia l te mperature = 3000C, Final temperature =1e -100C, Cool Sched () = 0.8% and ma ximu m number of generations = 1000. In each case study, 10 independent runs were made for each of the optimization methods

      Each SA approach was imp le mented in MATLA B

      7.1 and all the programs were run on a 2.4 GHz

      T (t) t (1

      .t)

      (10)

      Pentium IV processor with 512 M B of RAM (Random Access Memory).

      Where, takes a very low value.

3. SA Algorithm Implementation of ELD Proble ms

   Step 1: Initia lization of te mperature, T, para meter and ma ximu m. Find, randomly, an initia l feasible solution, which is assigned as the current solution Si and perform ELD in order to calculate the total cost, Fcost, with the preconditions (4) and (6) fulfilled.

   Step 2: Set the iteration counter to =1

   Step 3: Find a neighboring solution Sj through a random perturbation of the counter one and calculate the new total cost, Fcost.

   Step 4: If the new solution is better, we accept it, if it is worse, we calcu late the deviation of cost

   S=S -S and generate a random number

   1. Case study I

      This case study consists of 38 generating units. All units are within the convex fuel cost characteristics for the above system is taken fro m [16]. In this case, the load demand expected to be determined is PD = 6000 MW. The B mat rix of the transmission loss coeffic ient is not considered in this system.

      Table 2 shows the minimu m, mean cost, standard deviations and CPU time per iteration, cost achieved by the SA approach. As indicated in Table 2, the SA was the approach that obtained the min imu m cost for the ELD of 38 generating units. The best result obtained for solution vector Pi, i = 1 . . . 38 by SA with minimu m cost of 9153496.59 $/h is given in Table 2 and Table 2 also compares the results obtained with the SPSO, PSO_ Cra zy, Ne w PSO and PSO_TVA C [17] this paper with those of other studies reported in the literature and the convergence behavior of Case study I is shown in Figure 1.

      x 106

      j i 9.5

      Total Operating Cost

      uniformly distributed over (0, 1).

      9.4

      PD=6000 MW

      If e

      S / t

      (0,1)

      (11)

      9.3

      Accept the new solution Sj to replace Si.

      Step 5: If the stopping criterion is not satisfied, reduce temperature using para meter :

      T (t) =. t and return back to Step 2.

4. Results and Discussion

   In this paper, to evaluate the effectiveness o f the proposed SA approach, two case studies (38 and 110 generating units) of ELD proble ms were applied in which the objective functions were conve x fuel cost characteristics in the power system operation.

   9.2

   9.1

   0 200 400 600 800 1000

   Generations

   Figure 1.Convergence characteristics of 38 unit system (PD=6000MW)

   Generator Power O/P(MW)	SPSO	PSO_ Crazy	Ne w PSO	PSO_ TYAC	SA
   Pg1	519.097	366.631	550	443.659	405.6512

   Table2: Co mparison of results for case study I

   Pg2	437.92	550	512.263	342.956	405.6512
   Pg3	374.789	467.129	485.733	433.117	408.6681
   Pg4	394.877	370.471	391.083	500	408.6681
   Pg5	356.603	425.712	433.846	410.539	408.6681
   Pg6	380.358	415.226	358.398	482.864	408.6681
   Pg7	300.234	339.872	415.729	409.483	408.6681
   Pg8	335.871	289.777	320.816	446.079	408.6681
   Pg9	238.171	195.965	115.347	119.566	114
   Pg10	218.563	170.608	204.422	137.274	114
   Pg11	196.63	138.984	114	138.933	114
   Pg12	234.5	262.35	249.197	155.401	117.804
   Pg13	111.529	114.008	118.886	121.719	110
   Pg14	100.731	92.393	102.802	90.924	90
   Pg15	122.464	89.044	89.039	97.941	82
   Pg16	125.31	130.555	120	128.106	325
   Pg17	155.981	167.85	156.562	189.108	157.0614
   Pg18	65	65.754	84.265	65	65
   Pg19	70.071	65	65.041	65	65
   Pg20	263.95	199.594	151.104	267.422	272
   Pg21	245.065	272	226.344	221.383	272
   Pg22	191.702	130.379	209.298	130.804	260
   Pg23	99.123	173.544	85.719	124.269	123.6755
   Pg24	15.058	13.263	10	11.535	10
   Pg25	60.06	112.161	60	77.103	107.5567
   Pg26	91.14	105.898	90.489	55.018	84.8289
   Pg27	41.006	35.995	39.67	75	35.3695
   Pg28	20.399	22.335	20	21.682	20
   Pg29	34.65	30.045	20.995	29.829	20
   Pg30	20.957	24.112	22.81	20.326	20
   Pg31	20.219	20.494	20	20	20
   Pg32	25.424	20.011	20.416	21.84	20
   Pg33	26.517	27.44	25	25.62	25
   Pg34	18.822	18	21.319	24.261	18
   Pg35	9.173	8.024	9.122	9.667	8
   Pg36	26.507	25	25.184	25	25
   Pg37	24.344	20	20	31.642	21.0081
   Pg38	27.181	24.371	25.104	29.935	20.3849

   Table 3: Best results comparison for case study I

   1. Case study II

      In this case study a large scale data consisting of 110 unit generating unit system is emp loyed, having convex fuel cost characteristics without including line losses

      .The input data of the entire system is taken from S. O. Orero paper [18]. In this case study, there are three load demands, Low (PD = 10000 MW), Mediu m (P D = 15000 MW) and High (PD = 20000 MW) e xpected to be determined.

      Tables 5 shows the Best Power Output of 110 unit system for PD=10000MW, PD=15000MW and PD=20000MW. As indicated in Table 4, the SA was the approach that obtained the min imu m cost for the ELD of 110 generating units.

      The best result obtained by SA with minimu m cost for PD=10000MW, PD=15000MW and PD=20000MW of 131973.9018 $/h, 198352.6413 $/h and 313184.2522

      $/h respectively is given in Table 4 and also compares

      the results obtained with the SAB, SAF [19] in this paper with those of other studies reported in the literature and the convergence behavior of Case study II is shown in Figure 2, and Table 6 shows Standard deviation and CPU time for different test cases.

      Table 4: Best results comparison with diffe rent approaches for diffe rent loads for case study II

      Table 5: Best power output of 110 unit system for various loads

      Generator Power O/P	Power Demand (MW )			Generator Power O/P	Power Demand (MW )
      	10000	15000	20000		10000	15000	20000
      Pg1	2.4018	2.4031	12	Pg56	25.2	25.2	96
      Pg2	2.4	2.4	12	Pg57	25.2	50.0387	96
      Pg3	2.4	2.4	12	Pg58	35	35	100
      Pg4	2.4	2.4	12	Pg59	35	35.003	100
      Pg5	2.4	2.4	12	Pg60	45	45	120
      Pg6	4	4	20	Pg61	45	45	120
      Pg7	4	4	20	Pg62	45	45	120
      Pg8	4	4	20	Pg63	54.3	164.7334	185
      Pg9	4	4	20	Pg64	54.3	184.4727	185
      Pg10	15.2	15.778	76	Pg65	54.3	177.8478	185
      Pg11	15.2	76	76	Pg66	54.3	185	185
      Pg12	15.2	46.249	76	Pg67	70	70	197
      Pg13	15.2	49.3024	76	Pg68	70	70	197
      Pg14	25	25	100	Pg69	70	70	197
      Pg15	25	25	100	Pg70	150	360	360
      Pg16	25	25	100	Pg71	400	400	400
      Pg17	109.546	155	155	Pg72	400	400	400
      Pg18	95.5364	155	155	Pg73	60	104.8277	300
      Pg19	110.3678	155	155	Pg74	50	146.4736	250
      Pg20	105.0814	155	155	Pg75	35.2032	90	90
      Pg21	68.9	68.9	197	Pg76	50	50	50
      Pg22	68.9	68.9	197	Pg77	160	160	450
      Pg23	68.9	68.9	197	Pg78	150	272.0709	600
      Pg24	350	350	350	Pg79	50	147.4288	200
      Pg25	400	400	400	Pg80	20	120	120
      Pg26	400	400	400	Pg81	10	10	55
      Pg27	140	500	500	Pg82	12	12	40
      Pg28	140.3855	500	500	Pg83	20	20.0228	80
      Pg29	50.0496	200	200	Pg84	50	200	200
      Pg30	35.3865	100	100	Pg85	83.392	325	325
      Pg31	10	10	50	Pg86	269.215	440	440
      Pg32	12.2522	20	20	Pg87	10	35	35
      Pg33	20	80	80	Pg88	20	20	55
      Pg34	75	250	250	Pg89	20	75.0219	100
      Pg35	196.4132	360	360	Pg90	40	220	220
      Pg36	219.4718	400	400	Pg91	30.0073	45.2315	140
      Pg37	14.7661	40	40	Pg92	40	77.5334	100
      Pg38	20	70	70	Pg93	440	440	440
      Pg39	25	100	100	Pg94	370.1674	500	500
      Pg40	20	120	120	Pg95	600	600	600
      Pg41	40	180	180	Pg96	305.6974	457.955	700
      Pg42	50	220	220	Pg97	3.6	3.6	15
      Pg43	440	440	440	Pg98	3.6	3.6	15
      Pg44	560	560	560	Pg99	4.4	4.4	22
      Pg45	660	660	660	Pg100	4.4	22	22
      Pg46	421.9594	594.5063	700	Pg101	10	10	60
      Pg47	5.4	5.4	32	Pg102	10	10	80
      Pg48	5.4	5.4	32	Pg103	20	20	100
      Pg49	8.4	8.4	52	Pg104	20	20	120
      Pg50	8.4	8.4	52	Pg105	40	40	150
      Pg51	8.4	8.4	52	Pg106	40	40	166.0789
      Pg52	12	12	60	Pg107	50	50	131.9211
      Pg53	12	12	60	Pg108	30	30	150
      Pg54	12	12	60	Pg109	40	40	320
      Pg55	12	12	60	Pg110	20	20	200

      x 105

      Total Operating Cost

      6

      4

      2

      0

      PD=10000 MW

      PD=15000 MW

      PD=20000 MW

      Acknowledge ment

      The authors are thankful to Director, Madhav Institu te of Technology & Science (MITS), Gwa lior (M.P) India for providing support and facilit ies to carry out this research work.

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         0 200 400 600 800 1000

         Generations

         Figure 2.Convergence characteristics of 110 unit system with various loads

         Table 6: Standard deviation and CPU time fo r diffe rent test cases

         TES T CAS ES	Test case I	Test case II
         Standard De viation ($/hr)	0.1200	1.36
         CPU time /ite ration(sec)	0.237	0.245

5. Conclusions

This paper presents the Simu lated Annealing (SA) approach for optimizat ion of Economic Load Dispatch (ELD) proble ms. Practical generator operation is modeled using with piecewise quadratic cost functions . Algorith ms have been developed for the determination of the global or near-global optima l solution for the ELD proble ms. The proposed SA approach has produced results comparable or better than those generated by other evolutionary algorithms and the solutions obtained have superior solution quality and good convergence characteristics and the strength of the method was demonstrated by the change in load demands of the problems. Because of in the deregulated environment where cost min imization not only the objective, but at the same time profit ma ximizat ion is also concern. Fast and accurate economic load dispatch solution is as usual requirement in deregulated scenario as well. Therefore, results show that SA based optimization is a pro mising technique for solving complicated and large ELD proble ms in electrica l power system.

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