A Gravitational Search Algorithm for Solving Economic Load Dispatch Problem

A Gravitational Search Algorithm for Solving Economic Load Dispatch Problem

S. Khandualo 1 , A. K Barisal,

Asst Manager (El) CHEP, Elect Engg Dept., Chiplima, OHPC Ltd, VSSUT,Burla, Dist-Sambalpur,Odisha,India Sambalpur,Odisha

1. K Pradhan, P. K. Patro

   Mechanical Engg Dept, Manager (El) , VSSUT,Burla, CHEP,Chiplima,OHPCLtd Dist-Sambalpur,Odisha,India, Dist-sambalpur,Odisha,

   AbstractThis paper presents the application of a new optimization algorithm i.e Gravitational Search Algorithm (GSA) based on the law of gravity and mass interactions for solving the Economic load dispatch problem of a power system. The Economic load dispatch is the dispatch of available electricity generation resources to supply the load and losses in the transmission links in such a manner that total cost of production of thermal generation is minimized satisfying the constraints in the system. The GSA technique is applied to a six generator twenty-six bus test system and a twenty generator test system to illustrate the effectiveness of the proposed algorithm. Numerical results show that the proposed algorithm is capable of finding very nearly global solutions and achieves cheaper generation schedule in comparison to the other published methods.

   Keywords Economic Dispatch, Gravitational Search Algorithm, Prohibited Operating Zones

   1. INTRODUCTION (Heading 1)

      Economic load dispatch has become a vital task for proper operation and planning of power system. The Main objective of Economic Load Dispatch problem is to minimize the total

      from [12] and Test system-II adopted from [4] and the simulation results are compared to that of Bio geography based optimization [11], Hopfield Modeling framework [4], and Lambda Iteration Method [ 4], Intelligent Water Drop Algorithms[12], etc The results show the superiority of the proposed algorithm in solving the complex optimization problem in terms of minimization of cost, minimization of loss and computational time.

   2. ECONOMIC LOAD DISPATCH PROBLEM FORMULATION

      The main objective of Economic load dispatch problem is to minimize the total generation cost by economic loading of generators such that the operational and network constraints are satisfied.

      Objective function is to minimize

      m

      FTotal Fi Pi …………………………………………………..(1)

      i1

      system operating cost represented by the fuel cost required for the system thermal generation while satisfying all units and system operational constraints i.e equality and inequality

      constraints such as load balance constraint, ramp rate limits,

      The cost function of ith unit Fi Pi

      polynomial and is expressed as

      is a quadratic

      multi fuel options, prohibited operation zones etc , . Conventional methods based on Lagrangian multiplier, gradient search techniques [1], Dynamic Programming, [1] require models of thermal plants to be represented as piecewise linear or polynomial approximations of monotonically increasing nature. But such an approximation may lead to suboptimal solution resulting in huge loss of revenue over the time. Methods based on Dynamic Programming, Lamda Iteration methods, Gradient Search methods [1] to solve the Economic Load Dispatch problems was found that it provides solution but it will fail to obtaining solution feasibility and become more complex. Stochastic search algorithms like Tabu Search[2], Genetic Algorithm [3,8,12], Evolutionary Programming [5], Particle Swarm Optimization [7] have been proved to be very exciting in solving complex power systems problems, but these heuristic methods do not always guarantee the globally optimal solution. In this paper a new population based search algorithm called Gravitational search algorithm [6,10] is applied to two different test systems ,Test System-I adopted

      F P a b P c P2 ……………………………….(2)

      i i i i i i i

      where ai , bi , and ci are fuel cost coefficients of ith unit and m is the total number of committed units.

      The solution to the problem must satisfy the operational and network constraints in the system.

      The constraints are given below:

      A. Power Balance Constraint

      m

      The total generation Pi

      i1

      should be equal to the total

      D. Prohibited Operating Zones

      Prohibited operating zone means the unit is prohibited from generation due to some technical fault in the machine such as vibration in the shaft bearing, or steam valve operation etc.

      system demand PD and total transmission loss Ploss ., i.e

      m

      Pi PD Ploss ………………………………… (3)

      i1

      The transmission loss is represented as

      With prohibited zones, the unit has a fuel cost curve of discontinuous in nature [12].

      The additional constraints for Units with prohibited operating zones are

      Pmin

      P Pl …………………………………………………..(7

      m m m

      i i i,1

      )

      PL Pi Bij Pj Pi Bio Boo …… . (4)

      i1 j

      i1

      Pu P Pl

      , j 2,3,…n

      Bij : The Transmission loss coefficient.

      i, j1 i

      i, j

      i …………………………….(8)

      Pl P Pmax …………………………………………………(9)

      The template is used to format your paper and style the text. All margins, column widths, line spaces, and text fonts are

      i,ni

      i i,

      i, j

      prescribed; please do not alter them. You may note

      Where j is the number of prohibited operating zones of

      peculiarities. For example, the head margin in this template measures proportionately more than is customary. This

      unit i , Pl

      is the lower limit of jth prohibited unit. And

      measurement and others are deliberate, using specifications

      u

      P

      i, j 1

      is the upper limit of j 1th prohibited operating zone

      that anticipate your paper as one part of the entire proceedings, and not as an independent document. Please do not revise any of the current designations.

      B. Generator perating limits

      The output generation of each unit must be within minimum and maximum limit of generation. The optimized result must satisfy the following inequality constraint i.e

      of ith unit. ni is the total number of prohibited operating zone of unit i .

   3. GRAVITATIONAL SEARCH ALGORITHM

      A. Abbreviations and Acronyms

      GSA is one of the recent additions to heuristic algorithms was developed by Rashedi et al. in 2009 [6]. GSA is followed by the physical law of gravity and the law of motion. In the

      Pi min

      Pi Pi max ……………………………………………………..(5)

      proposed algorithm, agents are considered as objects and their

      performance is measured by their masses. All these objects attract each other by the gravity force and this force causes a

      Pi is the power output of ith unit,

      Pi min and

      Pi max are the

      global movement of all objects towards the objects with

      minimum and maximum real power output of ith generating unit.

      The cost function Fi Pi becomes discontinuous when following factors are considered

      C. Valve point loadings

      heavier masses. In the proposed algorithm, agents are considered as objects and their performance is measured by their masses. All these objects attract each other by the gravity force and this force causes a global movement of all objects towards the objects with heavier masses.

      Consider a system f m masses, in which the position of the ith mass by

      P P ,…Pd …, Pn ,i 1,2,…m … .. . (10)

      i i i i

      Large steam turbine generators are having a number of steam admission valves that are opened in sequence to obtain ever

      Where

      Pid

      represents the position of ith mass in the dth

      increasing output of the unit. When the valve is first opened, the throttling losses increases rapidly and the incremental heat rate rises suddenly. This causes a ripple in the heat rate curves, the cost

      function is no longer a quadratic function but a combination of

      dimension. At a specific time t, a gravitational force on mass

      M pi t M aj t

      i from mass j is given by

      F d t Gt Pd t Pd t … (11)

      sinusoidal function and quadratic function i.e represented as ij

      / Rij t j i

      F P a

      b P c P2

      . i i

      i i i

      i i ……………………………… .(6)

      Where

      M pi

      is the passive gravitational mass related to agent

      ei sin fi Pi min Pi

      i , M aj is the active gravitational mass related to agent

      j . Gt is the Gravitational constant at time t, is a small constant and Rij t is Euclidian distance between two agents i and j .

      compared with other published work, result reveals the superiority of the proposed algorithm.

      1. Test System-I

         Rij t || Pi t, Pj t||2

         (12)

         The system contains six thermal units, 26buses, and 46

         The total force acting on the agent i in the dimension d is calculated as follows

         transmission lines [12]. The load demand is 1263MW. The load balance constraint, generation limit constraint and

         m

         d

         d

         prohibited operating zone constraints are considered in the

         Fi t

         rand j Fij t

         j1, ji

         (13)

         system. The input data for test system-1 are furnished in Table-I, Table-II and the B-Coefficients taken from [12].

         Where rand j is a random number in the interval [0,1]

         The acceleration of the agent i at time t , and in direction dth ,

         i i ii

         ad t F d t/ M t, . (14)

      2. Test System-II:

      The Test System-II consists of Twenty Thermal units and supplies a total load demand of P =2500MW, B coefficient

      M ii is the inertia mass of ith agent.

      Velocity of a particle is a function of its current velocity added to its current acceleration. Therefore the next position and next velocity of an agent can be calculated as

      D

      matrix is adopted from [4].

      Uniti	ai ($/h)	bi ($/h)	ci ($/h)	Pimin (MW)	Pimax (MW)
      1	240	7	0.007	100	500
      2	200	10	0.0095	50	200
      3	220	8.5	0.009	80	300
      4	200	11	0.009	50	150
      5	220	10.5	0.008	50	200
      6	120	12	0.0075	50	120

      TABLE I. INPUT DATA OF TEST SYSTEM-I

      vd t 1 rand Pvd t ad t

      … (15)

      i i i i

      i

      Pid t 1 Pid t vd t 1

      .. (16)

      The Gravitational constant G, is initialized at the beginning and will be decreased with time to control the search accuracy.

      i.e G is a function of the initial value G0 and time t .

      Gt G0eiteration/ max imumiteration , is a constant, iteration is the current iteration and maximum iteration is the maximum number of iterations given.

      The masses of the agents are calculated using fitness evaluation. A heavier mass means a more efficient agent. This means that better agents have higher attractions and moves

      Units	ai ($/h)	bi ($/h)	ci ($/h)	Pimin (MW)	Pimax (MW)
      1	1000	18.19	0.00068	150	600
      2	970	19.26	0.00071	50	200
      3	600	19.8	0.0065	50	200
      4	700	19.1	0.005	50	200
      5	420	18.1	0.00738	50	160
      6	360	19.26	0.00612	20	100
      7	490	17.14	0.0079	25	125
      8	660	18.92	0.00813	50	150
      9	765	18.27	0.00522	50	200
      10	770	18.92	0.00573	30	150
      11	800	16.69	0.0048	100	300
      12	970	16.76	0.0031	150	500
      13	900	17.36	0.0085	40	160
      14	700	18.7	0.00511	20	130
      15	450	18.7	0.00398	25	185
      16	370	14.26	0.0712	20	80
      17	480	19.14	0.0089	30	85
      18	680	18.92	0.00713	30	120
      19	700	18.47	0.00622	40	120
      20	850	19.79	0.00773	30	100

      more slowly.

      TABLE II. INPUT DATA OF TEST SYSTEM-II

      fit i t is the fitness value of agent i at time t , bestt and

      worstt represents the strongest and weakness agent according to their fitness value.

      For a minimization problem, bestt min j 1,…m fit j t

      and worstt max fit t

      (17)

      j 1,…m j

      The gravitational and inertial masses are updating by the following equations

      mi t fiti t worstt/ bestt worstt

      (18)

      m

      M i t

      mi t / m j t

      j1

   4. RESULTS AND DISCUSSION

      (19)

      The proposed gravitational search algorithm was tested on two test systems having six units with 26 bus test system and twenty unit test systems. The GSA was programmed in MATLAB environment and executed on a 2.30GHz Pentium- III processor with 4GB RAM. The simulation results were

      TABLE III. COMPARATIVE RESULTS OF TEST SYSTEM-I

      Units (MW)	GA	PSO [13 ]	BBO [11]	IWD [12 ]	Proposed GSA
      1	474.80	447.497	447.399	450.13	462.6177
      2	178.63	173.322	173.239	173.62	184.9532
      3	262.20	263.474	263.316	260.61	256.6655
      4	134.28	139.059	138.000	139.49	133.0794
      5	151.90	165.476	165.410	159.7	152.3787
      6	74.181	87.128	87.0797	90.51	85.3885
      Total Gen	1276.0	1276.01	1275.44	1274.05	1275.083
      Ploss	13.021	12.9584	12.446	12.05	12.083
      Total Gen Cost ( $/h)	15459	15450	15446.0	15439	15373.83
      CPU time per iteration (s)	0.22	0.06	0.0638	0.0254	0.00807

      Fig.1.Convergence characteristics of GSA on Test System-I

      Fig.2: Convergence characteristics of GSA on Test System-II

      TABLE IV. OPTIMAL SOLUTION OF TEST SYSTEM-II

      Unit Generation (MW)	Lambda iteration method	Hopfield Model [4]	BBO [11]	Proposed GSA
      1	512.780	512.7804	513.089	570.7216
      2	169.103	169.1035	173.353	174.6018
      3	126.889	126.8897	126.923	98.8116
      4	102.865	102.8656	103.329	87.7545
      5	113.683	113.6836	113.774	123.2827
      6	73.571	73.5709	73.0669	71.0592
      7	115.287	115.2876	114.984	99.7516
      8	116.399	116.3994	116.423	98
      9	100.406	100.4063	100.694	122.1304
      10	106.026	106.0267	99.9997	106.9398
      11	150.239	150.2395	148.977	154.3317
      12	292.764	292.7647	294.020	298.2807
      13	119.115	119.1155	119.575	120.3959
      14	30.834	30.8342	30.5478	29.8224
      15	115.805	115.8056	116.454	115.6996
      16	36.2545	36.2545	36.2278	37
      17	66.859	66.859	66.8594	61.7785
      18	87.972	87.972	88.5470	85.0294
      19	100.803	100.8033	100.980	84.2168
      20	54.305	54.305	54.2725	51.4542
      Total Gen	2591.96	2591.967	2592.10	2591.062
      Ploss	91.967	91.9669	92.101	91.0624
      Total Gen Cost( $/h)	62456.6	62456.63	62456.7	62314.94
      CPU time in Sec	33.757	6.355	0.02928	2.525

      The convergence characteristics of the proposed method on Test system-I and Test system-II are provided in Figs. 1 and 2, respectively. The results obtained by the proposed GSA method on Test System-1 are compared with genetic algorithm (GA), particle swarm optimization (PSO), biogeography based optimization (BBO) and intelligent water drop algorithm (IWD) methods in Table III. The GSA method provides cheaper generation schedule in comparison to all other above methods in less execution time. Similarly in Table IV, the proposed GSA outperforms to other established methods in terms of quality of solution and execution time. Moreover, GSA method performs better to other methods reported in literature in both of the test systems considered for study.

   5. CONCLUSION

A new algorithm called Gravitational Search Algorithm was developed and demonstrated on two test systems to solve the economic load dispatch of two different test systems. Results show that GSA based algorithms are more capable of finding highly near-global solutions than lamda iteration method, Hopfield model, BBO etc. The optimal cost obtained by the GSA is quite cheaper than the other published work for the system adopted. In future, attempts can be made to apply the hybrid gravitational search algorithm to large thermal system in conjunction with hydro, wind energy by incorporating emission, spinning reserve and reliability constraints.

REFERENCES

1. A.J Wood, B.F Wollenberg B.F (1984) Power Generation Operation and Control, New York, Wiley.

2. W.M Lin,F.S Cheng, M.T Tsay,An improved Tabu Search for economic dispatch with multiple minima,IEEE Trans Power Syst 2002,17,pp. 108-112.

3. D.C Walters, G.B Sheble,Genetic Algorithm-Economic Dispatch Example,IEEE Trans Power Systems 1993,Vol 8, pp. 1325-32.

4. C.Tzong Su,C.T.Lin New Approach with a Hopfield Modeling Framework to Economic Dispatch IEEE Transactions on Power Systems, Vol.15, No. 2, May 2000, PP 541-545.

5. N.Sinha,R.Chakrabarti,P.K.Chattopadhyay Evolutionary Programming Techniques for Economic Load Dispatch IEEE Trans Evol Comput 2003,Vol 7, No. 1, pp. 83-94.

6. Rashedi E, Nezamabadi pour H, Saryazdi S (2009) GSA: A Gravitational Search Algorithm. Information Science, Vol-179, pp. 2232-2248.

7. Z.L.Gaing, Particle Swarm Optimization to solving the Economic Load Dispatch Considering the Generator Constraints IEEE Transactions on Power Systems,Vol.18, No.3,Aug2003, pp. 1187-1195.

8. R.Balamurugan and S.Subramanian An improved Dynamic Programming Approach to Economic Power Dispatch with Generator Constraints and Transmission losses Journal of Electrical Engineering & Technology,Vol. 3,Issue 3, 2008, pp. 320-330.

9. S.C.Swain,S.Panda,A.K.Mohanty etc Application of Computational Intelligence Techniques for Economic Load Dispatch International journal of Electrical Power and Energy Systems Engineering,Vol.3,Iss- 3, 2010, pp. 187-193.

10. S. Dugman, U. Guvenc, Yorukeren (2010) Gravitational Search Algorithm for Economic dispatch with valve-point effects. International review of Electrical Engineering, Vol. 5, No. 6, pp. 2890-289.

11. A.Bhattacharya, P.K.Chattopadhyay. Soling Complex Economic Load Dispatch Problems using Biogeographic-based Optimization Expert Systems with Application,Vol.37, 2010, pp. 3605-3615.

12. S.R.Rayapudi, An Intelligent Water Drop Algorithm for solving Economic Load Dispatch Problem International journal of Electrical and Electronics Engineering, Vol.5, No.1, 2011, pp. 43-49.

13. P.Jain, K.K.Swarnkar, S. Wadhwani, Prohibited Operating Zones Constraint with Economic Load Dispatch using Genetic Algorithm,International journal of Engineering and Innovative Technology,Vol.1,No.3, Mar-2012, pp,. 179-183.

14. Subramanian,K.Thanushkodi.and,P.N.Neelakantan,A Novel TANANs Algorithm to solve Economic Power Dispatch with Generator Constraints and Transmission losses International Journal of Engineering Research & Technology,Vol.2,Issue 8,Aug 2013, pp. 918- 923.