2013 Ninth International Conference on Natural Computation (ICNC) 978-1-4673-4714-3/13/$31.00 ©2013 IEEE 517 Hierarchical Economic Load Dispatch Based on Chaotic-particle Swarm Optimization Yu Zhu, Qianjun Li, Yongxin Feng，Weiming Han Electric Power Research Institute Guangdong Power Grid Corporation Guangzhou, China Feilong Liu, Chaobing Han, Jianxin Zhou, Fengqi Si School of Energy and Environment Southeast University Nanjing, China Abstract—The economic load dispatch (ELD) optimization is an important approach to decrease the power consumption. In this paper, a network-plant hierarchical economic load dispatch mode is proposed based on the conventional load dispatch mode. The concept of plant coal consumption is proposed, as well as the method to obtain it. The model to achieve the least coal consumption in the full network depending on the plant coal consumption characteristic is also proposed. And it is theoretically proved that the optimization result of hierarchical ELD is consistent with that of conventional ELD. The chaotic particle swarm optimization (CPSO) algorithm is used to solve the optimal dispatch problem, adopting the adaptive inertia weight to accelerate the convergence speed. The hybrid optimization algorithm is improved from particle swarm optimization (PSO) algorithm by chaotic searching in the neighborhood to avoid getting into the local optimum, with the algorithm steps listed in the paper. A numerical example is done and analyzed, verifying the validity of the hierarchical optimization mode and CPSO. Keywords-economic load diapatch, hierarchical optimization, chaotic-partical swarm optimization, plant coal consumption characteristic I. INTRODUCTION Economic load dispatch (ELD) is one of the most important and fundamental optimization task in power system for allocating power generation among the committed units. The optimization problem has been much accounted by generating plants and power dispatch centers [1-6]. Over the past decades, extensive research on optimization models and algorithms for ELD problems has been done. In addition to the traditional methods such as equal incremental method, dynamic programming (DP), Lagrangian relaxation method (LR), artificial intelligence methods such as genetic algorithm (GA) [7-8], chaos optimization algorithm (COA) [9-10], particle swarm optimization (PSO) [11-14] have been successfully employed to solve the ELD problems. However, with the dramatic increase in the number of generating units, existing stochastic algorithms may result in the “curse of dimensionality”, and the difficulty of optimizing calculation redoubled as well as the consuming-time, which is out of accord with the relevant indicators in “ two rules ” of power grid [15]. Thus conventional load optimal dispatch methods cannot be immediately applied in large-scale load dispatch optimization problems. In-depth study on dispatch mode and correlation models are requisite, as well as optimization algorithms with high precision and rapid convergence. This paper provides a network-plant hierarchical optimal load dispatching mode, which divides load dispatch into two layers, i.e. network-plant (virtual plant) and plant (virtual plant)-unit. Moreover, the method to obtain the coal consumption characteristic equation coefficient of the power plant and the procedures of solving hierarchical optimal load dispatch problems are proposed. In Section III, chaos is incorporated into particle swarm optimization to construct the chaotic particle swarm optimization (CPSO), Adaptive inertia weight is obtained to effectively accelerate convergence velocity and chaotic searching behavior is added to avoid premature convergence and local optimum point in CPSO. Section V is devoted to the discussion on the result of a numerical example followed by conclusion in Section VI. II. HIERARCHICAL LOAD DISPATCH A. Conventional load dispatching model The objective of conventional ELD problem is to find the optimum coal consumption of unit i. The load of power network P is allocated to N units depending on the different characteristics of each unit. The optimization model mathematically can be described as (1). Niiii=1minF(P)f(P)=∑ (1) i,minii,maxNii1PPPs.t.PP=≤≤⎧⎪⎨=⎪⎩∑ where Pi is the power output of the i-th unit, MW; Pi,min, Pi,max are the minimum and maximum power output of the i-th unit, MW. The unit cost function fi(Pi) is usually expressed as a quadratic polynomial. 2iiiiiiif(P)aPbPc=++ (2) where ai, bi and ci are the cost coefficients of the i-th unit. 

518 B. Hierarchical load dispatch model Fig. 1 shows the blueprint of the hierarchical load dispatch mode. The dispatch center firstly allocates load for each plant (virtual plant) depending on the plant coal consumption characteristic equations, then each plant allocates load among the units depending on the unit coal consumption characteristic equations. It’s a pretty important part in hierarchical ELD to obtain the plant coal consumption characteristic, whose accuracy is related directly to the final optimal result. All the unit coal consumption characteristic equations of a plant known, the sum of all the units’ minimum output is the plant’s minimum output, as the sum of all the units’ maximum output is the plant’s maximum output. A set number of operating points are chosen evenly between the plant’s minimum and maximum output, and each operating point is computed according to (1) with multiple computations and the minimum coal consumption on each point is recorded. After that we process all the minimum coal consumptions using the least square method (LSM) [16]. Like the unit coal consumption characteristic equation, the plant coal consumption characteristic equation is also expressed as a quadratic polynomial. Obviously, the denser the operating points are, the more accurate the plant coal consumption characteristic equation is. Figure 1. Blueprint of Hierarchical ELD It is assumed that there is N units in the power network belonging to n plants, each plant has mi (i = 1, … , n) units. The hierarchical ELD optimization model mathematically can be described as (3). n2iiiiiii1minF(P)(APBPC)==++∑ (3) i,minii,maxnii1PPPs.t.PP=≤≤⎧⎪⎨=⎪⎩∑ where F(Pi) means the sum coal consumption of all plants; Pi is the output of i-th plant, MW; Ai, Bi, Ci are the cost coefficients of the i-th plant; Pi,min, Pi,max are the minimum and maximum output of the i-th plant, MW. For each plant, the plant coal consumption equals to the sum of all the units’ coal consumption, which can be expressed as (4). iim22iiiiiijijijijijj1(P)minfAPBPC(aPbPc)==++=++∑ (4) iij,minijij,maxmiijj1PPPs.t.PP=≤≤⎧⎪⎨=⎪⎩∑ where f(Pi) means the coal consumption of the i-th plant; Pij means the output of the j-th unit in i-th plant, MW; Pij,min, Pij,max are the minimum and maximum output of the j-th unit in i-th plant, MW; aij, bij and cij are the cost coefficients of the j-th unit in i-th plant. For the same N units, the coal consumption of conventional ELD F(Pij’) is expressed as (5). If F(Pij) = F(Pij’) is established, then the result of hierarchical ELD is equal to the result of conventional ELD. Since the conventional ELD can get the best result in theory, which means there is no other Pij’ that makes F(Pij’) smaller. imn''ijijijijijiji1j1'2F(P)min(aPbPc)===++∑∑ (5) For the i-th plant, the plant’s coal consumption at load Pi’ must be smallest one, expressed as (6); otherwise there exists another Pij’ that makes f(Pi’) smaller for the i-th plant, and for the whole network, there exists a smaller F(Pij’), which contradicts (5). iim''2'ijijijijijj1(P)minf(aPbPc)==++∑ (6) Because the plant’s coal consumption characteristic equation is obtained based on the units’ coal consumption characteristic equations, as long as the plant coal consumption characteristic equation is accurate enough, the plant’s coal consumption must be the smallest, expressed as (7). iim''2''2'iiiiiijijijijijj1(P)fAPBPCmin(aPbPc)==++=++∑ (7) Then the coal consumption of the whole network equals to the sum of all the plants’ coal consumption, expressed as (8). n'2'iiiiii=1'iF()min(APBPC)P=++∑ (8) (8) and (3) are the same in essence, which means the result of hierarchical ELD is the same as the result of conventional ELD. III. CHAOTIC PARTICLE SWARM OPTIMIZATION Inspired by the social behavior of organisms such as fish schooling and bird flocking, Eberhart and Kennedy first introduced particle swarm optimization (PSO) in 1995 [17]. In 

519 order to search for the global optimum, each “particle” which stands for a potential solution adjusts its positions according to its own experience, as well as the experience of neighboring particles, using the best position encountered by itself and its neighbors. The basic characteristics of a particle in the search space are its position and velocity, which can be updated by the following equations [17-19]. k+1k11best,kk22best,kkvwvcr(px)cr(gx)=+−+− (9) k+1kk+1xxv=+ (10) where vk represents the velocity vectors of a certain particle; xk represents the current position of the particle; pbest,k represents the best previous position of the particle searched by itself; gbest,k represents the best previous position among all particles in the population. r1 and r2 are two pseudo-random numbers with range [0, 1]; wrepresents the inertia weight that indicates the effects of the previous velocity of the particle on its current one. c1 and c2 are two positive numbers which are called acceleration coefficients. Generally, we can set w = 1, c1 = c2 = 2. PSO algorithm is simple and is easily employed to solve convex and non-convex optimization problems. However, the performance of the traditional PSO greatly depends on its parameters, take w in (11) as example, if we set w a little higher, it may help to step out the local optima, but a smaller w may help to accelerate the convergence of the algorithm. In order to overcome the shortcoming of fixed parameters, we adopt adaptive inertia weight factor as follows. maxminminminavgavgminmaxavg(ww)(ff)w,ffffww,ff−−⎧−≤⎪−=⎨⎪>⎩ （11） where wmin and wmax represents the minimum and maximum value of w, respectively. f represents the current objective value of the particle, favg and fmin are the average and minimum objective values of all particles, respectively. PSO algorithm is prone to premature during the solution process. In order to avoid this disadvantage, the chaotic particle swarm optimization (CPSO) method based on the logistic equation has been proposed [19]. When the particles are being trapped in local optima, chaos disturbance is employed to jump out of the local optimum. In this article, we use the following logistic equation in the process of chaotic local search. j,k+1j,kj,kj,0zz(1z),k0,1,2,...,0z1μ=−=≤≤ (12) where μ is called control variate, we set μ = 4 generally. zj,k denotes the j-th chaotic variable and k represents the generation number. Obviously, zj,k is distributed in the interval (0, 1.0) under the conditions that the initial zj,0∈(0, 1) and that zj,0∉{0.25, 0.5, 0.75}. In the process of chaotic local search, the position of the particle can be formulated as follows: *jjjj,kxxzη=+ (13) where xj* represents the best solution at the current time; ηj is called the adjustment coefficient; zj,k is called chaos variable with range[-1, 1]. At the beginning of the search process, we set ηj a little bigger which can helps to jump out of the local optimum, As the solution gradually close to the optimal value, the value of ηj should be gradually reduced. So ηj is self-adaptive to the process of iteration as follows: 2*jmaxmaxj[(kk1)/k]xηγ=−+ (14) where γ is called radius of the neighborhood, γ = 0.1; kmax represents the maximum number of iteration, k represents the current number of iterations. The procedures of CPSO based on the logistic equation can be illustrated as follows: Step 1. Initialize the position and velocity of the particles in the population randomly; Step 2. Calculate the fitness values of every particles and save the pbest,k and gbest,k; Step 3. Determine the position and velocity for the next iteration using equations (9) and (10), Updating the inertia weight using equation (11); Step 4. Calculate the objective values of all the particles and save some of the best solutions; Step 5. Execute chaotic local search using equations (12) ~(14). Update the pbest,k and gbest,k; Step 6. If the iterations number reaches the maximum, saving the best solutions. Otherwise, let k = k+1 and go back to Step 2. IV. HIERARCHICAL ECONOMIC LOAD DISPATCH BASED ON CHAOTIC-PARTICLE SWARM OPTIMIZATION The basic procedures of hierarchical optimization economic load dispatch based on chaotic-particle swarm optimization algorithm are as follows: Step 1. Preparation. EMS divides all the units into several virtual power plants according to some certain principles. The coal consumption characteristics of each virtual power plant can be obtained by using chaos particle swarm optimization and least squares methods. Step 2. Modeling. Modeling in the network side according to the virtual power plants coal consumption characteristics. Step 3. Network – plant. EMS allocates the load among the virtual power plants using chaos particle swarm optimization while some equality and in-equality constraints are satisfied. 

520 Library of coal consumption characteristics of virtual power plantsPDetermine virtual power plant and unit commitment Obtain the coal consumption characteristics of each virtual power plantCalculate load demand of virtual power plantsLoad dispatch inside virtual power plantsA new load demand？YESModeling Step 4. Plant – units. Calculate the output of each unit using CPSO according to the output of each virtual power plant. Step 5. Return. If load demand of power network changes, go back to Step 2. Figure 2. Flow Chart of Hierarchical ELD V. NUMERICAL EXAMPLE We take 40 units divided into five virtual power plants as an example, and the unit parameters are shown in TABLE 1 [20]. The calculated plant-level parameters are shown in TABLE 2. The optimized load dispatch can be realized on the MATLAB software platform. When the load demand is 10000MW, the number of particles in PSO algorithm and CPSO algorithm with the type of conventional optimization ELD is set to 500, and the same in CPSO of hierarchical optimization ELD. The iteration number is 1000, the chaos disturbance is added at the 700th time, and the iteration times for chaos is 100. Fig. 3 presents the convergence curves of the three kinds of optimization methods. Obviously, CPSO algorithm of Hierarchical optimization ELD has a faster speed to converge at a lower coal consumption. The three methods have run for many times and the results are shown in TABLE 3. In comparison with results of immediately ELD of PSO and CPSO algorithm, the latter strategy divides the large-scale optimization problems into several small-scale problems, and each problem is simpler than the former one. So the computational efficiency enhances as well as the optimized results are more best than the direct load dispatch. TABLE 1. COAL CONSUMPTION CHARACTERISTIC COEFFICIENT OF THE UNITS Unit number a ($·h-1·MW-2) b ($·h-1·MW-1) c ($·h-1) P_min (MW) P_max (MW) #1-1 0.0069 6.73 94.705 36 114 #1-2 0.0069 6.73 94.705 36 114 #1-3 0.02028 7.07 309.54 60 120 #1-4 0.00942 8.18 369.03 80 190 #2-1 0.0114 5.35 148.89 47 97 #2-2 0.01142 8.05 222.33 68 140 #2-3 0.00357 8.03 278.71 110 300 #2-4 0.00492 6.99 391.98 135 300 #2-5 0.00573 6.6 455.76 135 300 #2-6 0.00605 12.9 722.82 130 300 #2-7 0.00515 12.9 635.2 94 375 #3-1 0.00569 12.8 654.69 94 375 #3-2 0.00421 12.5 913.4 125 500 #3-3 0.00752 8.84 1760.4 125 500 #3-4 0.00708 9.15 1728.3 125 500 #3-5 0.00708 9.15 1728.3 125 500 #3-6 0.00313 7.97 647.85 220 500 #3-7 0.00313 7.95 649.69 220 500 #3-8 0.00313 7.97 647.83 242 550 #4-1 0.00313 7.97 647.81 242 550 #4-2 0.00298 6.63 785.96 254 550 #4-3 0.00298 6.63 785.96 254 550 #4-4 0.00284 6.66 794.53 254 550 #4-5 0.00284 6.66 794.53 254 550 #4-6 0.00277 7.1 801.32 254 550 #4-7 0.00277 7.1 801.32 254 550 #4-8 0.52124 3.33 1055.1 10 150 #4-9 0.52124 3.33 1055.1 10 150 #4-10 0.52124 3.33 1055.1 10 150 #4-11 0.0114 5.35 148.89 47 97 #5-1 0.0016 6.43 222.92 60 190 #5-2 0.0016 6.43 222.92 60 190 #5-3 0.0016 6.43 222.92 60 190 #5-4 0.0001 8.95 107.87 90 200 #5-5 0.0001 8.62 116.58 90 200 #5-6 0.0001 8.62 116.58 90 200 #5-7 0.0161 5.88 307.45 25 110 #5-8 0.0161 5.88 307.45 25 110 #5-9 0.0161 5.88 307.45 25 110 #5-10 0.00313 7.97 647.83 242 550 TABLE 2. COAL CONSUMPTION CHARACTERISTIC COEFFICIENT OF THE POWER PLANTS Plant number A ($·h-1·MW-2) B ($·h-1·MW-1) C ($·h-1) P_min (MW) P_max (MW) #1 0.0078 3.4301 1528.4 212 538 #2 0.0043 1.0888 6807.1 719 1812 #3 0.0011 7.5366 9811.2 1276 3925 #4 0.0105 -49.019 82585 1843 4397 #5 0.0015 4.289 4613.6 767 2050 

521 02004006008001000115000120000125000130000135000140000Iteration Total Fuel Cost ($/h) Hierarchical CPSO CPSO PSOFigure 3. Cost convergence characteristic of three load dispatch strategies TABLE 3. OPTIMIZATION RESULTS OF CONVENTIONAL ELD AND HIERARCHICAL ELD optimum indexes Conventional PSO Conventional CPSO Hierarchical CPSO Coal Consumption （t·h-1） 118532.1 120109.3 120021.9 117638.2 117795.4 121106.0 117409.4 117430.5 116503.3 115841.3 116501.8 120141.4 116941.7 116542.8 115873.2 115175.9 115193.7 118950.5 Average Cost（t·h-1） 119200.5 117304.6 116446.3 Average consuming time（s） 19.5 21.2 9.8 VI. CONCLUSION In this paper, we have successfully proposed a new mode to solve ELD problems called hierarchical ELD on the basis of conventional ELD mode. 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