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Applied Soft Computing 13 (2013) 292–301 Contents lists available at SciVerse ScienceDirect Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc Augmented Lagrange Hopfield network initialized by quadratic programming for economic dispatch with piecewise quadratic cost functions and prohibited zones Vo Ngoc Dieu a,∗ , Peter Schegner b a b Department of Power Systems, Faculty of Electrical and Electronic Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam Institute of Electrical Power Systems and High Voltage Engineering, Faculty of Electrical and Computer Engineering, Technische Universität Dresden, 01069 Dresden, Germany a r t i c l e i n f o Article history: Received September 2011 Received in revised form July 2012 Accepted 11 August 2012 Available online 21 August 2012 Keywords: Augmented Lagrange Hopfield network Economic dispatch Piecewise quadratic fuel cost function Prohibited zones Quadratic programming a b s t r a c t This paper proposes a method based on quadratic programming (QP) and augmented Lagrange Hopfield network (ALHN) for solving economic dispatch (ED) problem with piecewise quadratic cost functions and prohibited zones The ALHN method is a continuous Hopfield neural network with its energy function based on augmented Lagrange function which can properly deal with constrained optimization problems In the proposed method, the QP method is firstly used to determine the fuel cost curve for each unit and initialize for the ALHN method, then a heuristic search is used for repairing prohibited zone violations, and the ALHN method is finally applied for solving the problem if any violations found The proposed method has been tested on different systems and the obtained results are compared to those from many other methods in the literature The result comparison has indicated that the proposed method has obtained better solution quality than many other methods Therefore, the proposed QP-ALHN method could be a favorable method for solving the ED problem with piecewise quadratic cost functions and prohibited zones © 2012 Elsevier B.V All rights reserved Introduction The operation cost in power systems needs to be minimized at each time satisfying operating constraints via economic dispatch (ED) problem In practical power system operation conditions, many thermal generating units, especially those units which are supplied with multiple fuel sources like coal, natural gas, and oil require that their fuel cost functions may be segmented as piecewise quadratic cost functions to represent for different types of fuel In addition, thermal generating units may have prohibited operating zones due to physical limitations on components of units Consequently, a unit with prohibited zones, its whole operating region will be broken into several isolated feasible sub-regions Therefore, the ED problem with piecewise quadratic cost functions and prohibited zones is to minimize the total fuel cost among the available fuels for each thermal unit subject to load demand, generation limits, and prohibited zone constraints This is a nonconvex and complicated optimization problem since it contains the discontinuous values at operating characteristics of generating units forming multiple local optima Therefore, the classical solution methods are difficult to deal with this type of problem One approach for solving the problem with such units having multiple fuel options is to linearize the segments and solve them by ∗ Corresponding author E-mail address: vndieu@gmail.com (V.N Dieu) 1568-4946/$ – see front matter © 2012 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.asoc.2012.08.026 traditional methods [1] A better approach is to retain the assumption of piecewise quadratic cost functions instead of linearized segments and proceed to solve them A hierarchical approach based on the numerical method (HNUM) has been proposed in [2] as one way to approach to the problem However, the major problem for the numerical methods is their exponentially growing time complexities for larger systems with non-convex constraints Since the first implementation for solving benchmark problem of Traveling Salesman [3], Hopfield neural network (HNN) has become very popular for solving constrained optimization problems The basic structure of the HNN is based on the highly connected networks of nonlinear analog neurons The advantage of the HNN is that it can solve complex optimization problems in fast manner since all of the neurons simultaneously and continuously change their analog state in parallel Moreover, the HNN also has simple technical implementation and properly handles the variables’ upper and lower limits using its sigmoid function However, the HNN only guarantees convergence to a local minimum of optimization problems Therefore, the HNN should be further improved for obtaining global optimal solution of practical problems The application of the HNN in [4] has created difficulties in handling some kinds of inequality constraints and dealing with large-scale problems with many constraints Moreover, the final solution of the HNN method is also sensitive to the choice of penalty factors associated with constraints in its energy function For solving the problem by the enhanced Lagrangian neural network (ELANN) [5] method, the dynamics of Lagrange multipliers associated with equality and inequality V.N Dieu, P Schegner / Applied Soft Computing 13 (2013) 292–301 Nomenclature aik , bik , cik cost coefficients for fuel cost function k of unit i Bij , B0i , B00 transmission loss formula coefficients N number of online generating units number of fuel cost functions for generating unit i ni mi number of prohibited zones of unit i PD total load demand of the system (MW) total network loss of the system (MW) PL Pi output power of unit i (MW) Pijl lower bound for prohibited zone j of generator i (MW) Piju upper bound for prohibited zone j of generator i (MW) Pi,min , Pi,max lower and upper generation limits of unit i (MW) Ui input of continuous neuron i corresponding to the output Vi U input of multiplier neuron corresponding to the output V Vi output of continuous neuron i representing for output power Pi V output of multiplier neuron representing Lagrangian multiplier penalty factor associated with power balance ˇ Lagrangian multiplier associated with power balance ˝ set of units with prohibited zones constraints were improved to guarantee its convergence to the optimal solutions and the momentum technique was also employed in its learning algorithm to achieve fast computational time However, both HNN and EALNN methods were involved a large number of iterations for convergence to an optimal solution and often showed oscillation during the transient process The adaptive Hopfield neural network (AHNN) proposed in [6,7] is an improvement of the HNN method by adjusting the slope and bias of neurons during the convergence process to speed up its performance With this improvement, the AHNN method can considerably reduce the number of iterations for obtaining a final solution compared to the conventional HNN Recently, heuristic optimization techniques have been applied for solving the problem such as genetic algorithm (GA) [8,9], evolutionary programming (EP) [10–12], differential evolution (DE) [13], and particle swarm optimization (PSO) [14] The GA method is critically dependent on the fitness function and sensitive to the mutation and crossover rates, the encoding scheme of its bits, and the gradient of the search space curve leading toward solutions With a parallel searching mechanism, the EP method has a high probability of finding an optimal solution However, for large-scale problems with complex constraints, a solution by this method may get trapped in a suboptimal state, where the variation operators cannot produce any offspring outperforming its parents These applications also involve a large number of iterations and are susceptible to the related control parameters [14] The DE method is a simple population based stochastic function minimizer which is very powerful for solving optimization problems This method makes few or no assumptions about the problem being optimized and has the ability to search in very large spaces of candidate solutions However, there is no guaranty for this method to always obtain global solution Moreover, the DE method becomes very slow when dealing with large-scale problems which may not be implementable for practical problems Recently, PSO has been widely applied in power system optimization problems Although 293 this technique can generate high-quality solutions within a shorter computational time and stable convergence characteristics compared to some other methods [15], it seems to be sensitive to the tuning of some weights or parameters Currently, this technique is still being improved to enhance its ability for solving complex problems In addition to the single methods, hybrid systems such as the hybrid of the EP, tabu search and quadratic programming (ETQ) [16], the hybrid real coded genetic algorithm (HRCGA) [17], and the hybrid integer coded differential evolution – dynamic programming (HICDEDP) [18] have been also proposed for solving the problem to overcome the difficulties suffered by single techniques and also obtain better solution quality However, the hybrid methods are usually slow due to the integration of single methods In this paper, a simple method based on quadratic programming (QP) and augmented Lagrange Hopfield network (ALHN) is proposed for solving ED problem with piecewise quadratic cost functions and prohibited zones The ALHN method is a continuous Hopfield neural network with its energy function based on augmented Lagrange function which can properly deal with constrained optimization problems There are three stages in implementation of the proposed method for solving the problem In the first stage, the equivalent fuel cost curve is established for each generating unit based on its different available cost curves and then the QP method is applied for obtaining optimal solution of the problem with the equivalent cost curves satisfying all constraints except prohibited zones In the second stage, based on the optimal solution obtained by QP method, the appropriate cost curve is determined for each unit and the ALHN method is applied for obtaining optimal solution In the last stage, a heuristic search is used for repairing the prohibited zone violations and the ALHN method is used again for obtaining optimal power dispatch if any violations found The proposed QP-ALHN method has been tested on various scale systems with different load demands and the obtained total costs and computational times are compared to those from other methods in the literature such as HNUM [2], HNN [4], ELANN [5], AHNN [6], adaptive real coded genetic algorithm (ARCGA) [8], conventional GA (CGA) and adaptive multiplier updating method (IGA AMUM) in [9], improved evolutionary programming (IEP) [11], EP [12], DE [13], modified particle swarm optimization (MPSO) [14], HRCGA [17], HICDEDP [18], and artificial immune system (AIS) [19] The remaining organization of this paper is as follows Section addresses the formulation of economic dispatch problem with piecewise quadratic cost functions and prohibited zones Proposed ALHN initialized by QP for solving the problem is described in Section Numerical results are followed in Section Finally, the conclusion is given Problem formulation The objective of the classic ED problem is to minimize the total cost of thermal generating units while satisfying various constraints including power balance and generator power limits In the ED problem with piecewise quadratic cost functions and prohibited zones, the generating units have different fuel types where each fuel type is represented by a piecewise quadratic function [4] and they have also prohibited zones in their whole operational region [20] Therefore, the objective of the ED problem with piecewise quadratic cost functions and prohibited zones is to find optimal solution for the problem based on appropriate selection of fuel type for each generating unit in its feasible operational regions so as their total cost is minimized while satisfying all constraints including power balance and generation limits The problem formulation in this paper is based on the quadratic cost function representing each fuel type of each generating unit from the mathematical model formulated by many researches which have been done in the 294 V.N Dieu, P Schegner / Applied Soft Computing 13 (2013) 292–301 literature Moreover, the problem considered here does not focus on how to exactly model the fuel cost function of units with exact cost coefficients Mathematically, the problem is formulated as follows: 3.1 Equivalent fuel cost coefficients N Min F = (1) Fi (Pi ) i=1 where the fuel types for each unit are represented by: Fi (Pi ) = convergence This proposed method is simple and effective for the problem with large number of generating units ⎧ ai1 + bi1 Pi + ci1 Pi2 , fuel 1, Pi,min ≤ Pi ≤ Pi1 ⎪ ⎪ ⎪ ⎨ ai2 + bi2 Pi + ci2 Pi , fuel 2, Pi1 < Pi ≤ Pi2 ⎪ ⎪ ⎪ ⎩ (2) The equivalent fuel cost coefficients for each unit can be calculated via composite generation production cost curve [21] or average value of all fuel cost curves In this paper, the equivalent fuel cost coefficients for each unit is based on the average value of their fuel cost curves The purpose of this approximate representation is to find the initial coefficients for generating units which will be further used for determination of fuel types The problem with the equivalent cost coefficients is reformulated as follows: aik + bik Pi + cik Pi2 , fuel k, Pik−1 < Pi ≤ Pi,max N subject to Fi,eq (Pi ) = ai,eq + bi,eq Pi + ci,eq Pi2 F = (1) Power balance constraint: The total real power output of generating units satisfies total real load demand plus power loss: (7) i=1 where the equivalent coefficients are calculated by: N Pi − PL − PD = (3) ai,eq = i=1 ni where the power loss PL in (3) is approximately calculated by Kron’s formula [21]: N N bi,eq = N PL = Pi Bij Pj + i=1 j=1 B0i Pi + B00 (4) i = 1, , N ⎧ l P ≤ Pi ≤ Pi1 ⎪ ⎨ i,min ⎪ ⎩ u Pij−1 ≤ Pi ≤ Pijl , u Pim i (8) bik (9) cik (10) ni k=1 j = 2, , mi ; ∀i ∈ ˝ ci,eq = ni ni k=1 (5) (3) Prohibited zones: For generating units with prohibited zones, their entire feasible operating zones are decomposed in feasible sub-regions and their feasible operating points should be in one of the sub-regions as follows: Pi ∈ aik k=1 i=1 (2) Generator operating limits: The real power output of generating units should be within their upper and lower bounds by: Pi,min ≤ Pi ≤ Pi,max ; ni ni (6) ≤ Pi ≤ Pi,max With this formulation, the ED problem with piecewise quadratic cost functions and prohibited zones has become a non-convex optimization problem with multiple minima For obtaining optimal solution for this problem, solution methods have to search for optimal solution in a very large search space, leading to time consuming Therefore, it is necessary to limit the search space of the problem to reduce computational time, especially for large-scale systems Implementation of QP-ALHN The applied methodology here is to convert the non-convex ED problem into a convex one so that it can be easily to be solved Firstly, an equivalent fuel cost curve for each unit is established based on their different cost curves and then QP method is applied for solving the newly formulated problem with all constraints except prohibited zones Secondly, the ALHN method is applied for obtaining the optimal solution for the problem with the fuel cost curve for each unit determined based on the optimal solution from QP Lastly, a heuristic search is used for handling prohibited zone violations and the ALHN is used for solving the final dispatch problem if any violations found In addition, the optimal solution by QP is used for initialization of neurons in the ALHN for a faster subject to constraints (3), (5), and (6) Although the equivalent coefficients not exactly reflect the fuel cost curves for generating units in the problem, it is used to approximately estimate the combined coefficients for units to reduce the search space of the problem Moreover, the precision of the solution by the proposed method is validated by comparing the obtained results to those from other methods in the literature 3.2 Fuel cost curve determination The newly formulated problem above can be easily to be solved by the QP method using optimization toolbox in Matlab [22] to obtain the initial operating point Based on the optimal solution obtained by the QP method, the appropriate fuel type will be appropriately selected for each unit as depicted in Fig For proper implementation of the average equivalent fuel cost coefficients in determination of fuel cost curves, a further modification is used for efficient fuel determination for the cases that the optimal solution by the QP method is in the vicinity of the border between two fuels In this paper, the lower fuel cost curve is selected if the optimal solution by the QP method belongs to the region of the upper fuel cost curve but the relative margin from the optimal solution to the border is less than 1% As shown in Fig 2, suppose that the initial operating point Pi0 by the QP method belongs to fuel cost curve 2, the margin Pi from the initial point to the border with fuel cost curve Pi1,max is determined by: Pi = Pi0 − Pi1,max (11) If the relative margin Pi /Pi0 is less than 1%, the fuel cost curve is selected This way can help to avoid selecting high fuel cost curves in sensitive cases, leading to cost savings V.N Dieu, P Schegner / Applied Soft Computing 13 (2013) 292–301 295 Fig Methodology for fuel cost curve selection 3.3 ALHN implementation After obtaining the appropriate fuel cost curve for each unit, the problem will become a classic ED problem with the objective as follows: N Fi (Pi ) = aik + bik Pi + cik Pi2 F = i=1 subject to constraints in (3), (5), and (6) (12) In this problem reformulation, the fuel cost curve k is fixed for each unit based on the optimal solution by the QP method as mentioned in Section 3.2 Therefore, this problem is easier to be solved than the original one To apply the ALHN method for solving the newly formulated problem, augmented Lagrange function is firstly formulated for the problem and then this function is directly used as the energy function for the continuous Hopfield neural network In the ALHN method, its energy function is simultaneously minimized with respect to the status change of the continuous neurons and maximized with respect to the status change of the multiplier neuron Moreover, the upper and lower limits of variables in the problem are easily handled by the sigmoid function of the continuous Hopfield network Therefore, the neural network will converge to the optimal solution in the search space of the problem via the dynamic change of the neurons The augmented Lagrange function L of the problem is formulated as follows: N N (aik + bik Pi + cik Pi2 ) + L= PD + PL − i=1 + ˇ Pi i=1 N PD + PL − Pi (13) i=1 Fig Modification for fuel cost curve selection To represent in ALHN, N continuous neurons representing power outputs of N generating units and one multiplier neuron representing the Lagrange multiplier are required The energy function E of the problem is formulated based on the augmented Lagrange 296 V.N Dieu, P Schegner / Applied Soft Computing 13 (2013) 292–301 The ALHN method will be used again for obtaining final optimal dispatch if there is any prohibited zone violations found Otherwise, the solution Pi∗ is the final optimal solution for the problem 3.5 Overall procedure The overall procedure of the proposed QP-ALHN method for solving the ED problem with piecewise quadratic cost functions and prohibited zones is as follows: Fig The relative positions between the initial solution and a prohibited zone function in terms of neurons as follows: N N (aik + bik Vi + cik Vi2 ) + V E = PD + PL − i=1 Vi i=1 + ˇ 2 N PD + PL − Vi i=1 N Vi + i=1 g −1 (V )dV (14) where the sums of integral terms are Hopfield terms where their global effect is a displacement of solutions toward the interior of the state space [23] The detailed steps of ALHN for solving the problem and its proof of convergence can be found in [24,25] Step 1: Determine equivalent cost coefficients for all generating units by using their average value of all fuel cost curves as in Section 3.1 Step 2: Apply the QP method to solve the problem with the determined equivalent cost coefficients in objective function using Matlab optimization toolbox Step 3: Determine appropriate fuel cost curve for each unit based on the optimal solution obtained by the QP method as in Section 3.2 Step 4: Apply ALHN which is initialized by the optimal solution from QP method to solve the problem with the determined fuel cost curve of each unit for obtaining optimal solution as in Section 3.3 Step 5: If the obtained solution by the ALHN method violates prohibited zones, the heuristic search in Section 3.4 is used for repairing the violations and the ALHN method is applied again for obtaining the final solution Otherwise, stop Numerical results 3.4 Prohibited zones violation repairing The optimal solution by the ALHN method for the problem with selected fuel cost curves may violate prohibited zones To properly handle the prohibited zone violations a heuristic search is used Suppose that the obtained optimal solution by the ALHN method for unit i is Pi∗ There may be three relative positions between Pi∗ and a prohibited zone in a selected fuel cost function of unit i as depicted in Fig • Case 1: There is no prohibited zone violation but the prohibited zone lies between Pi∗ and the maximum power output of the selected fuel function The newly maximum power output of the fuel cost function in this case is redefined by: new = Pijl Pik,max (15) • Case 2: The prohibited zone is violated To move P ∗ out of the i prohibited zone, a middle point Pijm of the prohibited zone is firstly determined as follows: Pijm = Piju + Pijl The proposed QP-ALHN is tested on the cases with and without prohibited zones For the case neglecting prohibited zones, the test systems include a 10-unit system with different load demands, large scale systems up to 2500 units based on the basic 10-unit system, and the IEEE 30-bus system For the case with prohibited zones, the test system includes 10 units In this case, two other mature metaheuristic search methods including differential evolution (DE) [26] and particle swarm optimization (PSO) [27] have been also implemented to solve the problem for the purpose of result comparison The algorithms of the QP-ALHN, DE, and PSO methods are coded on Matlab 7.8 platform and run on a 2.1 GHz PC with GB of RAM Unlike the metaheuristic search methods such as DE and PSO, the proposed ALHN needs only one run to obtain the optimal solution since it is not based on random factors and the final solution does not belong to its initializations Therefore, it is not necessary to run the proposed ALHN method many times since there is no difference among the best, worst, and mean values of the objective function while the standard deviation value is zero (16) 4.1 Neglecting prohibited zones The new limits of power output are adjusted as follows: new = Pijl , Pik,max if Pi∗ ≤ Pijm (17) if Pi∗ > Pijm (18) or new = Piju , Pik,min • Case 3: There is no prohibited zone violation but the prohibited zone lies between Pi∗ and the minimum power output of the selected fuel function The newly minimum power output of the fuel function in this case is redefined by: new Pik,min = Piju (19) 4.1.1 10-Unit system The test system consists of 10 generating units from [2] where each unit has two or three piecewise quadratic cost functions representing different fuel options The total demands are gradually changed from 2400 MW to 2700 MW with power loss neglected The total costs and average computational times obtained by the proposed QP-ALHN method are compared to those from HNUM [2], HNN [4], ELANN [5], AHNN [6], EP [12], ARCGA [8], IEP [11], DE [13], MPSO [14], real coded genetic algorithm (RCGA) [17], HRCGA [17], HICDEDP [18], and AIS [19] for various load demands of 2400 MW, 2500 MW, 2600 MW and 2700 MW as shown in Tables 1–4, respectively V.N Dieu, P Schegner / Applied Soft Computing 13 (2013) 292–301 Table Results comparison for 10-unit system with load demand of 2400 MW neglecting prohibited zones Method Total power (MW) Cost ($/h) CPU time (s) HNUM [2] HNN [4] ELANN [5] AHNN [6] EP [12] ARCGA [8] IEP [11] DE [13] MPSO [14] RCGA [17] HRCGA [17] HICDEDP [18] AIS [19] QP-ALHN 2401.2 2399.8 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 488.50 487.87 481.74 481.72 481.74 481.743 481.779 481.723 481.723 481.723 481.722 481.723 481.723 481.723 1.08 ∼60 11.53 ∼4 – 0.85 – 0.083 – 49.92 6.1 0.490 – 0.047 Table Results comparison for 10-unit system with load demand of 2500 MW neglecting prohibited zones Method Total power (MW) Cost ($/h) CPU time (s) HNUM [2] HNN [4] ELANN [5] AHNN [6] EP [12] ARCGA [8] IEP [11] DE [13] MPSO [14] RCGA [17] HRCGA [17] HICDEDP [18] AIS [19] QP-ALHN 2500.1 2499.8 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 526.70 526.13 526.27 526.230 526.25 526.259 526.304 526.239 526.239 526.239 526.238 526.239 526.240 526.239 – ∼60 12.25 ∼4 – 0.85 – 0.083 – 49.92 6.1 0.593 – 0.047 Table Results comparison for 10-unit system with load demand of 2600 MW neglecting prohibited zones Method Total power (MW) Cost ($/h) CPU time (s) HNUM [2] HNN [4] ELANN [5] AHNN [6] EP [12] ARCGA [8] IEP [11] DE [13] MPSO [14] RCGA [17] HRCGA [17] HICDEDP [18] AIS [19] QP-ALHN 2599.3 2599.8 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 574.03 574.26 574.41 574.37 574.75 574.405 574.473 574.381 574.381 574.396 574.380 574.381 574.381 574.381 – ∼60 9.99 ∼4 – 0.85 – 0.083 – 33.57 5.4 0.573 – 0.047 Table Results comparison for 10-unit system with load demand of 2700 MW neglecting prohibited zones Method Total power (MW) Cost ($/h) CPU time (s) HNUM [2] HNN [4] ELANN [5] AHNN [6] EP [12] ARCGA [8] IEP [11] DE [13] MPSO [14] RCGA [17] HRCGA [17] HICDEDP [18] AIS [19] QP-ALHN 2702.2 2699.7 2700 2700 2700 2700 2700 2700 2700 2700 2700 2700 2700 2700 625.18 626.12 623.88 626.24 626.26 623.828 623.851 623.809 623.809 623.809 623.809 623.809 623.809 623.809 – ∼60 21.36 ∼4 – 0.85 – 0.083 – 44.56 6.47 0.513 – 0.047 297 Table Result comparison for large-scale systems neglecting prohibited zones Method CGA [9] IGA AMUM [9] QP-ALHN No of units Total cost ($) CPU time (s) 30 60 100 30 60 100 30 60 100 1873.691 3748.761 6251.469 1872.047 3744.722 6242.787 1871.426 3742.855 6238.092 263.64 517.88 873.70 80.47 157.19 275.67 0.13 0.24 0.43 As observed from the tables, the proposed method obtains less total costs than HNUM and HNN for 2400 MW case and HNUM, HNN, AHNN, and EP for 2700 MW case The total costs of the proposed method are approximate to those from the others for the rest cases Note the power balance constraint in HNUM and HNN methods is not exactly satisfied for all cases For CPU time, it may not be directly comparable among the methods due to different computer processors and programming languages used However, a computational time comparison is to show the efficiency of the compared methods The computational times from the HNUM, HNN, AHNN, ELANN, HICDEDP, CIHBMO, DE, and ARCGA methods were from VAX 11/780, IBM PC-386, Compaq 90 MHz, 133 MHz Pentium PC, 2.8 GHz Pentium PC, 1.67 GHz with GB RAM PC, and 1.8 GHz CPU with 256 MB RAM PC, respectively The computer used for both RCGA and HRCGA methods was a Pentium III PC 500 MHz There is no CPU time reported for the EP, MPSO and AIS methods The CPU times from HNN, AHNN, DE, and ARCGA for all cases are about 60 s, s, 83.1 ms, and 0.85 s, respectively The chip frequency of the computer used for the proposed method is faster than that used by other methods such as HNUM, HNN, AHNN, ELANN, DE, and ARCGA but the computational times obtained by the proposed method are much faster In addition, the chip frequency of the computer used for the HICDEDP method is faster than the one used for the proposed QP-ALHN method but the computational time by this method are much slower Therefore, the proposed QP-ALHN is efficient for solving this problem in terms of total cost and computational time Note that the computational time from the proposed QP-ALHN method here is the total computational time from both QP and ALHN The optimal solutions by the QP-ALHN method for this test system are given in Table A1 The variation of energy function of the ALHN method for this test system with different load demands during the convergence process is given in Fig The figure shows the convergence behavior of the ALHN method for obtaining final solution 4.1.2 Large-scale systems The proposed QP-ALHN method is also tested on large-scale systems including 30, 60 and 100 units which are formed by duplicating the basic 10-unit system in Section 4.1 with the load demand of 2700 MW proportionally adjusted to the system size The total costs and computational times obtained by the proposed method are compared to those from conventional GA (CGA) and improved genetic algorithm with the adaptive multiplier updating method (IGA AMUM) in [9] as given in Table In all cases, the proposed QP-ALHN method can obtain less total costs than both the CGA and IGA AMUM methods with extremely faster computational times where the chip frequency of the computer used for the proposed method is around three times faster than that from the computer used for the CGA and IGA-AMUM method Note the computational 298 V.N Dieu, P Schegner / Applied Soft Computing 13 (2013) 292–301 Fig Variation of energy function of ALHN during convergence process for 10 unit system times of both CGA and IGA AMUM methods were from a PIII-700 PC In addition, for demonstration of its capability for solving very large-scale systems, the proposed method has also tested on systems up to 2500 units For obtaining the very large-scale systems in this case, the basic 10-unit system in Section 4.1 is duplicated with the load demand of 2700 MW proportionally adjusted to the system size similar to the case for large-scale systems up to 100 units above The obtained total costs and computational times for these systems are given in Table 6, in which the computational times for QP and ALHN are separated to show the efficiency of ALHN method The computational time from the QP method for the initial problem sharply increases while that of the ALHN method for the final problem slightly increases with the system size Obviously, the computational times from ALHN method are not considerable compared to those from QP method for different systems even though they solve the problem with the same sizes Therefore, the ALHN is very efficient for dealing with very large-scale systems 4.1.3 IEEE 30-bus system The IEEE 30-bus system with generating units is also used for testing the proposed method The system data and unit data are given in [28,7], respectively In this case, the total power loss of the system is calculated via power flow by Newton–Raphson method using Matpower [29] The obtained results by the proposed QP-ALHN for different load demands of 220 MW, 283.4 MW, and 380 MW are compared to those from AHNN [7] as shown in Table Obviously, the proposed method can obtain better results than AHNN in terms of total costs and power losses for all load demands Moreover, the number of iterations from the proposed method for each case is also vastly less than that from the AHNN method For computational times, the proposed method is also faster than the AHNN method However, the computational times obtained from the AHNN method were from a Compaq 90 MHz PC which is about more than twenty times slower than the one used for the proposed method Therefore, the computational times are approximate for the two methods for this test system Table Results for very large-scale systems neglecting prohibited zones No of units 500 1000 1500 2000 2500 Total cost ($) 31,190.46 62,380.92 93,571.37 124,761.83 155,952.29 CPU time for QP (s) CPU time for ALHN (s) Total CPU time (s) 9.406 60.047 171.859 374.359 674.688 0.266 0.578 0.969 1.422 1.875 9.672 60.625 172.828 375.781 676.563 Table Result comparison for the IEEE 30-bus system neglecting prohibited zones Load 220 MW Method AHNN [7] Total power (MW) Power loss (MW) No of iterations Total cost ($/h) CPU time (s) 227.086 7.086 3200 589.647 25 283.4 MW QP-ALHN 226.326 6.326 57 584.474 0.438 AHNN [7] 295.252 11.852 4000 810.278 30 380 MW QP-ALHN AHNN [7] QP-ALHN 294.497 11.097 165 806.885 1.031 396.324 16.324 3000 1390.997 25 395.162 15.162 61 1348.263 0.484 V.N Dieu, P Schegner / Applied Soft Computing 13 (2013) 292–301 Table Prohibited zones of units for 10-unit system Table 10 Sensitivity analysis for 10-unit system Unit Zone Zone 10 [215 225] [200 220] [230 255] [270 295] [305 335] [260 335] [365 395] [380 400] Zone [420 450] [390 420] [430 455] – 4.2 Considering prohibited zones The test system is the 10-unit system with different load demands from Section 4.1.1, in which units 3, 5, 7, and 10 have prohibited zones as shown in Table The obtained total costs and computational times from the QP-ALHN method for this system with different load demands are compared to those from DE and PSO methods as in Table Among the obtained solutions for the different load demands, the solution for the load demand of 2500 MW is not affected by the prohibited zones Therefore, the total cost for this case is the same that from Case 4.1.1 For the other load demands, the obtained total costs are higher than those from Case 4.1.1 due to the effect of prohibited zones For the result comparison, the proposed QP-ALHN method obtains less total costs with faster computational times than both DE and PSO methods for all load demands It has indicated that the proposed method is more efficient than DE and PSO methods for the ED problem with piecewise quadratic cost functions and prohibited zones 4.3 Robustness evaluation The ED problem with piecewise quadratic cost functions and prohibited zones is a nonconvex problem with multiple minima and almost the implemented solution methods for this problem is based on artificial intelligence such as neural networks and metaheuristic search methods The metaheuristic search methods are usually based on a population searching for a best solution and their best solution is obtained among several executions In the result comparisons in Tables 1–4, 5, 7, and 9, the results from the proposed method are compared to those from neural networks and the best results from metaheuristic search methods in the literature In the proposed method, there include two methods those are QP and ALHN In fact, there only the ALHN method is the main method contributing to the efficiency of the proposed method since the QP method contributes only to the initial solution which does not affect on the final solution of the problem On the one hand, the ALHN method dominates the conventional Hopfield neural networks for obtaining better optimum solution, faster computational time, and more convenience in implementation [30] On the other hand, the ALHN method has many advantages compared to the population based methods as follows: Table Result comparison for 10-unit system with different load demands considering prohibited zones Load demand 2400 MW 2500 MW 2600 MW 2700 MW 299 Result DE PSO QP-ALHN Total cost ($/h) CPU time (s) Total cost ($/h) CPU time (s) Total cost ($/h) CPU time (s) Total cost ($/h) CPU time (s) 482.0683 1.673 526.4616 1.673 575.1903 1.683 624.6675 1.655 482.0510 0.339 526.4546 0.331 574.9327 0.325 624.4452 0.335 481.7266 0.172 526.2388 0.141 574.7291 0.181 624.3212 0.171 Parameter Initialization Random Middle point Maximum point Minimum point ˛pi = × 10−6 ˛pi − ˛pi = 2.5 × 10−6 ˛pi + ˛pi = 3.5 × 10−6 ˛ = × 10−4 ˛ − ˛ = × 10−5 ˛ + ˛ = 1.5 × 10−3 Total cost ($/h) Total cost deviation (%) CPU time (s) 623.8092 623.8092 623.8092 623.8092 0.00 0.00 0.00 0.00 0.0635 0.0545 0.0785 0.0785 623.8092 623.8092 0.00 0.00 0.0630 0.0550 623.8092 623.8092 0.00 0.00 0.0705 0.1720 • The ALHN method does not base on population, thus it needs less memory and time to find an optimal solution • The ALHN method does not depend on its initialization and so it needs only one run to obtain the final solution • The ALHN method is a recurrent neural network with parallel processing and it can properly handle constraints by Lagrange function and the sigmoid function of the conventional Hopfield neural network Therefore, it can efficiently deal with large-scale systems In addition, the ALHN method obtains only one final solution for a system even though different initializations and updating step sizes which affect only on computational time are used For a concise evaluation of robustness of the proposed method, the 10unit test system with load demand of 2700 MW in Section 4.1.1 is used In the ALHN method, two parameters are easily to be fixed in advance including the slope of sigmoid function and penalty factor ˇ, in which the slope of sigmoid function can be fixed at any value greater than and the penalty factor can be usually fixed at 10−3 for all cases Therefore, the values of updating step sizes including ˛pi associated with continuous neurons representing power outputs of units and ˛ associated with multiplier neuron representing Lagrange multiplier of power balance constraint will be tuned based on the selected slope of sigmoid function and penalty factor and characteristics of test systems Therefore, a sensitivity analysis of the ALHN method can be based on the analysis of the updating step sizes without loss of generality The set of parameters of the ALHN method is selected as follows: = 106 , ˇ = 10−3 , ˛pi = × 10−6 , and ˛ = × 10−4 The sensitivity analysis here is based on different initializations and variations of updating step sizes for neurons as shown in Table 10 For the initialization, the initial value of output of continuous neurons is initialized by different ways such as random initialization between maximum and minimum power outputs of units, initialization at the average value of maximum and minimum power outputs of units, initialization at the maximum power output of units, and initialization at minimum power output of units As observed from the table, the final total cost from the proposed method is the same for different initializations and updating step sizes; that is there is no deviation between the total cost obtained from the selected parameters and that obtained from the different sensitivity analysis in Table 10 However, the only one different thing among the analyzed cases is the computational time, where the different parameters will lead the ALHN method to different computational times to converge to the optimal solution In the mentioned tables above, the proposed method can obtain better optimal solution than many other methods with faster computational time Moreover, the proposed method is also stable for convergence to the optimal solution Therefore, 300 V.N Dieu, P Schegner / Applied Soft Computing 13 (2013) 292–301 these features have revealed the robustness of the proposed method Table A2 Solutions by QP-ALHN for IEEE 30-bus system with different load demands neglecting prohibited zones Unit Conclusion In this paper, a combined QP and ALHN method has been efficiently and effectively implemented for solving the ED problem with piecewise quadratic cost functions and prohibited zones The ALHN method is the continuous Hopfield neural network with its energy function based on augmented Lagrange function The proposed ALHN can properly handle the problem constraints by the augmented Lagrange function and the sigmoid function from the conventional continuous Hopfield neural network Moreover, the ALHN is a recurrent network with parallel processing which can obtain optimal solution for large-scale problems in a very fast manner For implementation of the proposed method to the nonconvex problem, the fuel cost curve of each unit is predetermined by using QP to solve the problem with equivalent average cost coefficients and then the ALHN method is applied for solving the problem with the determined fuel cost curve for each unit Then a heuristic search is used for repairing the prohibited zone violations if the optimal solution obtained by ALHN violates prohibited zones and the ALHN method is applied again for solving the final optimal dispatch problem The proposed QP-ALHN has been tested on various systems from 10 units to 2500 units and the IEEE 30-bus system with different load demands The results obtained by the proposed QPALHN method for these systems have been compared to those from many other methods in the literature The result comparison has indicated that the proposed method can obtain better total costs and faster computational times than many other methods for the test systems Therefore, the proposed QP-ALHN method can be a very favorable method for solving the nonconvex ED problem with piecewise quadratic cost functions and prohibited zones, especially for the large-scale systems Appendix A For the cases neglecting prohibited zones, the optimal solutions by QP-ALHN for 10-unit system with different load demands of 2400 MW, 2500 MW, 2600 MW, and 2700 MW are given in Table A1 and for the IEEE 30-bus system with different load demands of 220 MW, 283.4 MW, and 380 MW are given in Table A2 For the case considering prohibited zones, the optimal solutions for the 10-unit system by the QP-ALHN method are given in Table A3 Table A1 Solutions by QP-ALHN for 10 unit system with different load demands neglecting prohibited zones Unit 10 2400 MW 2500 MW 2600 MW 2700 MW Fuel Pi (MW) Fuel Pi (MW) Fuel Pi (MW) Fuel Pi (MW) 1 3 1 189.740 202.343 253.895 233.045 241.829 233.045 253.275 233.045 320.383 239.399 1 3 1 206.518 206.457 265.739 235.953 258.018 235.953 268.863 235.953 331.487 255.058 1 3 1 216.543 210.906 278.544 239.097 275.520 239.097 285.717 239.097 343.493 271.987 1 3 3 218.250 211.663 280.723 239.632 278.498 239.632 288.585 239.632 428.518 274.866 283.4 MW 380 MW Fuel 220 MW Pi (MW) Fuel Pi (MW) Fuel Pi (MW) 1 1 138.0222 35.0000 16.3038 10.0000 15.0000 12.0000 2 1 1 188.4058 47.9348 19.4736 11.6833 15.0000 12.0000 3 2 199.9999 76.5188 25.0000 30.0000 30.0000 33.6428 Table A3 Solutions by QP-ALHN for 10 unit system with different load demands considering prohibited zones Power output P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) P71 (MW) P8 (MW) P9 (MW) P10 (MW) Load demand 2400 MW 2500 MW 2600 MW 2700 MW 189.5592 202.2488 253.6250 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United approach to optimal real and reactive power dispatch, IEEE Transactions on Power Apparatus and Systems PAS-104 (1985) 1147–1153 [29] R.D Zimmerman, C.E Murillo-Sánchez, R.J Thomas, MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education, IEEE Transactions on Power Systems 26 (2011) 12–19 [30] V.N Dieu, P Schegner, Real power dispatch on large scale power systems by augmented Lagrange Hopfeld network, International Journal of Energy Optimization and Engineering (2012) 19–38 ... follows Section addresses the formulation of economic dispatch problem with piecewise quadratic cost functions and prohibited zones Proposed ALHN initialized by QP for solving the problem is described... (QP) and augmented Lagrange Hopfield network (ALHN) is proposed for solving ED problem with piecewise quadratic cost functions and prohibited zones The ALHN method is a continuous Hopfield neural network. .. problem with piecewise quadratic cost functions and prohibited zones, the generating units have different fuel types where each fuel type is represented by a piecewise quadratic function [4] and

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