A load economic dispatch based on ion motion optimization algorithm

Chapter 12 A Load Economic Dispatch Based on Ion Motion Optimization Algorithm Trong-The Nguyen, Mei-Jin Wang, Jeng-Shyang Pan, Thi-kien Dao and Truong-Giang Ngo Abstract This paper presents a new approach for dispatch generating powers of thermal plants based on ion motion optimization algorithm (IMA) Electrical power systems are determined by optimization in power balancing, transporting loss, and generating capacity The scheduling power generating units for stabilizing different dynamic responses of the control power system are mathematically modeled for the objective function Economic load dispatch (ELD) gains as the objective function is optimized by applying IMA In the experimental section, several cases of different units of thermal plants are used to test the performance of the proposed approach The preliminary results are compared with the other methods in the literature shows that the proposed plan offers higher effect performance Keywords Ion motion optimization · Electric power generating plant outputs · Economic load dispatch 12.1 Introduction Recently, electrical modes of renewable energy sources have increased rapidly [1] Fast rate load and fluctuation in the power grids needs stable balance Economic load dispatch (ELD) [2] refers to a scheduling method that rationally allocates the productive output of each generating unit to meet the constraints of power system T.-T Nguyen · M.-J Wang · J.-S Pan (B) Fujian Provincial Key Laboratory of Big Data Mining and Applications, Fujian University of Technology, Fuzhou 350118, Fujian, China e-mail: jengshyangpan@gmail.com T.-T Nguyen e-mail: vnthe@hpu.edu.vn M.-J Wang e-mail: meijinwang0608@gmail.com T.-T Nguyen · T Dao · T.-G Ngo Department of Information Technology, Haiphong Private University, Haiphong, Vietnam © Springer Nature Singapore Pte Ltd 2020 J.-S Pan et al (eds.), Advances in Intelligent Information Hiding and Multimedia Signal Processing, Smart Innovation, Systems and Technologies 157, https://doi.org/10.1007/978-981-13-9710-3_12 115 116 T.-T Nguyen et al operation [3] A common single-objective optimization problem is to minimize power generation costs and to maximize power generation efficiency Economic scheduling problems are the nonlinear problems that constrained optimization due to excessive dependence on initial values and gradient information Traditional methods such as Lagrangian method, linear programming [4, 5], and internal penalty function [6] are not suitable for solving nonlinear-like economic scheduling problems In response to the shortcomings of traditional methods, recent techniques like metaheuristic algorithms have been applied to solve the power system optimization problems successfully [7, 8, 9] The algorithms such as particle swarm optimization (PSO) [10], genetic algorithm (GA) [11], simulated annealing (SA) [12], and neural network method (ANN) [13] refer to the metaheuristic These methods have proven to be very effective in solving nonlinear and constrained problems [14] A recent metaheuristic algorithm, ion motion optimization algorithm (IMA) [15] that is inspired based on the essential characteristics of attracting and mutually exclusive anions and cations IMA’s structure is simple but effective, and easy to understand and to programming This paper takes advanced IAM to consider its factors for solving the optimization of the economic load scheduling problem of the power system 12.2 Related Work 12.2.1 Economic Load Dispatch Economic load dispatch (ELD) is a kind of economic scheduling problems that minimize the total power generation cost under the operating constraints of the power system It is a critical mathematical optimization problem in power systems ELD is a great significance for the economic and reliable operation of the power system [2] The main objective of the economic loading on generators is a minimum cost of producing and simultaneously met power demand (PD) under constraints of generator output limits, system losses, ramp rate limits, and prohibited operating zones The optimal dispatch is formulated as minimizing cost: m F= fi (Pi ) (12.1) i=1 According to the analysis the cost function of ith generating unit, fi (P) is a quadratic polynomial function that is described a: fi (Pi ) = + bi Pi + ci Pi2 $ /h (12.2) For different variables and notations are used, here, PD is power load demand Pi is active power deliver Pimin is the minimum real power output Pimax is maximum 12 A Load Economic Dispatch Based on Ion Motion … 117 power Pi0 is the previous real power output , bi and ci are coefficients of fuel cost ei , fi are coefficients of the power grid system with valve-point effects m is a number of the committed units Ploss denotes the transporting load loss Bij , B0i , B00 are Bmatrix coefficients for transmission power loss Uri and Dri are the up ramp limits and down ramp limits Pimax is lower limits of the prohibited zone for generating pzk unit PiU is upper limits of kth is the prohibited zone for ith generating unit Imax is a maximum number of iteration I is current iteration According to the equality constraints, total power generation m i=1 Pi is equivalent to load demand PD and total loss as the following equation: m (Pi ) = PD + PLoss (12.3) i=1 For the loss coefficient on transition load, the total loss may be derived as m m n PLoss = Pi Bij Pj + i=1 j=1 B0i Pi + B00 (12.4) i=1 The cost function of generation is also satisfied to below inequality constrained: Pimin ≤ Pi ≤ Pimax (12.5) It is desired to control the generation power of each committed online generator smoothly and should be within the generator limits But the ramp rate limit restricts the limit for controlling the operation of generation in two operating periods The ramp rate limit of ith generation unit is Max Pimin , Pi0 − Dri ≤ Pi ≤ Min Pimax , Pi0 + Uri (12.6) • (a) if generation increases, Pi − Pi0 ≤ Uri (12.7) Pi0 − Pi ≤ Dri (12.8) • (b) if generation decreases, The input–output characteristic of a generator is varied mainly when it has some valves in its steam turbine The ripples produced in the heat rate curve are the primary cause of valve point, and they are not expressed by a polynomial function 118 T.-T Nguyen et al 12.2.2 Ion Motion Optimization Algorithm The ion motion optimization algorithm (IMA) [15] simulated the motion of the ions with anion (negative ion charge) set and cation (positive ion charge) set These two sets are employed as ion candidate solutions in the operation process They perform different evolutionary strategies in the liquid phase and the solid phase and circulate between the liquid and the solid phase to achieve the purpose of optimizing the ions Ions in the IMA algorithm can move toward best opposite charges It means that anions move toward the best cation; on the other hand, cations move toward the best anion The movement of ions in this algorithm can guarantee the improvement of all ions throughout iterations Their movement power depends on the attraction/repulsion forces between them The amount of this force specifies the momentum of each atom The following steps represent the process of the algorithm Initialization An initial random population is randomly generated according to a uniform distribution within the lower and upper boundaries with D dimensions Liquid phase In the liquid phase, the anion group (A) and the cation group (C) updated according to the following patterns, respectively Ai,j = Ai,j + AFi,j × Cbestj − Aj (12.9) Ci,j = Ci,j + CFi,j × Abestj − Cj (12.10) where Cbest and Abest are cation and anion optimization, respectively Subscript i = 1, 2, 3, , NP/2, (NP/2 is the size of ions population), and j = 1, 2, 3, , D The optimal anion and cation are the anion and cation with the lowest fitness value in the entire anion group and the Cation group, respectively, for a minimization problem The resultant of anions attracted force AFi,j and CFi,j are mathematically modeled as follows: 1 + e−0.1/ADi,j CFi,j = + e−0.1/CDi,j AFi,j = (12.11) (12.12) where ADi,j and CDi,j are the distances of ith anion from the best cation, and cation from the best anion in dimension, respectively ADi,j = Ai,j − Cbestj , and CDi,j = Ci,j − Abestj Solid phase The ion is gradually gathered with iteration near the optimal ion by the gravitational force The solid phase was set for breaking the phenomenon of excessive concen- 12 A Load Economic Dispatch Based on Ion Motion … 119 tration, and also providing diversity for the algorithm in case of over-concentration of ions to make the algorithm fall into a local optimum The ion motion gradually slows down like the physical process as the iteration proceeds from the initial intense motion, and gradually, the liquid state ions will recrystallize into crystals The process of recrystallization was simulated in IMA was known as a solid phase Aj = Aj + ϕ1 × (Cbest − 1), if rand > 0.5 otherwise Aj + ϕ1 × (Cbest), (12.13) Cj = Cj + ϕ2 × (Abest − 1), if rand > 0.5 otherwise Cj + ϕ2 × (Abest), (12.14) Termination condition Completion of the solid phase evolution strategy to determine whether to achieve the termination conditions of the algorithm The termination conditions include the presupposition accuracy, the number of iterations, and so on If it is reached, the optimal ion is directly output; otherwise, the anions and cations are returned to the liquid phase from the solid phase and continue to be iterated In such a process, anions and cations are circulated in the liquid phase and solid stage, and the optimal solution is gradually obtained with iteration 12.3 Scheduling Load Power Optimization Based on IMA Search space optimization of the ELD includes both feasible and unfeasible scenarios that the main work is to identify the feasible points which produce close optimum results within the boundary framework It means the possible points have to satisfy all the constraints, while the unworkable aspects violate at least one of them As mentioned in the above section, the power system economic scheduling problems have multiple constraints, such as power balance constraints, operational constraints, slope limits, and prohibited operating space These constraints make the feasible domain space of the problem very complicated [3] Therefore, the solution or set of optimized points must necessarily be feasible, i.e., the points must satisfy all constraints So, it is essential to design a suitable objective function, which results in success of an optimization problem The performance indices utilize in the area of optimization purposes with high acceptance rate The objective function characterized by the given different execution conditions and constraints [3] To handle constraints, we use the penalty functions to deal with unfeasible points We attempt figuring out an unconstrained problem in the search space points by modifying the objective function in Eq (12.1) The formula function is as follows: Min f = if Pi ∈ F f (Pi ), f (Pi ) + penalty(Pi ), otherwise (12.15) 120 T.-T Nguyen et al Start Calculate The objectives Modelling Dispatch Space No Feasible Yes Ions mapping ELD model Search feasible points for Optimization No Success i< iterMax i=i+1 Yes Yes The global best Optimization Update local and global best End Fig 12.1 Flowchart of the proposed IMA for dispatch power generation (ELD) where F is optimum dispatch For dealing with constraints of prohibited zones, a binary variable is used for adding to objective formula as follows: Vj = 1, if Pj violates the prohibited zones otherwise (12.16) The nearest distance points in the possible areas measure the effort to refine the solution n Pi − PL − PD Fi (Pi ) + q1 Min f = i=1 n i=1 n + q2 Vj j=1 (12.17) 12 A Load Economic Dispatch Based on Ion Motion … 10 121 Case study with a six-unit system The mean value of fitness function 1.489 1.488 1.487 1.486 1.485 1.484 1.483 20 40 60 80 100 120 140 160 180 200 Iterations Fig 12.2 Comparison of the proposed IMA for dispatching load scheduling generators with FA, GA, and PSO approaches in the same condition Parameters of penalty factors and constants associated with the power balance are used to tune practically with values 1000 set to q1 , and the value one set to q2 in the simulation section The necessary steps of IMA optimization for scheduling power generation dispatch: Step Initialize the IMA population that associated model dispatch space Step Update the Anion group (A) and the Cation group (C) updated according to Eqs (12.10) and (12.11) as the patterns, respectively Step Calculate ions according to the fitness value of the function as Eq (12.17), figure current nearest solutions and then update the position as feasible archives Step If the termination condition met (e.g., max iterations), go to step 2, otherwise, terminate the process and produces the result (Fig 12.1) 12.4 Experimental Results To evaluate the performance of the proposed approach, we use the case study of six-unit and fifteen-unit systems to optimize the objective function in Eq (12.17) The outcome of the case testing for dispatch ELD is compared to other approaches, i.e., genetic algorithm(GA) [11], firefly algorithm (FA) [16], and particle swarm optimization (PSO) [10] Setting parameters for the approaches: population size N is set to 40, and the dimension of the solution space D is set to and 16 for the six-unit system he fifteen-unit system, respectively The max-iteration is set to 200, 122 T.-T Nguyen et al Table 12.1 Coefficients setting for a six-unit system Units γ $/MW2 0.0075 β $/MW α$ Pmin MW Pmax MW 7.60 250.0 110.0 500.0 0.0093 10.20 210.0 51.0 200.0 0.0091 8.50 210.0 82.0 300.0 0.0092 11.50 205.0 51.0 150.0 0.0082 10.50 210.0 51.0 150.0 0.0075 12.20 125.0 61.0 140.0 and number of runs is set to 15 The final obtained results averaged the outcomes from all runs The compared results for ELD are shown in Fig 12.2 A Case study of six units The features of a system with six thermal units are listed in Table 12.1 The power load demand is set to 1200 (MW) The coefficients as Eq (12.2) for a six-unit system in the operating normally with capacity base 100 MVA are given as follows: ⎡ ⎤ 0.15 0.17 0.14 0.19 0.26 0.22 ⎢ 0.17 0.60 0.13 0.16 0.15 0.20 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.15 0.13 0.65 0.17 0.24 0.19 ⎥ −3 Bij = 10 × ⎢ ⎥, ⎢ 0.19 0.16 0.17 0.71 0.30 0.25 ⎥ ⎢ ⎥ ⎣ 0.26 0.15 0.24 0.30 0.69 0.32 ⎦ 0.22 0.20 0.19 0.25 0.32 0.85 B0 = 10−3 [−0.390 − 0.129 0.714 0.059 0.216 − 0.663], B00 = 0.056, and PD = 1200 MW Table 12.2 shows the comparison results of the proposed approach with the FA, GA, and PSO approach The solution has six generator outputs, including P1–P6 The average results of the runs for generating power outputs, making total cost, total power loss load, and total computing times, respectively Figure 12.2 depicts the comparison of the proposed IMA for dispatch power generating outputs of a system six units with FA, GA, and PSO approaches in the same condition B Case study of 15 units The given coefficients for a system has 15 thermal units as its feature is listed in Table 12.3 The power load demand is set to 1700 (MW) The features of a system with 15 thermal units are listed in Table 12.3 There are 15 generator power outputs in each solution listed as P1 , P2 , …, P15 The dimension D of the search space equalizes to 15 12 A Load Economic Dispatch Based on Ion Motion … 123 Table 12.2 The best power outputs for six-generator systems Outputs FA GA PSO IMA P1 459.54 459.54 458.01 459.22 P2 166.62 166.62 178.51 171.57 P3 258.04 253.04 257.35 255.49 P4 117.43 117.43 120.15 119.83 P5 156.25 153.25 143.78 154.72 P6 85.89 85.89 76.76 73.77 Total power output (MW) 1239.76 1235.76 1234.55 1234.53 Total generation cost ($/h) 14891.00 14861.00 14860.00 14844.00 Power loss (MW) 37.76 35.76 34.56 34.54 Total CPU times (sec) 296 286 271 272 Table 12.3 Coefficients setting for a fifteen-unit system Units γ $/MW2 β $/MW α$ Pmin MW Pmax MW 0.00230 10.51 671.12 150.0 445.0 0.00185 10.61 574.82 155.0 465.0 0.00125 9.51 374.98 29.0 135.0 0.00113 8.52 37.50 25.0 130.0 0.00205 10.51 461.02 149.0 475.0 0.00134 10.01 631.12 139.0 460.0 0.00136 10.76 548.98 130.0 455.0 0.00134 11.34 228.21 65.0 300.0 0.00281 12.24 173.12 25.0 165.0 10 0.00220 10.72 174.97 24.0 169.0 11 0.00259 11.39 188.12 23.0 85.0 12 0.00451 8.91 232.01 22.0 85.0 13 0.00137 12.13 224.12 22.0 85.0 14 0.00293 12.33 310.12 25.0 61.0 15 0.00355 11.43 326.12 19.0 56.0 Bi0 = 10−3 × [−0.1 − 0.22.8 − 0.10.1 − 0.3 − 0.2 − 0.20.63.9 − 1.70.0 − 3.26.7 − 6.4]; B00 = 0.0055, PD = 1700 MW Table 12.4 depicts the comparison of the proposed approach with the other procedures, e.g., FA, GA, and PSO methods in the same condition for the optimization 124 T.-T Nguyen et al Table 12.4 The best power output for fifteen-generator systems Outputs FA [14] GA [13] PSO [15] IMA P1 455.21 455.01 455.01 455.01 P2 91.98 93.98 120.03 85.00 P3 90.06 85.06 84.85 84.83 P4 89.97 89.97 75.56 45.29 P5 156.00 150.00 162.94 152.00 P6 350.76 350.76 322.48 357.49 P7 226.36 226.36 165.70 242.22 P8 60.00 60.00 60.34 60.56 P9 52.37 52.37 91.84 29.60 P10 26.10 25.10 45.10 50.40 P11 25.96 25.96 42.70 30.60 P12 74.01 74.01 77.97 80.00 P13 61.99 66.99 45.38 66.27 P14 36.22 34.22 47.37 26.24 P15 52.05 51.05 55.00 55.00 Total power output (MW) 1846.81 1837.81 1828.27 1827.60 Total generation cost ($/h) 1241.09 1236.09 1235.61 1234.61 Power loss (MW) 147.84 137.84 129.27 127.60 Total CPU time (sec) 411 378 313 314 system with 15 generators The statistical results involved the generation cost, evaluation value, and average CPU time are summarized in the table Observed over Tables, the results of quality performance in terms of the cost, power loss and time consumption of the proposed method also produced better the other approaches The proposed IMA outperforms other methods The observed results of quality performance in terms of convergence speed and time consumption show that the proposed method of parallel optimization outperforms the other methods 12.5 Conclusion In this paper, we presented a new approach based on ion motion optimization algorithm (IMA) for dispatching power generators outputs Economic load dispatch (ELD) is optimized with different responses of the control system in balancing, transporting loss, and generating capacity The linear equality and inequality constraints were employed in modeling objective function The experimental section, 12 A Load Economic Dispatch Based on Ion Motion … 125 several cases of different units of thermal plants are used to test the performance of the proposed approach The preliminary results are compared with the other methods in the literature such as FA, GA, and PSO in the same condition that shows that the proposed approach provides better quality performance and runs less time than the other criteria References Tsai, C.-F., Dao, T.-K., Pan, T.-S., Nguyen, T.-T., Chang, J.-F.: Parallel bat algorithm applied to the economic load dispatch problem J Internet Technol 17 (2016) https://doi.org/10.6138/ JIT.2016.17.4.20141014c Al-Sumait, J.S., Sykulski, J.K., Al-Othman, A.K.: Solution of different types of economic load dispatch problems using a pattern search method Electr Power Compon Syst 36, 250–265 (2008) https://doi.org/10.1080/15325000701603892 Dao, T., Pan, T., Nguyen, T., Chu, S.: Evolved bat algorithm for solving the economic load dispatch problem In: Advances in Intelligent Systems and Computing, pp 109–119 (2015) https://doi.org/10.1007/978-3-319-12286-1_12 Vajda, S., Dantzig, G.B.: Linear programming and extensions Math Gaz (2007) https://doi org/10.2307/3612922 Nguyen, T.-T., Pan, J.-S., Chu, S.-C., Roddick, J.F., Dao, T.-K.: Optimization localization in wireless sensor network based on multi-objective firefly algorithm J Netw Intell 1, 130–138 (2016) Yeniay, Ö.: Penalty function methods for constrained optimization with genetic algorithms Math Comput Appl (2005) Soliman, S.A.-H., Mantawy, A.-A.H.: Modern Optimization Techniques with Applications in Electric Power Systems (2012) https://doi.org/10.1007/978-1-4614-1752-1 Nguyen, T.-T., Pan, J.-S., Wu, T.-Y., Dao, T.-K., Nguyen, T.-D.: Node coverage optimization strategy based on ions motion optimization J Netw Intell 4, 1–9 (2019) Xue, X., Ren, A.: An evolutionary algorithm based ontology alignment extracting technology J Netw Intell 2, 205–212 (2017) 10 Sun, J., Palade, V., Wu, X.J., Fang, W., Wang, Z.: Solving the power economic dispatch problem with generator constraints by random drift particle swarm optimization IEEE Trans Ind Informatics 10, 222–232 (2014) https://doi.org/10.1109/TII.2013.2267392 11 Chiang, C.L.: Improved genetic algorithm for power economic dispatch of units with valvepoint effects and multiple fuels IEEE Trans Power Syst 20, 1690–1699 (2005) https://doi org/10.1109/TPWRS.2005.857924 12 Suppapitnarm, A., Seffen, K.A., Parks, G.T., Clarkson, P.J.: Simulated annealing algorithm for multiobjective optimization Eng Optim (2000) https://doi.org/10.1080/03052150008940911 13 Du, K.L.: Clustering: a neural network approach Neural Netw (2010) https://doi.org/10.1016/ j.neunet.2009.08.007 14 Nanda, S.J., Panda, G.: A survey on nature inspired metaheuristic algorithms for partitional clustering (2014) https://doi.org/10.1016/j.swevo.2013.11.003 15 Javidya, B., Hatamloua, A., Mirjalili, S.: Ions motion algorithm for solving optimization problems Appl Soft Comput J 32, 72–79 (2015) http://dx.doi.org/10.1016/j.asoc.2015.03.035 16 Apostolopoulos, T., Vlachos, A.: Application of the firefly algorithm for solving the economic emissions load dispatch problem (2011) https://doi.org/10.1155/2011/523806 ... iteration near the optimal ion by the gravitational force The solid phase was set for breaking the phenomenon of excessive concen- 12 A Load Economic Dispatch Based on Ion Motion … 119 tration, and... optimization algorithm (IMA) [15] simulated the motion of the ions with anion (negative ion charge) set and cation (positive ion charge) set These two sets are employed as ion candidate solutions... metaheuristic algorithm, ion motion optimization algorithm (IMA) [15] that is inspired based on the essential characteristics of attracting and mutually exclusive anions and cations IMA’s structure