A hybrid genetic algorithm and bacterial foraging approach for dynamic economic dispatch problem

Dynamic economic dispatch (DED) is one of the most significant non-linear complicated problems showing non-convex characteristic in power systems. This is due to the effect of valve-points in the generating units’ cost functions, the ramp-rate limits and transmission losses. Hence, proposing an effective solution method for this optimization problem is of great interest. The original bacterial foraging (BF) optimization algorithm suffers from poor convergence characteristics for larger constrained problems. To overcome this drawback, a hybrid genetic algorithm and bacterial foraging (HGABF) approach is presented in this paper to solve the dynamic economic dispatch problem considering valve-point effects, ramp-rate limits and transmission losses. The HGABF approach can be derived by integrating BF algorithm and genetic algorithm (GA), so that the BF’s drawback can be treated before employing it to solve the complex and high dimensioned search space of the DED problem. To illustrate the effectiveness of the HGABF approach, several test systems with different numbers of generating units are used. The results of HGABF approach are compared with those obtained by other published methods employing same test systems. These results show the effectiveness and the superiority of the introduced method over other published methods.     

一种基于混沌序列的动态差分进化算法在电力系统动态经济调度中的应用

针对电力系统动态经济调度(DED)问题,引入差分进化算法,提出一种基于混沌序列的动态差分进化算法(ＡＤＤＥＣＳ)。该算法采用混沌序列动态调整差分进化算法的参数设置,保持种群的多样性。动态搜索策略被用于提高算法的整体搜索性能,它由全局搜索策略和局部搜索策略2部分组成。为了加速收敛和解决DED复杂的约束处理问题,采用基于多目标概念的约束处理机制,并提出一种根据机组调节能力来按比例分摊不可行解约束违反量的新方法。同时在搜索过程中,通过采用不同的变异策略结合改进的随机搜索策略来避免算法早熟,增强全局最优解的搜索能力。提出的方法的可行性和有效性由10机测试系统来证明,和其他方法相比,ADDECS方法计算速度快,计算精度高且鲁棒性强。     

Dynamic Economic Dispatch with Wind Power Penetration Based on Non-Parametric Kernel Density Estimation

Abstract

In order to analyze the randomness of wind power in dynamic economic dispatch (DED) with wind power, based on non-parametric kernel density estimation (KDE) technology, the probability distribution of wind power output and wind power forecast error is accurately modeled. A segmented statistical method on wind power forecast data is adopted to construct the confidence interval of the wind power output, the upper and lower bounds of the forecast errors. According to the established wind power output probability model, forecast confidence interval and forecast error upper and lower bounds, a DED model with wind power is formulated in this paper. A hybrid algorithm combining the evolutionary advantages of bat algorithm (BA) and particle swarm optimization (PSO) algorithm is designed to solve the proposed model. A crossover mechanism, which can solve the problem of falling into local optimum easily existed in BA and PSO, is introduced in the evolution of the algorithm. Finally, the effectiveness of the proposed model and algorithm is verified by simulation examples.     

Dynamic economic dispatch using harmony search algorithm with modified differential mutation operator

Dynamic economic dispatch (DED), which is a complex non-linear constrained optimization problem, has a pivotal role in power system operation. It is one of the prime functions of power generation and control, where the aim is to operate an electrical power system most economically while the system operation is within its security limits. This problem possess non-convex characteristic when generation unit valve-point effects are considered. This paper proposes to solve DED problem with valve-point effects, using a modified form of recently developed differential harmony search algorithm. A five- and ten-unit system with non-smooth fuel cost function is used to establish the effectiveness of the proposed method over various other methods. It is shown that the proposed method is capable of providing better quality solutions.     

Control area performance improvement by extended securityconstrained economic dispatch

An algorithm for solving the extended security constrained economic dispatch (ESCED) problem with real-time economic dispatch grade speed and reliability is presented. The ESCED problem is formulated by adding a regulating margin and ramp rate constraints to the network security constrained economic dispatch problem previously solved by the CEDC algorithm. Starting with Newton's method to optimize the Lagrangian, the ESCED is developed by superimposing on Newton's method eight major components called tracking start initialization, hessian pre-elimination, implicit dual variable calculations, regulating margin sensitivity coefficient calculations, traumatic event evaluation, constraint relaxation, implicit ramp rate constraint implementation, and relaxed incremental cost calculations. Test results are also presented     

Reserve constrained multi-area economic dispatch employing differential evolution with time-varying mutation

For a power pool that involves several generation areas interconnected by tie-lines, the objective of economic dispatch (ED) is to determine the most economical generation dispatch strategy that could supply the area load demands without violating the tie-line capacity constraints. The objective of multi-area economic dispatch (MAED) is to determine the generation levels and the interchange power between areas which would minimize total fuel cost while satisfying power balance constraint, upper/lower generation limits, ramp rate limits, transmission constraints and other practical constraints. In reserve constrained MAED (RCMAED) problem inter-area reserve sharing can help in reducing the operational cost while ensuring that spinning reserve requirements in each area are satisfied. The tie-line limits too play a pivotal role in optimizing the cost of operation. The cost curves of modern generating units are discontinuous and non-convex which necessitates the use of powerful heuristic search based methods that are capable of locating global solutions effectively, with ease. This paper explores and compares the performance of various differential evolution (DE) strategies enhanced with time-varying mutation to solve the reserve constrained MAED (RCMAED) problem.The performance is tested on (i) two-area, four generating unit system, (ii) four area, 16-unit system and (iii) two-area, 40-unit system. The results are found to be superior compared to some recently published results.     

Chaotic self-adaptive differential harmony search algorithm based dynamic economic dispatch

Dynamic economic dispatch (DED) is one of the main optimization problems in electrical power system operation and control. DED problem is a non-smooth and non-convex problem when valve point effect, ramp-rate limits and prohibited operating zones of generation units are taken into account. This paper proposes an efficient chaotic self-adaptive differential harmony search (CSADHS) algorithm to solve the complicated DED problem in the presence of valve point effect, ramp-rate limits and prohibited operating zones constraints. In the proposed algorithm, chaotic self-adaptive differential mutation operator is used instead of pitch adjustment operator in the harmony search (HS) algorithm, to enhance the searching performance to find the quality solution. The effectiveness of the proposed algorithm is demonstrated on 10, 15 and 30 unit systems for a period of 24 h. The simulation results obtained by the proposed algorithm are compared with the results obtained, using differential harmony search (DHS) algorithm, chaotic differential harmony search (CDHS) algorithm, and also with the results of other methods available in the literature. In terms of solution quality, the proposed algorithm is found to be better than other algorithms and in terms of speed of convergence, standard deviation of generation cost, and computational time, the proposed algorithm is better than DHS and CDHS algorithm.     

经济调度中考虑温室气体排放约束的新方法

随着2005年2月16日《京都议定书》的正式生效,温室气体排放问题引起了电力工业界的广泛重视。《京都议定书》等新出台的温室气体排放政策本质上体现了配额-交易的排放控制策略,这种策略曾在美国被用来控制氮氧化合物(NOX)以及二氧化硫(SO2)等污染气体的排放。配额-交易排放策略下,温室气体排放不再受到刚性的排放上限约束,取而代之的是一定数量的可交易的排放配额限制。针对这一变化,本文提出了一种在配额-交易排放控制策略下经济调度中考虑温室气体排放的新方法。该方法通过引入排放影响因子将排放产生成本与其他发电成本 (例如燃料成本)一并计入到扩展的发电成本中,在寻求扩展发电成本最小的同时,自动确定系统排放配额的使用量,从而,在整体上确保了系统发电的经济性。文章采用Sorted table based lambda-iteration方法对所提出模型进行了求解,该方法可以同时处理发电成本曲线连续与分段连续的机组。算例分析通过对一个10机系统的测试,表明了本文方法的有效性。另外,文中还对系统运行的经济性与排放之间的折中关系进行了分析。     

Solving dynamic economic dispatch problem with security constraints using bacterial foraging PSO-DE algorithm

This paper presents a heuristic optimization methodology, namely, Bacterial foraging PSO-DE (BPSO-DE) algorithm by integrating Bacterial Foraging Optimization Algorithm (BFOA), Particle Swarm Optimization (PSO) and Differential Evolution (DE) for solving non-smooth non-convex Dynamic Economic Dispatch (DED) problem. The DED problem exhibits non-smooth, non-convex nature due to valve-point loading effects, ramp rate limits, spinning reserve capacity, prohibited operating zones and security constraints. The proposed hybrid method eliminates the problem of stagnation of solution with the incorporated PSO and DE operators in original bacterial foraging algorithm. It achieves global cost by selecting the bacterium with good foraging strategies. The bacteria with good foraging strategies are obtained in the updating process of every chemo-tactic step by the PSO operator. The DE operator fine tunes the solution obtained through bacterial foraging and PSO operator. A 3- and 7-unit systems for static economic dispatch, a 26-bus, 6-generator test system and an IEEE 39-bus, 10-unit New England test systems are considered to show the effectiveness of the proposed method over other methods reported in the literature.     

Differential evolution algorithm for emission constrained economic power dispatch problem

In this paper, a differential evolution (DE) algorithm is developed to solve emission constrained economic power dispatch (ECEPD) problem. Traditionally electric power systems are operated in such a way that the total fuel cost is minimized regardless of emissions produced. With increased requirements for environmental protection, alternative strategies are required. The proposed algorithm attempts to reduce the production of atmospheric emissions such as sulfur oxides and nitrogen oxides, caused by the operation of fossil-fueled thermal generation. Such reduction is achieved by including emissions as a constraint in the objective of the overall dispatching problem. A simple constraint approach to handle the system constraints is proposed. The performance of the proposed algorithm is tested on standard IEEE 30-bus system and is compared with conventional methods. The results obtained demonstrate the effectiveness of the proposed algorithm for solving the emission constrained economic power dispatch problem.     

A security constrained environmental/economic power dispatch technique using F-MSG algorithm for a power system area including limited energy supply thermal units

A security constrained non-convex environmental/economic power dispatch problem for a lossy electric power system area including limited energy supply thermal units is formulated. An iterative solution method based on modified subgradient algorithm operating on feasible values (F-MSG) and a common pseudo scaling factor for limited energy supply thermal units are used to solve it. In the proposed solution method, the F-MSG algorithm is used to solve the dispatch problem of each subinterval, while the common pseudo scaling factor is employed to adjust the amount of fuel spent by the limited energy supply thermal units during the considered operation period. We assume that limited energy supply thermal units are fueled under take-or-pay (T-O-P) agreement.The proposed dispatch technique is demonstrated on IEEE 30-bus power system with six thermal generating units having non-convex cost rate functions. Two of the generating units are selected as gas-fired limited energy supply thermal units. Pareto optimal solutions for the power system, where the constraint on the amount of fuel consumed by the limited energy supply thermal units is not considered, are calculated first. Later on, the same Pareto optimal solutions for the power system, where the fuel constraint is considered, are recalculated, and the obtained savings in the sum of optimal total fuel cost and total emission cost are presented. The dispatch problem of the first subinterval of the test system was solved previously by means of differential evolution (DE), and a hybrid method based on combination of DE and biogeography based optimization (BBO) for the best cost and the best emission cases in the literature. The results produced by these methods are compared with those of produced by the proposed method in terms of their total cost rate, emission rate and solution time values. It is demonstrated that the proposed method outperforms against the evolutionary methods mentioned in the above in terms of solution time values especially when the exact model of the test system is considered.     

Optimal dynamic economic dispatch of generation: A review

This paper presents a review of the research of the optimal power dynamic dispatch problem. The dynamic dispatch problem differs from the static economic dispatch problem by incorporating generator ramp rate constraints. There are two different formulations of this problem in the literature. The first formulation is the optimal control dynamic dispatch (OCDD) where the power system generation has been modeled as a control system and optimization is done in the optimal control setting with respect to the ramp rates as input variables. The second one is a later formulation known as the dynamic economic dispatch (DED) where optimization is done with respect to the dispatchable powers of the committed generation units. In this paper we first outline the two formulations, then present an overview on the mathematical optimization methods, Artificial Intelligence (AI) techniques and hybrid methods used to solve the problem incorporating extended and complex objective functions or constraints. The DED problem in deregulated electricity markets is also reported.     

A simple two stage optimization algorithm for constrained powereconomic dispatch

A simple two stage optimization algorithm is proposed and investigated for fast computation of constrained power economic dispatch control problems. The method is a simple demonstration of the hierarchical aggregation-disaggregation (HAD) concept. The algorithm first solves an aggregated problem to obtain an initial solution. This aggregated problem turns out to be classical economic dispatch formulation, and it can be solved in 1% of overall computation time. In the second stage, a linear programming method finds optimal solution which satisfies power balance constraints, generation and transmission inequality constraints and security constraints. Implementation of the algorithm for IEEE systems and EPRI Scenario systems shows that the two stage method obtains an average speedup ratio of 10.64 as compared to the classical LP-based method     

A hybrid HNN-QP approach for dynamic economic dispatch problem

This paper introduces a solution of the dynamic economic dispatch (DED) problem using a hybrid approach of Hopfield neural network (HNN) and quadratic programming (QP). The hybrid algorithm is based on using enhanced HNN; to solve the static part of the problem; the QP algorithm for solving the dynamic part of the DED. This technique guarantees the global optimality of the solution due to its look-ahead capability. The new algorithm is applied and tested to an example from the literature and the solution is then compared with that obtained by some other techniques to prove the superiority and effectiveness of the proposed algorithm.     

Dynamic economic dispatch using artificial immune system for units with valve-point effect

Dynamic economic dispatch (DED) is an important dynamic problem in power system operation and control. The objective of the problem is to schedule power generation for the online units over a time horizon, satisfying the unit and ramp-rate constraints. In this paper, clonal selection based artificial immune system (AIS) algorithm is used to solve the dynamic economic dispatch problem for generating units with valve-point effect. The feasibility of the proposed method is validated with ten and five unit test systems for a period of 24 h. Results obtained with the proposed approach are compared with other techniques in the literature. The results obtained substantiate the robustness and proficiency of the proposed methodology over other existing techniques in terms of solution quality and computational efficiency.     

动态系统实现火电厂机组负荷优化分配

利用连续型Hopfield神经网络(CHNN)可以将负荷优化分配问题转化为求解多变量非线性动态系统的稳态值。文中直接将负荷上下限约束条件作为神经元激活函数，构造出了通用的网络拓扑结构，理论和仿真试验证明了该方法能够全局收敛。并通过动态仿真手段获取稳态负荷优化分配结果。通用软件包开发和仿真过程表明该方法简单、可靠且适用。     

Reserve Constrained Dynamic Dispatch of Units With Valve-Point Effects

This paper addresses a hybrid solution methodology integrating particle swarm optimization (PSO) algorithm with the sequential quadratic programming (SQP) method for the reserve constrained dynamic economic dispatch problem (RCDEDP) of generating units considering the valve-point effects. The cost function of the generating units exhibits the nonconvex characteristics, as the valve-point effects are modeled and imposed as rectified sinusoid components. The hybrid method incorporates the PSO algorithm as the main optimizer and SQP as the local optimizer to fine-tune the solution region whenever the PSO algorithm discovers a better solution region in the progress of its run. Thus, the SQP guides PSO for better performance in the complex solution space. To validate the feasibility of the proposed method, a ten-unit system is taken and studied under three different load patterns. The effectiveness and computation performance of the proposed method for the RCDEDP of units with valve-point effects is shown in general.     

A parallel HAD-textured algorithm for constrained economic dispatchcontrol problems

In our earlier papers we developed a parallel textured algorithm to solve the constrained economic dispatch control (CEDC) problems. The exact convergence theorem and its proof were provided to guarantee the convergence of the algorithm to the true solution; and some examples were given to show the impact of exact convergence conditions. In this paper, we incorporate the hierarchical aggregation-disaggregation (HAD) concept and the textured concept to solve the CEDC. The algorithm is then implemented on an nCUBE2 machine and tested on a modified IEEE 14-bus system, a modified IEEE 57-bus system, a 114-bus system, and a 228-bus system. Some test results are given to show the speedup advantage of the proposed algorithm over the one without textured decomposition. Even when the proposed algorithm is executed sequentially, the speedup is still essential     

Evolutionary programming based economic dispatch for units withnon-smooth fuel cost functions

This paper develops an efficient, general economic dispatch (ED) algorithm for generating units with nonsmooth fuel cost functions. Based on the evolutionary programming (EP) technique, the new algorithm is capable of determining the global or near global optimal dispatch solutions in the cases where the classical Lagrangian based algorithms cease to be applicable. Effectiveness of the new algorithm is demonstrated on two example power systems and compared to that of the dynamic programming, simulated annealing, and genetic algorithms. Practical application of the developed algorithm is additionally verified on the Taiwan power (Taipower) system. Numerical results show that the proposed EP based ED algorithm can provide accurate dispatch solutions within reasonable time for any type of fuel cost functions     

应用近似动态规划算法求解安全约束随机动态经济调度问题

针对大电网安全约束随机动态经济调度(DED)问题的求解时间太长,提出了应用近似动态规划算法快速求解不含抽水蓄能电站电网的安全约束随机DED问题的方法。建立了随机DED问题的虚拟存储器模型,以系统的正旋转备用容量作为存储变量,构建系统相邻时段的状态转移方程,并考虑了各输电线路和断面的安全约束。以风电场日前功率预测曲线为基础,通过拉丁超立方抽样产生风电场出力的误差场景,并逐一场景递推求解每个时段的二次规划模型以对各个时段的值函数进行训练,形成收敛的值函数,再代入预测场景求解以获得最终的优化调度方案。该方法实现了对随机DED模型各个场景和各个时段的解耦求解,将一个大规模优化问题分解为一系列的小规模优化问题,有效提高了对大电网随机DED模型的求解速度。以某一实际省级电网为算例,通过与场景法和鲁棒优化调度方法的比较验证了所提出模型和求解方法的正确有效性。