Novel Hybrid Algorithm for Hydrothermal Generation Scheduling

Abstract

Electric power systems have experienced continuous growth in all three major sectors of the power system, namely, generation, transmission, and distribution. Electricity generation cannot be stored economically, but it involves a dynamic system that needs a balance of its supply and demand for 24 hours. Conventionally, nearly 70-75% of total installed capacity, the power generation systems include both fixed head hydro and fuel-fired thermal units world-wide. Owing to the low operational cost and minimum start-up time, the hydro units facilitate peak load shaving by replacing high fuel cost thermal units. The designed infrastructure of hydroelectric power plants considers the units to be located on the same stream or/and on different streams. On the other hand, the selection of the thermal units as a base, middle or peak one is based on their efficiency, fuel cost and the dynamic response. The efficient scheduling of available energy resources for satisfying load demand has become an important task in the modern power system. The generation scheduling problem aims at determining the optimal operation strategy for the next scheduling interval subject to a variety of constraints. For a short-term hydrothermal generation, planning period is an hour or a day or half or a week as the load demand on the power system exhibits cyclic variations. Short-term hydrothermal scheduling is an important planning task in power system operation with two aspects; a day-ahead scheduling for short-term profitable market contracts and hourly scheduling for power balance requirements, hence it is the subject of intensive investigation. Major hydro units have high capital cost, whereas run-off river plants and pumped storage plants are not to be relied upon over short term planning of generation allocation to meet the load demand. The well-timed allocation of hydro energy resources requires probabilistic analysis and long-term considerations, as the excess water utilization in the present period, may lead to optimum availability in the future, increasing in this way the future operating costs. Hydrothermal generation scheduling is more complex than thermal unit commitment owing to various physical and technical operational constraints. Other than system constraints and specifications of hydrothermal units, the stochastic nature of water availability, limited energy storage capability of water reservoirs, and uncertainty in load demand over the scheduling period make its solution a more challenging task. Also, the increased concern of clean environment requirements, the low priced fuel of thermal units cannot meet the guidelines of environment act. The harmful emission production limit from the thermal unit must be satisfied simultaneously. This non-linear problem aims at determining the optimal hourly water discharge from hydro reservoirs and power generation from thermal units to meet load demand over the operation period while utilizing the limited amount of water available to its fullest extent, such that the total system production cost and pollutants emission of committed thermal units are minimized while satisfying thermal, hydraulic and other network constraints. Short term hydrothermal generation scheduling is a constrained multi-objective optimization problem with conflicting objectives. The conventional search methods are applied to solve the hydrothermal coordination problem made use of straightforward single point search procedures of exploration through pattern moves, to make the optimization problem simpler. But simplifying assumptions to a practical problem may extend to a suboptimal solution. The input-output characteristics of practical models of thermal-units and hydro-units are non-smooth, non-convex, and discontinuous. Direct search and /or gradient methods are although efficient ones, yet find a tedious task to manage non-linearity of the hydrothermal coordination problem. On the other hand, population-based heuristic (P-heuristics) search methods have paved a way to address various issues of optimization problems like complexity, multimodality, dimensionality, constraints handling, global solution search, convergence speed, etc.. However, the fitness measure of P-heuristics is influenced by the potential of the initial population, selection of control parameters, mutation and crossover techniques to balance exploration and exploitation and method of selection of a population for next generation. The interactive optimization techniques generate a set of non-inferior solutions and search the best possible solution for conflicting objectives of the problem with a trade-off between the computational effort and the quality of the solution. In recent years, a growing interest and awareness in the successful, inexpensive, and efficient application of stochastic algorithms are witnessed. According to no free lunch theorem, no algorithm is universally the best optimizer. There is a scope of improvement in optimization tools to improve convergence behaviour, the robustness of the global solution, dependence on control parameters, and strategies to prevent entrapping in sub-optimal solutions and optimal management of computational space and time for large scale real-world optimization problems. The scope of this thesis work is focused on the proposal of a novel multi-objective metaheuristic optimization technique integrated with promising strategies for better desirable results to constrained multi-objective real-time problems. In the first phase of the investigation, the exploitation capability of a global search technique is enhanced to identify the potential and quality solution search space. In the second phase, the global search technique is integrated with local search techniques for a better exploration and problems like premature convergence and stagnation to local minima are alleviated through adaptive learning. In the third phase, a strategy is adopted to archive elicit solutions. The differential evolution (DE) is undertaken as a global search technique. The DE technique is based on a population-based evolutionary algorithm with two control parameters, evolved through three evolutionary operators’ mutation, crossover, and selection. Basic DE suffers from occasional stagnation and premature convergence, where the solution converges to suboptimal solutions, but it generates a new candidate solution during evolution. To overcome the drawbacks of basic DE to some extent, three modifications in the DE strategy are proposed. Regeneration of promising initial population is proposed by exploring the uniformly distributed population members and their opposite ones simultaneously. The chaotic search strategy is one of the promising mimetic approaches to strengthen the exploitation ability of DE for a quality solution and to accelerate its convergence speed. Among various chaotic sequence variations, namely Logistic map, Tent map, Gauss map, and Sine map, Circle map etc., a chaotic sequence of control parameters of DE, mutation constant, and crossover rate, is generated by Logistic mapping in this study. Differential evolution is a self-referential algorithm. In the first attempt of modification to DE strategy, crisscross mutation and crisscross crossover operations are proposed in both horizontal and vertical directions alternately of the population for a better exploration and to escape from local optima. This strategy suggests a dual mechanism of interaction across the three-dimensional problem space and supports the generation of qualified potential solutions through competition. This supplementary attribute to chaotic DE eliminates the comparatively weak solutions from a parent population to increase the convergence rate and global search capability with accuracy. Further, a competitive computational intelligence strategy, opposition-based learning is incorporated to avoid stagnation of solution in local minima. The jumping rate is selected based on pre-specified non-dominated solutions in the current generation. The proposed concept contributes to enhancing the efficient convergence of the search algorithm. The proposed optimization techniques are applied to search optimum solution of the fixed head, multi-chain short-term hydrothermal generation scheduling problems. Multi-objective generation scheduling problem undertakes two conflicting objectives that are the operating cost and emission of gaseous pollutants Compromised solution is selected by finding the aggregated effect of participating objectives using their membership functions. In this work, the constraints are satisfied by perturbing the decision variables in a systematic procedure. The proposed approach is tested on four fixed head hydrothermal power system. The proposed optimization techniques are also validated on generalized benchmark test functions. In the research work, an interactive fuzzy method is used to solve the multi-objective optimization problem with two conflicting objectives. The decision-maker decides the solution interactively by considering the unified effect of participating objectives by exploiting their membership functions. The highest cardinal priority ranking provides maximum satisfaction level of the participating objectives. The penalty function method lacks flexibility and undergoes multiple runs to fine-tune the penalty factor leading to a high computational cost. In the second phase of solving the multi-objective constrained optimization problem, the equality constraints are handled using a two-step approach in hydrothermal system problems. In the first step, a variable elimination approach with heuristic repair strategy is applied to handle equality constraints and subsequently in second step fuzzy model is recommended to handle unpredicted residues in violations of operational constraints; those result in due to the stochastic nature of variables and algorithm-dependent parameters. Residues are fuzzy quantified and are considered as an objective to be optimized. The best-compromised solution of the three interdependent objectives of the hydrothermal generation scheduling (HTGS) problem is achieved by an interactive fuzzy decision model. The results obtained are satisfactory and demonstrate the appropriateness of each algorithm not only for handling non-convexities, discontinuity, and non-differentiable functions but are also well suitable for constrained optimization problems of the power sector. Stochastic optimization population-based algorithms generate several random solutions and improve during the process of optimization. This bounded search space may contain discontinuities, more than one local minima, global optimum located on the boundaries of search space, deceptive valleys, etc.. It is important to equip the optimization algorithms with suitable mathematical functions to update the position of random solutions in the promising regions for handling these difficulties to find the global optimum. In the second attempt, the Sine-Cosine algorithm (SCA) that uses trigonometric functions to redefine the search space and allows to guarantee the exploitation of space between two solutions is incorporated. The range of distance and direction of movement in the cyclic pattern of the sine-cosine function is updated adaptively to emphasize the exploitation as the iteration counter grows. Chaotic differential evolution algorithm is hybridized with the sine-cosine algorithm. The performance of the hybrid algorithm is tested through simulations on generalized benchmark test functions and hydrothermal systems. In the third attempt, the self-adaptive differential evolution algorithm, a global search technique is integrated with a Simplex search method, a local search technique. Chaotic differential evolution algorithm is responsible for diversification whereas Simplex search is dedicated to the exploitation of search space. The accuracy of the elite solution is improved by invoking the simplex technique for neighbourhood search. The proposed hybrid algorithm identifies the solution close to the Pareto front and supports the selection of suitable compromising solution among the available options of conflicting objectives hydrothermal system problem. In the fourth attempt, the artificial neural network (ANN) training algorithm is hybridized with differential evolution. ANN-based methodologies possess some interesting qualities like learning from experience, ability to handle uncertainties and ambiguity in data set, modelling the nonlinear and complex relationship between input-output data, and store information in the form of the weights of connections between the network layers. The major challenge with ANN is in dealing with adaptive learning and to maintain the balance between recently acquired knowledge with information already embodied in the network. The selection of geometrical configuration, learning parameter values, and learning strategy in multi-layered neural networks greatly influence the convergence rate and accuracy of an optimal solution. In this study, the feedforward neural network (FNN) model is proposed for the mapping of variables. The training time and global search capability of FNN are influenced by the initial population set and the learning parameters. The proposition of a salp chain mechanized behaviour of interaction is enabled for a search around the individual, individual best, and the global best position to generate a promising population that accelerates the convergence speed. The chaotic differential evolution is responsible for diversification and is adopted to optimize the learning strategy of FNN. The external elite approach is used to archive the non-dominant solution. In the thesis, the validity and effectiveness of the proposed novel metaheuristic techniques developed to solve multi-objective constrained optimization problems have been extensively verified and numerically tested on two categories of optimization problems. In the first category, a set of twenty diverse benchmark test functions unimodal and multimodal groups is selected to investigate the exploitation and exploration capability respectively, of proposed hybrid techniques. The quantitative performance analysis of each problem is estimated for its fitness value in terms of the best, average, the worst, the standard deviation, whereas qualitative performance is observed from the search history, trajectory of a population member, average fitness of the population, and convergence behaviour in the relevant parametric search space. The capability of the proposed algorithm is evaluated against well-established state-of-the-art meta-heuristic algorithms, reported in the literature for a fair comparison. In the second category of numerical optimization of real-world application, hydrothermal generation scheduling problems of varied dimensions are selected. Fuzzy model of handling uncertain violations in the equality and inequality constraints due to the stochastic nature of decision variables and variations in load demand and water availability in reservoirs is the contribution of this thesis work. For a fair comparison of the efficacy of the proposed algorithms, the quantitative results obtained after the application of the proposed variants of DE in terms of the minimum cost, minimum emission are compared with the state-of-the-art metaheuristic techniques. Cardinality ranking operator is preferred to rank the best-compromised solutions obtained from the compared the most utilized meta-heuristic stochastic optimization algorithms applied to respective test systems. The robustness and the accuracy of the global solution obtained from the suggested algorithms are measured through statistical significance tests. Through the investigations in this study, it is concluded that no single strategy can improve all the performance metrics of differential evolution when applied to a variety of optimization problems. However, the major contributions to P-heuristics when applied to solve non-linear multi-objective optimization problem are summed up as: Three strategies to improve the initial population are applied. First, exploring the opposite population that improves the diversification of population, Second randomization of the population with the sine-cosine strategy that cyclically reposition the population member around the other and supports exploitation of space between two solutions. This strategy takes care of the problem of isolation of global optimum. Third, using the Salp chain behaviour, the individual member of the population is guided towards the global best position. This strategy has drastically increased the convergence rate. Two strategies at mutation and crossover levels of DE are explored. Chaotic tuned mutation constant and crossover rate controls diversification and intensification within feasible space. Crisscross mutation and crisscross crossover operations are employed in orthogonal directions alternately in three-dimensional search space. This strategy prevents the stagnation of local minima. To archive elite solutions, opposition based learning is incorporated to generate potential Pareto set of non-dominant solutions. In the second phase of the study, the investigation of differential evolution hybrid with Simplex search confirms the robustness of the algorithm. In the third phase, feedforward neural network a nonlinear optimizer is trained using differential evolution. This meta-optimization strategy will justify the computational expenses in terms of space and time when applied to large scale optimization problems, in case the issue of parameter values is resolved.