New Algorithms for Some Economic Dispatch Problems

Abstract

An electric power system consists of several generating stations which cater to the load demands of various regions. The prime function of any generating utility is to optimally schedule the real power output of its generating units to meet any specified real power demand subject to various constraints on the operation of the units and the system. The optimal scheduling of individual generators at the least possible cost is referred to as the economic dispatch or economic load dispatch (ELD) problem. The ELD studies play a vital role in the day­to­day operation of the power system and in formulating economic operating strategies, besides ensuring the stability and security of the system. This thesis work makes an effort to probe deeper into some aspects of the economic load dispatch problem and the underlying mathematical formulations and attempts to come up with generalized algorithms that can effectively handle different types of systems under different operating conditions and, to a certain extent, to try and resolve some aspects hitherto unresolved. The primary focus is on developing efficient computational techniques to solve some Specific types of ELD problems in a simple and systematic manner. In the course of this investigation, we highlight some imperfect assumptions involved in the solutions proposed for the ELD problems for systems with complicated constraints like prohibited operating zones (POZ). We also set forth new concepts and strategies and develop new techniques in this investigation to help resolve some of these incorrect propositions and ambiguities. The first chapter introduces the ELD problem in general and proceeds to discuss the effects of the transmission losses and the presence of POZs on both the scheduling of the generators and the complexity of the ELD analysis. It also provides a brief review of some relevant aspects of the state-of-the-art solution techniques and clearly spells out the motivation for the present work. The second chapter presents a generalized algorithm for solving the ELD problem efficiently. The algorithm is effectively applicable to any system comprising power generating units with any type of well-defined, smooth and monotonic cost functions, besides quadratic cost functions usually considered in conventional algorithms. The proposed method first identifies the units that are forced to operate at their generating limits for any given value of the system demand. Subsequently, it limits the ELD problem to calculating The system's incremental cost of received power and the power output of only those units operating within their normal feasible range. The specific improvement introduced here is the development of an efficient computational scheme for calculating the value of the system incremental cost accurately. In addition to quadratic and higher order polynomial cost functions, the proposed algorithm can easily be generalized to include units with smooth, monotonic, non­ polynomial cost functions. The major advantages of the proposed ELD scheme are its inherent simplicity, scalability, rapid convergence and high computational efficiency. These characteristics are particularly important for real-time online implementation. The results obtained for test cases from the literature and some new ones as well are presented to illustrate the effectiveness of the proposed scheme. The third chapter proposes an algorithm for solving the ELD problem considering power losses in the transmission network. The losses are computed using the transmission loss­ formula A coefficients method suggested by Nanda and Bijwe as an alternative to the conventional B­loss coefficients approach popularized by Kirchmayer. The proposed ELD­with­ Losses scheme builds upon the ELD scheme developed in the second chapter for the lossless case. The specific contribution of the third chapter is the computational approximation suggested for the iterative procedure involving The Newton­Raphson (NR) method with the losses considered, while still retaining the elegant solution scheme developed in the earlier chapter. The results obtained using test cases from the literature are presented to demonstrate the precision and effectiveness of the proposed technique. The fourth chapter presents a novel algorithm for efficiently solving the ELD problem for systems having generators with prohibited operating zones. The proposed ELD­ POZ scheme partitions the no convex solution space into simpler convex intervals in which the ELD scheme developed in the second chapter can be applied directly. The improvisation lies in the optimal ordering­cum­sorting strategy adopted to systematically determine the output levels of the units constrained by POZs and to adjust the output power of the remaining units appropriately. The proposed scheme also recognizes and exactly computes the multiple, equivalent optimal solutions wherever applicable–– another significant contribution of this thesis work. It also seeks to clarify and set right some unintentionally imperfect propositions and assumptions currently prevalent in the literature regarding the formulation and analysis of the ELD problem considering POZs. The results generated for a number of systems using test cases from the literature along with some new ones are presented to clearly illustrate the validity as well as the Simplicity and superiority of the proposed scheme for different types of systems. The final chapter briefly recounts the work done in this thesis work. It also presents a summary of the significant results obtained using the schemes proposed in the earlier chapters, along with the conclusions drawn in support of the validity and superiority of the proposed algorithms. More areas for further investigation and some possible avenues for future applications of the proposed techniques are also indicated.