OPTIMAL POWER SYSTEMS OPERATION USING MATHEMATICAL PROGRAMMING

Abstract

Optimal operation of power systems is very important because of huge system operating costs. The idea of economic dispatch is used by engineers in power utilities right from the inception of this vital indus try. With the increase in size and complexity, the simple criterion of 'equal incremental cost' operation of generators does not give optimal operating conditions with desired accuracy and hence more sophisticated methods are now used to find the optimal flow of activeand reactive-power in the system. In the present study different criteria are chosen to describe the steady state optimal operating strategy. The active- and reactive powers are allocated amongst the system sourees such that the defined criterion is optimally satisfied. The criteria or the objectives for optimization could be total system operating cost,system losses, and other functions to describe a desired behaviour of the system. The objectives are optimized such that the system power flow equations and limit constraints imposed upon the variables by the system operating conditions and design considerations are satisfied. Because of large number of variables and constraints involved, and both the objective function and constraints being nonlinear, the problem is quite a challenging one from computational considerations. The complexity and size of the Optimal Power Dispatch problem has motivated the decomposition of the problem variables, namely node voltage magnitudes and node phase angles, into two groups comprising of state and control variables. The use of sensitivity matrix is made for up dating the state variables after the control variables have been modified by the optimization process. The complete problem of optimal power dispatch happens to be a constrained nonlinear programming problem. A new approach is suggested in this work to transform the constrained problem into an unconstrained one by using Moving Exterior Truncation Transformation. It involves the choice of one single parameter known as Truncation Level which can always be calculated from the results of load flow study. The technique therefore avoids the arbitrary selection of penalty parameters and/or Lagrangian Multipliers to be associated with constraints. In order to reduce the size of the problem, the complete problem of optimal power dispatch has been decomposed into two sub-problems of active-and reactivepovver optimization. The decomposition utilizes the physical properties of the electric power networks and helps in solving the two important problems independently. The active-power optimization is done by minimizing system operating costs., with node voltage magnitudes held constant at optimal system voltage level evaluated by minimizing the real-power losses, with node phase angles as problem variables, subject to the above defined equality and inequality constraints. These constrained sub-problems are transformed into unconstrained problems by the use of yet another new transformation known as Least Squares Formulation. This formulation is again very simple and requires the setting of one single parameter, the initial value of which can always be calculated from the results of L.F. study. These problems have also been solved by using Moving Exterior Truncation Transformation. The problem of reactive-power optimization has been tackled both from economic and system reliability consi derations. In the first case, total system real-power losses are minimized with node phase angles held constant, thus giving the optimal system voltage level or optimal reactive power flow. In the second case an altogether new objective function is suggested for the allocation of reactive powers amongst the system generating sources. The total reactive power generation is minimized such that the generators share reactive powers proportional to their reactive power capability limits. Such reactive power sharing would improve the stability of machines which are likely to fall out-ofstep because of excessive under-excited operation when the reactive power allocation is done without taking into consi deration relative stability of the generators in the system. This approach for optimal reactive power allocation is presented for the first time in this thesis. A new approach is suggested by the author for another important aspect of reactive power allocation in the power system. The problem of capacitor allocation by o-l programming is presented which helps in determin ing the size and location of the capacitors in the system. The suggested technique dees not require back tracking hence the storage requirement is minimum. As it requires. less number of solution vectors to be tested. hence is more efficient than other enumeration methods. In this study use is made for the first time of new formulations unknown to the field of power systems. Simple and reliable solution algorithms with assured convergence are used for the solution of unconstrained optimization problems formed with the help of these trans formations. The transformations are general and can be used for solving other problems in the area of power systems. For load flow studies many efficient methods exist which are widely used because of their fast convergence. Yet under certain conditions some of these methods fail to converge and it is not possible to ascertain if it is because of instability in the solution method or there exists no solution. A new formulation for the load flow problem is presented in this study. The proposed method transforms the load flow problem into an unconstrained nonlinear programming problem having the squared mismatches of powers as the objective function. The method would provide solution to the load flow problems under all conditions and shall therefore be a useful tool for system planners. This happens to be the major advantage of the suggested technique for the solution of load flow problem. The general purpose computer programs have been developed for all the problems undertaken in the present study and are tested on IBM 1620 and run on IBM 360. Due to limited computational facilities systems upto 23 buses have been taken for carrying out the various studies relating to the problems discussed above and the results are reported in different chapters. It is expected that the new techniques suggested for solving optimal power dispatch and other important problems will find a great utility in power industry.