OPTIMAL LOAD FLOWS USING LINEAR PROGRAMING SAID AHMED-ZAID Power Affiliates Program Department of Electrical Engineering ” University of Illinois at Urbana·Champaign Urbana, Illinois 61801 PAP-TR—80-3 December 1980 iii ACKNOWLEDGMENT The author wishes to thank his advisor, Professor P. W. Sauer for his constant help and guidance during the preparation of this thesis. iv TABLE OF CONTENTS Page CHAPTER I INTRODUCTION ...................... I l.I Research Motivation ...................... I I.2 Literature Review ....................... Ä CHAPTER II LINEAR OPTIMAL LOAD FLOWS ·............... 7 2.I Introduction ......................... 7 2.2 System Description ...................... 8 2.3 Linearized Load Flow ..................... lO 2.Ä Simultaneous Power Interchange Capability (SPIC) ....... IÄ CHAPTER III MAXIMUM POWER SUPPLY CAPACITY ............. l9 3.l Problem Definition ...................... I9 3.2 Iterative Solution of the MPSC Problem ............ 22 3.3 Additional Comments ...................... 2Ä 3.Ä Economic Levels of Loadability ................ 26 CHAPTER IV NUMERICAL RESULTS ................... 29 Ä.I_ Iterative Solution of the MPSC Problem ............ 29 Ä.2 Addition of Economic Constraints ............... 37 CHAPTER V CONCLUSIONS AND RECOMMENDATIONS ............. Äl 5.l Conclusions .......................... ÄI 5.2 Recommendations ........................ Äl REFERENCES ............................. Ä3 ' CHAPTER l INTRODUCTION l.l Research Motivation The main objectives of an electric energy systdnareto meet load demands with adequacy and reliability and to keep at the same time eco- logical and economic prices as low as possible. Electric energy demand has been shown to be an exponential function doubling its rate over every decade. This ever-increasing load has led to larger and more complex systems. interconnections throughout the United States and . Canada are growing and expanding. The main advantages of such inter- connections are continuity of service and economy of power production. Power interchanges between interconnected systems are scheduled to take ad- vantage of hour·apart peak demand periods or available lower cost capacity. During emergencies, spinning reserve capacity is shared, con- tributing to the continuity of service. This extensive interconnection of large scale power systems has resulted in the formulation of many new concepts in power system plan- ning and operation. The Commonwealth Edison CompanylsSystem Planning _ Department has done extensive studies in interchange power calculations ll,2]. Linear programming in conjunction with linearized load flows proved to be effective tools in solving the Simultaneous Power interchange Capability of a thermally limited system [21. The objective function to be maximized was an index ofperformance expressing the total power that could be brought into a single area system through neighboring ties subject to generation and transmission limitations. The ninth annual review of 2 overall reliability and adequacy of the North American bulk power systems published in August 1979 by the National Electric Reliability Council (NERC) brings forth the "emergency transfer capabilities" that represent the ability of the transmission network to transfer power from one area to another. These numbers expressed in megawatts (Mw) are shown in Fig. l-l and represent the interregional transfer capa- bilities between companies or groups of companies pooled together. These numbers are for nonsimultaneous transfers and were determined in [3];. accordance with the following NERC definition Emergency Transfer Capability: The total amount of power (above the net contracted purchases and sales) which can be scheduled, with an assurance of adequate system reliability, for interregional or multiregional trans- fers over the transmission network for periods up to several days, based on the most limiting of the following: (a) All transmission loadings initially with long- time emergency ratings and all voltages initially within acceptable limits. (b) The bulk power system capable of absorbing the initial power swings and remaining stable upon the loss of any single transmission circuit, transformer, or generating unit. " (c) All transmission loadings within their respec- tive short-time emergency ratings and voltages within emergency limits after the initial power swings follow- · ing the disturbance, but before system adjustments are made. (ln the event of a permanent outage of a facility, transfer schedules may need to be revised.) _ ‘ The major controversial issue raised in this paper is the degree of validity of those numbers since they do not correspond to a realistic situation. That is, these numbers are indices of performance calculated by bringing (or transferring) power into a network subject to trans- mission and generation constraints. They do not represent a realistic situation inasmuch as power is not brought into the system in order to WL W .„ Aa äw NIU °° ··¢¢ Ü '7 '·" ¤° L. I'.;gvzlz A' ~- wg MAAt°• Ü „:•.„•-. 740 év „, .' pfypß-" SE RFigl.-l. National Electric Reliability Council (NERC) Emergency Transfer Capabilities (Mw). (l979 Summen) h take up some of the load either in an emergency or a normal situation. The only case where power is wheeled across the network from one area to another seems to be the justifiable case where such numbers would be A meaningful. A related problem is that of Maximum Supply Capability which is defined in this paper as the maximm load a generating system can serve, neglecting losses, and with all units operating. Reference [Ä] formu- l lates this problem as a linear program with the additional constraint that loads vary linearly as a function of the total system load. This paper will review briefly the Simultaneous Power Interchange problem and will focus mainly on the Maximum Power Supply Capacity devising an algorithm to find the exact optimal solution and discussing other com- putational aspects using digital computers. . l.2 'Literature Review V In this section a brief review of the pertinent literature will be given. Also a few papers using related methods will be discussed al- P though not directly related to the subject of this thesis. E Early development of tie lines did not involve high loading levels E as the power transfers were essentially for emergency assistance. As E these interties continued to grow and expand, they gave rise to a complex, closely interconnected network permitting an increased relia- bility and economy of operation. Reference [I] considers the transmis· sion interchange capability between two companies and defines the new concepts of jimiting element, key element, power transfer distribution factor, line outage distribution factor, incremental interchange · 5 capability, waximum interchange capability and first Contingency inter- change Gapability. This analysis is limited to two simultaneous trans- fers and is done using simple graphical approaches. Reference [2] con- siders the simultaneous import of power into one company from several surrounding neighbors. The problem is solved using linear programming in conjunction with a linearized load flow that makes use of power transfer distribution factors [l]. A related problem is that of Maximum Power Supply Capacity which is treated in Reference [N]. The problem is defined as being the maxi- mum load a generating system can serve neglecting the losses. The authors make use of the D.C. load flow assumptions and formulate the problem asa linear program. Reference [5] gives a mathematical justification for the successful use of current distribution factors referenced to the swing bus and arbitrary ground tie modifications as opposed to current distribution factors referenced to the ground bus. References [6], [7], [8], and [9] discuss the applications of linear programming to power system security control calculations. Reference [6] presents the linear programming formulation of the re- T scheduling of active power required in corrective and preventive real- time security control and also in off-line studies. Reference [7] discusses the applications of such a method to practical power systems showing its versatility in implementing a variety of objectives, con- trolled devices and priorities. Reference [8] is an extension of the previous two reports to handle piecewise linear objective functions, a ” feature that is interesting in dealing with the security—constrained economic dispatch/controlior multiple-valved—turbine units. Finally reference [9] discusses the real·time corrective control of reactive power to relieve abnormal voltage levels and excessive reactive power flows. The main conclusions to be drawn are that linear programming together with incremental network models proved to give acceptable accuracies for operational purposes. Additional features are compu- tational reliability, fast speed of calculation and ability to handle large systems efficiently. 7 CHAPTER Il LINEAR OPTIMAL LOAD FLOWS 2.l Introduction The term Optimal Load Flow refers to an operating state or load flow solution where some power system quantity is optimized subject to constraints on the problem variables and on some functions of these variables. The constraints are usually classified under two categories: load constraints and operating constraints. The load constraints require that the load demands be met by the system and can be expressed in the form of the familiar load flow equa- tions. The operating constraints impose minimum or maximum operating limits on system variables and are associated with both steady-state l and transient stability limitations. These restrictions are imposed on various power system quantities such as equipment loadings (mainly for _ transmission lines and transformefs), bus voltages, phase angle differ- ences, real and reactive injected powers, etc. In this chapter, an optimal linear load flow is one in which the objective function to be optimized and the constraints are linear „ functions of the system variables. These linear programs usually have several drawbacks and yield only approximate results to the exact solution. Also, many operating constraints cannot be handled by these programs and in most cases a general nonlinear formulation is needed to represent the model adequately. Several methods have been devised to solve nonlinear programs but none exhibit the efficiency and re- ° liability of the Simplex method.