The Existence of Multiple Power Flow Solutions in Unbalanced Three-Phase Circuits

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 2, MAY 2003 605 The Existence of Multiple Power Flow Solutions in Unbalanced Three-Phase Circuits Yuanning Wang, Student Member, IEEE, and Wilsun Xu, Senior Member, IEEE Abstract This paper presents a new phenomenon found for unbalanced power flow solutions the existence of two voltage solutions at the neutral points of unbalanced three-phase circuits. The phenomenon is different from that found in traditional singlephase balanced power flow solutions. It is dependent on the degree of system or load unbalance. If the unbalance is reduced to zero, the two solutions will merge into one. This paper presents our observations and analysis on the problem. A test system is used to illustrate the practical significance of the phenomenon. We hope this work will stimulate further research on this challenging and interesting problem. Index Terms Multiple solutions, power quality, unbalanced power flow. I. INTRODUCTION WITH the increased awareness on power quality, many algorithms have been proposed and developed for power quality analysis. One of the notable developments is the calculation of three-phase power flows for unbalanced power system conditions [1] [6]. Although these algorithms generally work well for solving unbalanced power flow problems, our understanding on the characteristics of the solution results is still limited. For example, we have found that there exist two solutions for the neutral voltage of an unbalance three-phase load. The problem of multiple load flow solutions is not new for single-phase (balanced) power flow analysis. It is typically associated with voltage instability of the study system. Although this type of multiple solutions may still exist for a stressed unbalanced system, we have observed a different type of cases with multiple solutions those associated with the degree of supply and load unbalance. If the degree of unbalance is reduced, the two solutions will merge into one solution. This is a new phenomenon. The problem could affect the interpretation of unbalanced power flow results. The purpose of this paper is to present our observations and preliminary analysis on the phenomenon. Our intention is to raise awareness on the peculiarity of unbalanced power flow solutions. By summarizing and analyzing several simple cases, we hope this paper can serve as a step stone for other researchers interested in investigating the problem further. Fig. 1. Sample system that could lead to power flow multiple solutions. II. STUDY SYSTEM The problem of multiple unbalanced power flow solutions was first observed when we analyzed a 35-bus unbalanced system. The power flow converged to two sets of nodal voltage solutions. Careful analysis of the results revealed that the difference between the solution sets is concentrated at the neutral voltages of a few star-connected components. Further investigation helped us to isolate and simplify the problem into the one shown in Fig. 1. If one connects three constant power, individual-phase loads into a star form depicted in Fig. 1, two solutions could be obtained for the voltage at the neutral point. Since the objective of this paper is to conduct analytical investigations on the phenomenon, the simple system of Fig. 1 will be used as the study case. A simple system also makes it easier to understand the phenomenon. To compute the voltage at the neutral point, the following equation can be established: Substituting the loads and source voltages yields This is the load flow equation that has multiple solutions. In this paper, the problem is analyzed using two types of unbalanced system conditions, load unbalance and supply voltage unbalance. (1) (2) Manuscript received August 7, 2002. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and in part by Alberta Heritage Foundation for Science and Engineering Research. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: ynwang@ee.ualberta.ca; wxu@ee.ualberta.ca) Digital Object Identifier 10.1109/TPWRS.2003.810898 III. LOAD UNBALANCE RELATED MULTIPLE SOLUTIONS In this case, we assume that the only source of unbalance is the load. One of such operating conditions is 0885-8950/03$17.00 2003 IEEE

606 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 2, MAY 2003 Fig. 2. Solution curves of load unbalance case. (a) where is a variable that defines the degree of load unbalance. characterizes the balanced condition. With these assumptions, the general power flow equation (2) can be simplified as This equation is a quadratic equation. There are two solutions for it in theory. In order to view the features of the solutions, function is defined (3) (4) The plot of as a function of is shown in Fig. 2. Each curve in the figure represents the function with a different unbalance parameter. The load flow solutions are the intersection points of the curves with the -axis, that is,. As shown in the above figure, there exist two real solutions for some values of parameter. If the degree of unbalance becomes zero, the two solutions merge into one. It is also interesting to note that there is another point where the solutions merge into one too. If the -curve does not intersect with the -axis, the solutions are complex numbers. To view all possible solutions of as a function of, (4) is solved directly as follows: It can be seen from (5) that there are always two solutions for. The condition under which the two solutions merge into one is or which leads to (the balanced solution) and, which leads to. is a highly unbalanced, practically impossible, and theoretically valid case. If is within the range from 0 to 4, will have two real solutions, otherwise complex solution pairs would be expected. Equation (5) also shows that when equals to 3, in addition to the solution of, there is another solution of. One may interpret this case as having one solution. Further analysis shows that when, has two real solutions, one is positive and the other is negative; while when, has two negative real solutions. (5) (b) Fig. 3. Solution characteristics of V. (a) Trajectory of real solutions as a function of. (b) Entire solution root locus. The solution trajectory verifies the analysis shown before, which is shown in Fig. 3. Fig. 3(a) shows the trajectory of the real solutions with -axis as the parameter. Fig. 3(b) shows the trajectory of the entire solutions in a complex plane (root locus). Inspection of the curves and (3) and (5) leads to the following preliminary conclusions 1) the solution points could be independent of the load size. They are more related to the degree of load unbalance; 2) if (i.e., the loads are balanced), the two solution points merge into one point. The two solutions are quite close if the degree of unbalance is small. 3) The two close solutions obtained when is small can be either two real solutions or two complex solutions. It would be more difficult to differentiate the two complex solutions since both have the same magnitude. 4) For the case of, although the load is not balanced, the two solutions also merge into one point. This is another unique and hard-to-explain phenomenon of the unbalanced power flow solution problem. To interpret the physical meaning of the two solutions, one unbalanced case with (i.e., zero phase a load), is considered. This zero load condition can be modeled as either open

WANG AND XU: EXISTENCE OF MULTIPLE POWER FLOW SOLUTIONS IN UNBALANCED THREE-PHASE CIRCUITS 607 (a) (b) Fig. 4. Interpretation of two solutions when =1. (a) Open circuit solution. (b) Short-circuit solution. (a) Fig. 5. Solution curves of voltage magnitude unbalance case. circuit or short circuit of phase, as shown in Fig. 4. Accordingly, there are two solutions for the problem. The solution corresponding to the open circuit case is, an average of and. The solution corresponding to the short-circuit case is, the voltage of. IV. VOLTAGE UNBALANCE RELATED MULTIPLE SOLUTIONS The cases in this section are related to the unbalance of the supply voltage. The supply voltage unbalance consists of both magnitude and phase angle unbalance. In order to simplify the analysis, three types of cases with increased complexities are considered. A. Magnitude Unbalance The first case involves the situation where only the source voltage magnitudes are unbalanced where is a parameter to characterize the voltage magnitude unbalance; and corresponds to the balanced case. With these conditions, (2) can be further simplified into the following equation: Again, there are two solutions expected for this quadratic equation. The features of the solutions can be viewed from Fig. 5, which shows the curves with different unbalance parameter. Function is defined as (6) (b) Fig. 6. Solution characteristics of V. (a) Trajectory of real solutions as a function of. (b) Entire solution root locus. As shown in the figure, there exist two real solutions for some values of parameter. Solving (6) directly, we can get the solutions of as a function of the voltage magnitude unbalance degree Similarly, (7) shows that two solutions of exist for most values of. When, two solutions merge into one. One of such solutions is and, which corresponds to the balanced case. Another solution is,. Although this is a theoretically possible solution, it is unlikely to happen since the supply voltage would be quite unbalanced. will have two real solutions if is greater than 3 or less than 0, otherwise complex number can be expected. When, has one positive and one negative solutions; while the condition of leads to two negative solutions. The trajectory of the solutions is shown in Fig. 6. The real solutions of are given in Fig. 6(a). Since there also exist complex solutions, the whole solution locus of is plotted in complex plane in Fig. 6(b). Study results show that most conclusions drawn for the case A apply to this case as well. The solutions are directly related to the degree of unbalance, and are independent of the load condition. (7)

608 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 2, MAY 2003 Fig. 7. Solutions locus of V. When, the sources are balanced, two solution points merge into one point (i.e., ). can also give one solution,. A pair of conjugate complex solutions exist for. B. Phase Angle Unbalance The case of phase angle unbalance is then investigated. The degree of unbalance is modeled using the following equations: where is a variable that defines the degree of phase angle unbalance. characterizes the balanced condition. Using the condition shown before, (2) can be simplified as Again, this equation is a quadratic equation. The solutions of are shown in (9) and plotted in Fig. 7. where. There are two solutions for any given parameter. It can be seen from Fig. 7 that most of the solutions are complex numbers except when, where only one solution exists. For the angle unbalance case, one can therefore conclude that there is only one condition where the two solutions merge into one. For the two-solution cases (i.e., ), there will be difficulties to differentiate the two solutions since both have the same magnitude. C. Combined Magnitude and Phase Angle Unbalance This is the most general case. Since there are two variables (magnitude and phase angle), an alternative method to analyze the impact of voltage unbalance on solution trajectories is needed. The power flow equation for this case can be rewritten as (8) (9) Fig. 8. Solutions locus of V. where is a complex variable expressed in polar coordinates. The solution for takes the form of The solution trajectory can be visualized in a complex plane of with different values, as shown in Fig. 8. There are three trajectories in the figure, each corresponding to a different magnitude of (i.e., the valu). The trajectories are formed when the phase angle of varies from 0 to 360. It can be seen that the solution of consists of two arcs that form an egg-shaped trajectory. The size of the trajectory reduces when the magnitude of unbalance is reduced. As long as 0, there will always exist two solutions. The two solutions go clockwise along the corresponding trajectories. When the phase angle equals to 180, the two solutions are real-valued solutions. It is also noticed that a pair of phase angles of, such as 90 and 270 always generates conjugate complex solutions for. It is worth pointing out that the same procedure can be used to analyze the load unbalance that involves both active and reactive powers, namely, is a complex number. We have found that the solution trajectories are similar to those involving the voltage unbalance they also consist of two arcs. V. DISCUSSIONS AND CASE STUDY The analysis presented for the study system has revealed many interesting and unique characteristics of the unbalanced power flow solutions. Although the system is simple, it includes the characteristics of large size systems. For example, the system shown in Fig. 9, a more practical case, is electrically similar to the study system when the grounding impedance is infinite. It can be shown that this system has two solutions for the neutral point voltage if the ground impedance is not zero. The above case can be further extended for unbalanced systems with ungrounded or impedance grounded generators and transformers, including delta-connected transformers. Furthermore, the different neutral voltages will lead to different phase voltages and branch currents. The investigation on multiple power flow solutions is therefore not only of academic interest but also of practical value. Our results obtained so far indicate that the two solutions are close if the degree of

WANG AND XU: EXISTENCE OF MULTIPLE POWER FLOW SOLUTIONS IN UNBALANCED THREE-PHASE CIRCUITS 609 TABLE I POWER FLOW RESULTS OF CASE 1 ZERO PHASE C LOAD Fig. 9. Another system configuration leading to multiple solutions. Fig. 10. Example with multiple solutions. unbalance is small. But it is hard to tell at this stage that this conclusion is applicable to all large systems. One should note that when the system size increases the number of solutions is expected to increase as well. This conclusion is derived from the following observation: suppose an unbalanced system has two load buses. Both buses have a zero phase C load. Since a zero load can be represented as either an open circuit or a short circuit, there are four possible combinations for the two phase-c loads. As a result, there are four possible solutions for this case. It is important to note that the multiple solutions are strongly related to the constant power load modeling assumption. This model is also the cause of multiple solutions in the traditional balanced power flow analysis. In reality, the assumption may not be true and the system may just have one solution. Unfortunately, changing the load model is not the best method to avoid the multiple solution problems, since many distribution feeders loads are single phase and are best understood as constant power loads (as shown in the following example). In order to assess the impact of multiple solutions on overall power flow results of a system, a simple test system has been constructed. The system, shown in Fig. 10, is a representative distribution feeder involving ungrounded neutral. There are three loads each connected to one of the phases at the feeder end. Other loads are omitted to highlight the main characteristics of the system. Two case studies were conducted for the system. Case 1 assumes a zero phase C load and Case 2 assumes a 0.4869-MW phase C load. The solution results are summarized in Tables I and II. It can been seen that the two solution points are quite different for Case 1. This is understandable since one solution is related to phase C open circuit and the other related to short circuit. For the short-circuit solution, voltages on phase C is very low. However, other phases have a higher than normal voltage. This situation is similar to a single-phase to ground fault case. What surprised us is that the two solution points for Case 2 are quite different as well. This system is fully balanced except that TABLE II POWER FLOW RESULTS OF CASE 2 UNBALANCED PHASE C LOAD the phase C load is different from the other phases by MVA. The reason for such a large difference in the two solution results is likely related to the large difference between the phase angles of loads. Phase C load has a zero phase angle and the other two phases have a phase angle of 21.8. According to the analysis of Section IV-C, certain phase angle unbalance can drive two solution points far apart for the same magnitude of unbalance. In summary, the case study results have shown that multiple solutions do exist in real systems and do have practical significance. It is not easy to detect which one is the real solution. With our limited knowledge on the subject, one possible guide-

610 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 2, MAY 2003 line we could offer to find the right solution is to ground the neutrals of the system at first. The power flow results of this modified system are then used as the initial values for solving the actual power flow case involving ungrounded or impedance grounded neutrals. VI. CONCLUSIONS This paper has presented the phenomenon of multiple solutions in unbalanced three phase circuits. It was found that the solutions are dependent on the degree of network unbalance. For any specific unbalance parameter, a pair of solutions exists: some are real, some are conjugate complex numbers. There also exist points that the multiple solutions merge into one, such as when the degree of unbalance becomes zero. Based on our experiences on running many unbalanced power flow cases, it seems that systems without solidly grounded neutrals will encounter multiple solution situations. Although the individual phase constant power loads are the main cause of multiple solutions investigated in this paper, we have recently found that commonly accepted three-phase generator models could also lead to multiple solutions. It is still not clear to us which solution the power flow generally converges to and what conditions govern the convergence characteristics. We hope the preliminary results presented in this paper will stimulate further research on the subject and eventually lead to useful guidelines to unbalanced power flow studies. REFERENCES [1] Arrillaga, C. P. Arnold, and B. J. Harker, Computer Modeling of Electrical Power Systems. New York: Wiley, 1983. [2] D. Reichelt, E. Ecknauer, and H. Glavitsch, Estimation of steady-state unbalanced system conditions combining conventional power flow and fault analysis software, IEEE Trans. Power Syst., vol. 11, pp. 422 427, Feb. 1996. [3] X.-P. Zhang, Fast three phase load flow methods, IEEE Trans. Power Syst., vol. 11, pp. 1547 1554, Aug. 1996. [4] W. Xu, J. R. Marti, and H. W. Dommel, A multiphase harmonic load flow solution technique, IEEE Trans. Power Syst., vol. 6, pp. 174 182, Feb. 1991. [5] P. A. N. Garcia, J. L. R. Pereira, S. Carneiro Jr., V. M. da Costa, and N. Martins, Three-Phase power flow calculations using the current injection method, IEEE Trans. Power Syst., vol. 15, pp. 508 514, May 2000. [6] B. C. Smith and J. Arrillaga, Improved three-phase load flow using phase and sequence components, Proc. Inst. Elect. Eng. C, vol. 145, no. 3, pp. 245 250, May 1998. Yuanning Wang (S 01) received the M.Sc. degree in electrical engineering from Harbin Institute of Technology, Harbin, China, in 1995. She is currently pursuing the Ph.D. degree in the Department of Electrical and Computer Engineering at the University of Alberta, Edmonton, Canada. Her research interests are power system voltage stability and electricity market modeling. Wilsun Xu (M 88 SM 93) received the Ph.D. degree from the University of British Columbia, Vancouver, Canada, in 1989. Currently, he is a Professor with the Department of Electrical and Computer Engineering at the University of Alberta, Edmonton, Canada, where he has been since 1996. From 1989 to 1996, he was an electrical engineer with BC Hydro, where he was responsible for power quality and voltage stability projects. His main research interests are in the areas of power quality and voltage stability.