Fast Weak-Bus and Bifurcation Point Determination using Holomorphic Embedding Method Daniel Tylavsky, Vijay Vittal

Transcription

1 1 Fast Weak-Bus and Bifurcation Point Determination using Holomorphic Embedding Method Shruti ao Daniel Tylavsky, Vijay Vittal General Electric Arizona State University GE Power School of Electrical, Computer and Energy Engineering Schenectady, NY, 1345, USA Tempe, AZ, 8587, USA Weili Yi The Energy Authority Strategic nnovations & Analytics Jacksonville, FL, 30, USA Abstract--A new method of solving the power-flow problem, the holomorphically embedded load-flow method (HELM) is theoretically guaranteed to find the high-voltage solution, if one exists, up to the saddle-node bifurcation point (SNBP), provided sufficient precision is used and the conditions of Stahl s theorem are satisfied. Sigma (σ) indices, have been proposed as estimators of the distance from the present operating point to the SNBP, and indicators of the weak buses in a system. This paper investigates the theoretical foundation of the σ method and shows that the σ condition proposed in [10] will not produce reliable results and that a modified requirement can be used to produce a tight upper bound on the SNBP. A new HEM-based method is then proposed that can be used to estimate the weak buses in a system (from a steady-state voltage stability perspective) at all operating points through to the SNBP, using a single power-flow solution. Numerical results for the proposed approach are compared to traditional modal analysis for the EEE 14-bus and 118-bus systems. ndex Terms Holomorphic embedding, power-flow, sigma algebraic approximants, analytic continuation. T. NTODUCTON he HELM, proposed by Dr. Antonio Trias, involves a specific implementation of the holomorphic embedding method (HEM) applied to the power-flow problem and is theoretically guaranteed to converge to the operable solution, if it exists, for any given power-flow problem [1] provided Stahl s conditions are satisfied [8] [9]. From its introduction to the power system community in 01, the holomorphic embedding method (HEM) has been applied to an everexpanding list of applications. The holomorphically embedded power flow formulation (HEPF) was first applied to the power-flow problem with only PQ buses []. A model for PV buses was proposed in [3] along with an algorithm which considered the discrete changes in the system such as bus-type switching and tap-changing transformers. The algorithm was extended to dc systems in [4] where it was shown that the method could accommodate the nonlinearities that characterize the -V curves of power electronics devices. A formulation aimed at finding the low-voltage solutions of a power system was proposed in [7] using the HEPF. n a later patent from the inventor of HELM, indices called σ algebraic Di Shi, Zhiwei Wang GE North America PMU & System Analytics Group Santa Clara, CA, 95134, USA approximants were proposed to be used to estimate the distance from the present operating point to the SNBP of a system and to detect the weak buses in the system, buses that directly impact the voltage stability limit [10]. The HEPF theory, adapted for ZP-load models, was first presented in [14], such that the solution obtained at different values of embedding parameter, α, represented the bus voltages when the loads were scaled by α, and the smallest real pole or zero of the Padé approximants was shown to give the load-scaling value corresponding to the SNBP. HEPF provides the voltage solution as Padé approximants, a ratio of two (analytical) polynomials in the embedding parameter α. The analytical nature of this solution expression was used to advantage in [1] to estimate the SNBP using a single power-flow solution. The analytical nature was also critical to generating nonlinear reduced-order equivalent networks that remain accurate even as the operating conditions change [5], [6]. Knowledge that the poles of the close-to-diagonal Padé approximants accumulate on Stahl s compact set [8], [9] was critical to efficient SNBP estimation. This analytical nature of the scalable HEPF solution is exploited in this paper, for fast estimation of the weak buses in a system, from a voltage stability perspective. This paper makes two contributions: First, by investigating the underpinnings of the theory behind the σ indices, we explain systematically why the procedure is unreliable in sections and. Section V provides numerical results that support the theory of sections and, and demonstrate shortcomings of the σ indices in estimating the weak buses and the SNBP of a system. Secondly, an alternative theoretically-grounded approach that can be used for fast estimation of weak buses in a system is proposed in section V. Section V provides numerical results for the EEE 14-bus and 118-bus systems, comparing the proposed HEM-based fast weak-bus estimation approach with traditional modal analysis. Finally, the conclusions are presented in section V.. PHYSCAL NTEPETATON OF SGMA NDCES The idea behind the σ indices is to, in essence, develop a

2 two-bus nonlinear equivalent for each bus of the power system, spanning that bus and the slack bus [10], an equivalent that preserves the slack bus voltage and the voltage at the retained bus. The parameters for this proposed reduced equivalent are calculated to be consistent with the simple twobus system comprised of a slack bus and a PQ bus as shown in Fig. 1, where Z is the line impedance, S is the complex-power injection at the PQ bus, V 0 is the slack bus voltage and V is the PQ bus voltage. Fig. 1 Two-bus system diagram The power balance equation for the PQ bus in Fig. 1, given by (1), can be re-arranged to obtain (), V V0 S (1) Z V U 1 U (). where U=V/V 0 is the normalized voltage and σ is defined as ZS. (3) V 0 The roots of (), which is a quadratic equation, are: U j (4) where σ and σ are the imaginary and real parts of σ, respectively. f the slack bus voltage is assumed to be controlled at 1.0 pu, the two roots represent the high- and low-voltage solutions for the given two-bus network. The two solutions meet at the point at which the radicand in (4) becomes zero, i.e. the SNBP. To ensure the existence of the high-voltage solution in a two-bus system, it is necessary that the radicand in (4) be positive. Thus, the condition to ensure that the two-bus system is short of or at its static voltage collapse point, called the σ condition, is given by: (5) For a multi-bus system, [10] in essence finds, for all system buses, two-bus equivalents structurally equivalent to Fig. 1 spanning the slack bus and bus of interest, where the PQ bus power injection is the native injection at this bus in the full network. One immediately sees a problem when trying to map a voltage-preserving two-bus equivalent (described below) onto the structure of Fig. 1: n a realistic system model, voltages with real parts less than 0.5 are common, but cannot occur for the high voltage solution in a two-bus system (assuming the slack bus voltage angle is 0 ) as shown in (4). (n fact, the low voltage solution for the two-bus equivalent constructed below (low voltage solution from (4)) corresponds to the high voltage solution of the full network for cases when the real part of the normalized voltage is below 0.5.) Because [10] does not account for this incompatibility, or that taking the low voltage solution would have resolved this incompatibility, their theory leads to inaccurate predictions. n order to reveal the underlying problem more clearly, the approach used by [10] (which in essence calculates the equivalent parameters of this two-bus-equivalent analog, along with the proximity to voltage collapse) is explained next. n the subsequent section, problematic numerical examples resulting from that approach are provided before a revised approach is proposed.. TWO-BUS EQUVALENTS USNG HEM AND NDCES The normalized voltages at all buses in a full system model solution can be calculated using HEM as power series of the embedding parameter, α, given by U(α). A two-bus equivalent may be constructed, and the σ index may then be obtained as a series of α, using (6) (which is structurally identical to ()). ( ) U ( ) 1 (6) U ( ) The σ series can then be evaluated at α = 1.0 using Padé approximants [11], to obtain σ indices. f a parabola in the complex σ plane is plotted as defined by the expression in (5), the distance from the σ indices in the plane to the surface of the parabola can be used to qualitatively estimate the distance from the SNBP, and the buses whose σ indices are closer to the surface are the weak buses according to [10]. To show it is incorrect to claim that the proximity of the σ condition to zero is a measure of voltage stability margin [10], consider the following. The U and σ series can be evaluated at α = 1.0, where the system loading may be adjusted so that α = 1.0 corresponds to any loading condition up to and including the SNBP, and (6) can be split into real and imaginary parts to obtain: UU U (7) U U U & U where U and U are the real and imaginary parts of the fullmodel solved bus normalized voltages respectively. The expressions for σ and σ from (7) can be substituted into the σ condition, (5), to obtain (8) U U 0.5 U U U U U Thus, it is seen from (8) that the proximity of the σ condition to zero, is equivalent to the proximity of the real part of the normalized bus voltage to 0.5. For a simple two-bus (non-equivalent) system, indeed the real part of the normalized voltage is 0.5 at the SNBP and hence the proximity of the σ condition to zero is an indicator of the proximity to SNBP. However, for a multi-bus system (and hence its proper twobus equivalent), this is not true and consequently, the magnitude of the σ condition is not a measure of closeness to the SNBP, but measures closeness to the point where the alternative root of (4) should be selected. Nor is the radicand of (4) approaching zero a reliable indicator of a weak bus as (8)

3 3 claimed in [10]; however, the buses whose σ conditions first approach zero as the system load increases, have larger phase angles, which cause the real part of the normalized voltage to drop below 0.5. V. NUMECAL ESULTS FO SGMA NDCES The inadequacy of the approach in [10] for estimating the SNBP when the system loading is lower than the SNBP loading can be demonstrated numerically using the EEE 118 bus system. The SNBP of this system occurs when all injections are multiplied by 3.18, i.e., at λ = 3.18, as obtained from MATPOWE [13], where λ is the scaling factor for all complex-power injections in the system. The complex σ indices at λ = 1.88 are plotted for all the buses in the complex σ plane and shown in the left plot of Fig.. At λ = 1.88, which corresponds to 60% of the maximum allowable load-scaling factor, many of the σ indices are very close to the surface of the parabola, i.e., limit imposed by the σ condition. This occurs because some of the normalized bus voltages have real parts very close to 0.5. As the system loading increases further, the real parts of the full-model normalized voltages decrease below 0.5, and hence the numerical value of the expression in (8) increases, causing the σ indices to move away from the surface of the parabola. This behavior is shown in the right plot of Fig., where the σ indices are plotted at λ = 3.1, a point much closer to the SNBP. Fig. Plot of σ vs. σ at λ = 1.88 and λ = 3.1 for the 118-bus system This behavior is confirmed by plotting the σ condition expression against λ as shown in Fig. 3 for the first 10 buses (based on native bus numbering) of the 118-bus system. t is seen that some of the buses come very close to violating the σ condition as λ increases, far before the SNBP is reached, and then start increasing in value again, which agrees with Fig.. Fig. 3 Plot of σ condition vs. λ for the 118-bus system f one searches for the point at which the σ condition is nearly violated (near zero), one will find that to occur for different buses at loading conditions well below the SNBP, and hence using the proximity of the σ condition to zero to estimate closeness to the SNBP would be misleading; however, if one searches for the smallest loading at which the σ condition becomes negative (which is theoretically impossible for voltage values short of the SNBP loading) using the Padé approximant of σ(α), one will obtain a reasonable estimate of the SNBP for the following reasons (provided a scalable form of the HEM formulation is used [1]). The Padé approximants of the bus voltage magnitudes are found to be largely monotonic up to the SNBP, beyond which the voltage function no longer exists, and the value of the Padé approximants begins to oscillate wildly as α increases due to the poles and zeros (Stahl s compact set) of the approximants. The location of closest real-valued pole/zero when surveyed over all system bus voltages has been found to be a tight upper bound on the true SNBP [1]. Beyond the SNBP, the oscillations in the Padé approximants of the bus voltages as α increases (a quantity which has no physical interpretation) also appear as oscillations in the value of the σ index (oscillations in σ(α) vs. α). These oscillations inevitably lead to a σ condition which becomes negative just beyond the SNBP. Thus the lowest load at which a negative value occurs in the σ condition, when surveyed over all buses in the system, is a good indicator of the location of the SNBP. This search method (for the negative σ(α) values) proposed by the authors can be thought of as being analogous (though not equivalent) to solving multiple power-flow problems, at increasing load levels, while searching for the point at which the powerbalance equations are not satisfied. When the modeled operating condition is even infinitesimally beyond the SNBP, the voltage functions do not exist but the Padé approximants of those voltage functions contain information that can be used to advantage. Beyond the SNBP the high power-balance mismatches are an indication that the system is now modeled to be beyond the SNBP. While it is true that the voltage function does not exist beyond the SNBP, the unacceptable power-balance mismatches can nevertheless be used as an indirect indication of this non-existence. We have validated this approach for finding the SNBP on systems with up to 6000 buses [1]. V. A DFFEENT APPOACH TO ESTMATE THE WEAK BUSES N A SYSTEM, USNG HEM The traditional method of estimating the weak buses in a system is modal analysis which involves calculating the eigenvalues of the portion of the reduced Jacobian that retains only the Q-V relationships and determining the buses that have the highest participation factors in the critical modes, with smallest eigenvalues [14]. An alternative to using modal analysis is to simply use Q V sensitivities to estimate the weak buses at any given operating point. The Jacobian matrix

4 4 (evaluated at that operating point) can be used to perform sensitivity analysis using the following linearized equation: P J P J PV (9) Q J Q J QV V When comparing the order of weakest to strongest buses from V-Q sensitivities and modal analysis, one does not expect a perfect one-to-one correspondence between the two, since modal analysis method does not provide the order of the buses with the highest to the lowest sensitivity but instead provides the order of buses for which the lowest eigenvalue has the largest contribution to V-Q sensitivity. However, as the smallest eigenvalue decreases (i.e., as the system load increases), the V-Q sensitivities at the buses with higher participation factors for the weakest mode depend to a greater extent on this smallest eigenvalue. Hence near the SNBP, and when the smallest eigenvalue is significantly numerically smaller than the next greatest eigenvalue, one can expect a strong correlation between the buses with high V-Q sensitivities and those with high participation factors in the weakest mode. However, both modal analysis as well as simply sensitivitybased analysis, have the drawback that they are based on linearization of the system at the given operating point. As the operating condition changes, the set of weakest buses can change, especially as the operating condition gets closer to the SNBP where the nonlinearity of the P-V curve is high. This section will exploit the scalable HEPF formulation presented in [1] to calculate the Q V sensitivities as a function of the load-scaling factor α, thus determining the weak buses of a system at all operating points through to the SNBP, by solving only one nonlinear power-flow problem. n the following discussion, first the use of HEM to obtain Q V at a given operating point will be explained, followed by an explanation of how this idea can be extended to use HEM to capture the nonlinearity of Q V sensitivities as a function of the operating condition, and estimate the weak buses at all operating points through to the SNBP, by simply evaluating the resultant Padé approximants. f one wants to calculate the Q V sensitivities using HEM for each bus with the injections at other buses remaining constant at a given operating point, one can use the HEM direction-of-change scaling formulation [1] to estimate the weak buses by scaling only the incremental reactive power injection of one bus at a time and calculating the sensitivity of its voltage magnitude at that bus with respect to its incremental reactive power injection, given by (10) (for PQ buses). Note that it is necessary that the incremental reactive power injection be the same for all buses, in order to ensure a fair comparison and in this work it is assumed to be 1 MVAr (1.0 pu on a 1 MVA base). Since the reactive power injection Q i(α) for PQ buses using direction-of-change scaling is given by (Q i+α), where Q i is the native reactive power load, its derivative w.r.t. α is 1.0. The sensitivity of the voltage magnitude with respect to the reactive power injection for the i th bus at the given operating point is obtained by evaluating (10) at α = 0 (since α = 0 corresponds to the base-case in the direction-of-change scaling formulation). Vi Vi Vi, i m (10) Q Qi i This process is repeated for all buses, with a new directionof-change for each bus that scales only the incremental reactive power injection at that bus in order to obtain the sensitivities. The computational complexity to determine the sensitivity at a given operating point is the same as that of using (9) to calculate the sensitivities, which can be explained as follows. When using HEM to determine the bus sensitivity at a given operating point, one needs to calculate only the constant term of the power series for V i(α) / Q i(α) (since it is evaluated at α = 0). To do this, one needs to calculate only two terms in the voltage power series, since at α = 0, the derivative V i(α) / α depends only on the α 1 term in V i(α). This is done for all the buses with a different direction-of-change for each bus, however the calculations are simple and the computation for different buses is completely parallelizable. Note that the constant term of V i(α) would be the same for all direction-ofchange cases since it depends only on the base-case operating condition, which is the same irrespective of the direction-ofchange of the scaling. Also in order to get the second term in the series in each case, only a linear system of equations is solved. Additionally, since only the constant term of V i(α) / Q i(α) is used, Padé approximants are not needed. However, if one wants to estimate the weak buses of a system over a range of operating conditions (for example, weak buses as the flow over a critical interconnection is increased), one can use the scalable HEPF formulation proposed in [1], to obtain the voltages as a function of the direction-of-change scaling factor α. One can then calculate the V-Q sensitivities using (10) as a function (Padé approximants) of α. From our experiments using test cases of sizes as large as 6000 buses, no more than 40 terms are usually needed to accurately estimate the SNBP [1]. Once the power series for V(α) is obtained, the calculation of the V i(α) / Q i(α) series and their Padé approximants is completely parallelizable. Once the Padé approximants of the V-Q sensitivities are obtained, one simply needs to evaluate them at different values of α, to obtain the sensitivities at an operating point where the injections in a specified direction of change are scaled by α (such as increasing the load in predefined sink area and the generation in pre-defined source area so as to increase the flow along a critical interface). Thus, the HEPF-based sensitivity formulation has the advantage that the resultant Padé approximant of the V-Q sensitivities is valid for all operating conditions through to the SNBP (provided a sufficient number of series terms are used, typically 40). The key computational advantage is that only one power-flow problem needs to be solved using HEM and then the sensitivities (as function of α) are obtained by simply plugging

5 5 different values of α in the Padé approximants. V. NUMECAL ESULTS FO WEAK-BUS ESTMATON USNG HEM-BASED SENSTVTY APPOXMANTS For the 14-bus system modified to contain only PQ buses, the top five weakest buses (in decreasing order of weakness ) obtained using modal analysis (using bus participation factors for the smallest eigenvalue) and the five buses with highest V- Q sensitivities (in decreasing order of sensitivity) obtained using the HEM direction-of-change scaling formulation are listed in Table 1 at the base-case and when the system is very close to the SNBP. The results for modal analysis presented in this section are obtained using VSAT (DSATools) [15], a widely used software package in the power system industry for voltage stability studies and weak-bus determination. Note that for the HEM method, the buses with positive reactive power injections (i.e., with local VAr support) are not considered, since they are unlikely to be the weak buses of the system from a steady-state voltage-stability perspective. Note that the order of the buses changes with the system operating condition, which is expected. Also note that the order of the top five weakest buses obtained using both modal analysis and V-Q sensitivities, (10), obtained using the HEM is identical for both operating conditions. The authors also tested the proposed method at five other operating conditions for this system and observed perfect correspondence in all cases. Loading level TABLE 1 WEAK BUS DETEMNATON USNG DECTON-OF-CHANGE HEM SCALNG FOMULATON AND MODAL ANALYSS (modal analysis) (HEM) Smallest eigenvalue Base-case 14, 1, 13, 11, 10 14, 1, 13, 11, All base-case Q i , 10, 13, 9, 11 14, 10, 13, 9, For a stressed 118-bus system with all loads and real-power generation scaled by 3.1, the set of ten weakest buses obtained from VSAT and the ten buses with highest V-Q sensitivities obtained using the HEM direction-of-change scaling formulation are rank listed in Table. The entries shown in bold are either in both sets or adjacent to a bus in both sets. Note that the SNBP for this system occurs when all injections are multiplied by As explained in the previous section, since the modal analysis method does not provide the buses with highest V-Q sensitivities but instead provides the buses for which the smaller eigenvalues have the largest contribution to sensitivity, the slight differences observed between the set of buses is to be expected. Loading level All base-case injections x 3.1 TABLE WEAK BUS DETEMNATON USNG DECTON-OF-CHANGE HEM SCALNG FOMULATON AND MODAL ANALYSS (modal analysis) 1,,0, 44, 43, 45,3, 38, 30, 5, (HEM) 44, 1,,117,43, 5, 0, 45, 53,86 Smallest eigenvalue V. CONCLUSON n summary, a simple modification of the σ condition (search for negative radicand rather than proximity to zero) can be used to estimate a tight upper bound on the SNBP with reasonable accuracy. This condition is satisfied only when the modeled load is such that the U and σ series no longer have a physical interpretation. Additionally, a different approach to estimating the weak buses in the system using HEM is proposed which is efficient and correlated well with traditional modal analysis, as demonstrated numerically for the 14-bus and 118-bus systems. However, the HEPF direction-of-change scaling formulation has the advantage that one can calculate the V-Q sensitivities at all operating conditions through to the SNBP, by solving a single power-flow problem. V. ACKNOWLEDGMENT The authors would like to thank Prof. V. Ajjarapu of owa State University for his invaluable feedback provided during the preparation of this manuscript. X. EFEENCES [1] A. Trias, System and method for monitoring and managing electrical power transmission and distribution networks, U.S. Patents 7,519,506 (009) and 7,979,39 (011). [] A. Trias, The Holomorphic Embedding Load Flow Method, Power and Energy Society General Meeting, pp. 1-8, July 01. [3] S. ao, Y. Feng, D. J. Tylavsky and M. K. Subramanian, "The Holomorphic Embedding Method Applied to the Power-Flow Problem," in EEE Transactions on Power Systems, vol. 31, no. 5, pp , Sept [4] A. Trias and J. L. Marín, The Holomorphic Embedding Loadflow Method for DC Power Systems and Nonlinear DC Circuits, EEE Transactions on Circuits and Systems : egular Papers, vol. 63, no., pp , Feb [5] Shruti ao, and Daniel Tylavsky, Nonlinear Network eduction for Distribution Networks using the Holomorphic Embedding Method, Presented at North American Power Symposium (NAPS), 016, Denver, CO, 016, pp [6] Y. Zhu, D. Tylavsky and S. ao, "Nonlinear structure-preserving network reduction using holomorphic embedding," in EEE Transactions on Power Systems, vol. PP, no. 99, pp [7] Y. Feng and D. Tylavsky, A novel method to converge to the unstable equilibrium point for a two-bus system, North American Power Symposium (NAPS), 013, Manhattan, KS, 013, pp [8] H. Stahl, On the convergence of generalized Padé approximants, Constructive Approximation, vol. 5, pp. 1 40, [9] H. Stahl, The convergence of Padé approximants to functions with branch points, Journal of Approximation Theory, vol. 91, no., pp , [10] A. Trias, Sigma algebraic approximants as a diagnostic tool in power networks, U.S. Patent 014/ , 014. [11] G. Baker and P. Graves-Morris, Padé approximants, Series: Encyclopaedia of Mathematics and its applications, Cambridge University Press, 1996, pp , 130. [1] Shruti D. ao, Daniel J. Tylavsky, and Yang Feng, Estimating the saddle-node bifurcation point of static power systems using the holomorphic embedding method, nternational Journal of Electrical Power and Energy Systems, vol. 84, pp. 1-1, 017. [13]. D. Zimmerman, C. E. Murillo-Sánchez and. J. Thomas, MATPOWE: Steady-State Operations, Planning and Analysis Tools for Power Systems esearch and Education, EEE Transactions on Power Systems, Feb. 011, vol. 6, no. 1, pp [14] B. Gao, G. K. Morison and P. Kundur, Voltage stability evaluation using modal analysis, EEE Transactions on Power Systems, vol. 7, no. 4, pp , Nov [15] Voltage Security Assessment Tool, DSATools, accessed in April 015, available at: