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Hybrid power flow analysis : Combination of AC and DC models Soobae Kim, Student Member, IEEE and Thomas J. Overbye, Fellow, IEEE Department of Electrical and Computer Engineering University of Illinois Urbana-Champaign Urbana, IL 61801 USA Email : [email protected] and [email protected] Abstract—This paper presents a hybrid approach with AC and DC power flow models for power flow analysis. Power flow problems are solved with the AC model in a part of system, whereas with the DC model in other parts. The proposed methodology can obtain benefits from both AC (non-linear) and DC (linear) models. In other words, this approach formulates both nonlinear and linear algebraic equations, and deals with tradeoff issues between accuracy of the AC model and faster solution of the DC model. A case study with an IEEE 14-bus system is provided to compare accuracy between the proposed and the DC model. It is shown that the hybrid approach can be used for applications requiring more accuracy than DC model allows. Index Terms—Power system analysis, Power flow model, AC power flow, DC power flow I. INTRODUCTION Power flow analysis is used to solve steady-state operating conditions of electric power systems. It is the most essential tool for power system analysis and is applied to planning, economic scheduling, system operation and control areas. Therefore, it has been constantly required to develop the efficient power flow analysis method with the widespread of computer technologies [1] and especially, as the increase of online application for wide area monitoring and control in the field, it might be more stressed to develop the reliable and efficient power flow methodology than before. The classic power flow problem (AC model) can be formulated based on Kirchhoff’s circuit law in electric power networks and is a set of nonlinear algebraic equations representing real, reactive power injection flow at each node. The nonlinear equations can be solved with an iterative algorithm. The Newton-Raphson (NR) method is the most powerful algorithm using a linearizing technique. However, because of inherent characteristics of nonlinear equations, the AC model has convergence difficulties and is time consuming, especially when contingency analysis is considered [2]. To overcome these problems, several approximate models using physical properties of power systems have been studied [3-6]. Decoupled and DC power flow models are representative. The decoupled model is based on the observation that the 978-1-4244-8052-4/11/$26.00 ©2011 IEEE interactions are weak between real power and voltage magnitude and between reactive power and voltage angle. This weakness allows related elements of the Jacobian matrix to be ignored, and thus the computation expense for the power flow solution can be reduced [5]. The DC model additionally assumes that voltage magnitudes at all buses are flat and reactive power flows are neglected. This further approximation of the DC model creates a linear problem and thus it has a very fast solution. According to [7], the DC model is 60 times faster than the AC model. Thanks to the fast solution, the DC power flow model has become a commonly used analysis technique in power systems and market applications. Examples include contingency analysis, calculation of power transfer distribution factor (PTDF), line outage distribution factors (LODF), transmission interchange limit analysis, market clearing engine (MCE) and financial transmission right (FTR) [7]. However, because of inherent approximations, the DC model may cause large errors in these applications [6]. In [8], guidelines are presented to reduce the amount of error. Lu, et al. [7] proposed method to enhance DC model with a correction term, based on historical data while still maintaining its computational efficiency and linear formulation. Notwithstanding much effort, precision of the DC method is fully dependent on configuration and operating conditions of individual power systems. Thus it is crucial to carefully implement DC model. This paper presents a hybrid approach by combining the AC with the DC models in the problem formulation. Both the time requirements of the AC model and the inaccuracies of the DC model are reduced. Thus, the hybrid model is stronger than either the AC or DC models operating alone. The paper is organized as follows. Section II presents a brief analytic basis of the power flow problem. The proposed approach is presented in Section III. And Section IV illustrates the results from the IEEE 14-bus system. The conclusion is made in Section V. II. FORMULATION OF THE POWER FLOW PROBLEM This section briefly explains the formulation of the AC and the DC power flow models. A. AC power flow model AC power flow equations can be formulated with four variables at each bus: voltage magnitude, angle, and real and reactive power injection. At each bus, two of four above variables are known and the remaining two are obtained by solving a set of nonlinear power flow equations. Based on nodal analysis and with specified constraints on power and/or voltage at bus k, the complex power can be derived as: Sk = Pksp + jQksp = Vk I k * = Vk [∑YknVn ]* (1) As these complex equations are not analytic functions of the complex voltages, it is necessary to separate their real and imaginary parts in order to apply the NR method. Depending on bus types: load or generator bus, the power problems are as follows: At load bus (PQ) k, Pk = Vk ∑Vn [Gkn cos(θk − θn ) +Bkn sin(θk − θ n )] (2) Qk = Vk ∑Vn [Gkn sin(θk − θ n ) − Bkn cos(θk − θn )] (3) At generator bus (PV) k, Pk = Vk ∑Vn [Gkn cos(θk − θn ) +Bkn sin(θk − θ n )] (4) Vk = Vksp Pk = ∑ Bkn (θ k − θn ) (6) The real power system does not always match these assumptions. So, [6] analyzed the assumptions in order to apply the DC power flow model to a real field. To contain error to be within 5% on average, the paper reports that the ratio (X/R) should be greater than 4 and the standard voltage deviation should be less than 0.01. In addition, it showed that the assumption of a small angle difference is quite reasonable. However, even following the guidelines, some line power flows have significant inaccuracy compared to AC results. III. PROPOSED METHODOLOGY The proposed approach formulates the power flow problem by combining the AC with the DC models. In other words, power flow problems are solved with the AC model for certain buses and the DC model for others. Power flow equations which require accurate solution are formulated with the AC model and the rest are done with the DC model for a faster but less accurate solution. Then, NR method can be used to solve the reduced set of nonlinear equations. Figure 1 shows the procedure. (5) Where Pk / Qk is a real/reactive power injection at bus k, Ykn = G kn + jB kn is an element of the system admittance matrix, Vk and Vn are the voltage magnitude of the two end buses of a line, and θk / θn are the voltage angles at bus k/n respectively. The NR algorithm is a common method to solve a set of nonlinear equations based on the Taylor series expansion. It is characterized by quadratic convergence. Most problems with the NR method converge around five iterations [1]. The computation time is fully dependent on system size, and a single solution for a large system, such as NERC 43,000, may take as long as a few seconds [2]. B. DC power flow model The DC power flow model simplifies the AC model with the following assumptions: 1. Voltage magnitudes on all buses are 1 p.u. 2. Voltage angle differences are small: sin(θk − θn ) ≈ θk − θn , cos(θ k − θ n ) ≈ 1 3. Line resistance is negligible: Rkn << X kn , Gkn ≈ 0 4. Reactive power injections on all buses are ignored. Based on the above assumptions, the real power flow in (2) can be approximated as: Figure 1. Procedure for the hybrid approach From this approach, we hope to take advantages from both the AC and the DC models. Accuracy in areas of interest can be maintained with exact nonlinear equations, and computation expense can be reduced with approximate linear equations. This could result in an improved power flow model with a faster solution without sacrificing accuracy in areas of interest. A. AC and DC Area selection The area can be divided depending on the importance of a certain section with a viewpoint of power system operation and control. This work calls the important area the AC region and the other the DC region. Some buses in the DC region which have a connection with buses in the AC region are called BD buses (Boundary bus). Figure 2 shows the AC/DC/BD division of a simple power system. If we assume that Buses 1, 2 and 3 are in the AC region, then Buses 4 and 6 may be BD buses, which are connected with one of buses in the AC region. Bus 5 is in DC region. 13 12 14 6 9 C 11 5 10 G C 1 4 2 7 8 3 G C Figure 3. IEEE 14-bus system Figure 2. AC, BD and DC area selection B. Power problem formulation and boundary assumption For the problem formulation, this approach assumes that each bus in the AC region has real and reactive power injection or voltage magnitude information depending on the bus type. The buses in the DC region have only real power injection as a given condition. Then power flow equations in the AC and DC regions can be formulated with (2), (3), (4), (5) and (6). The BD bus has a crucial impact on the accuracy of the states in AC region because they are fully dependent on each other in power flow equations. But, the given condition for a BD bus is only real power injection. Therefore, a proper guess for reactive power injection or voltage magnitude of the BD bus is required to improve accuracy in AC region. A best guess may be a flat voltage approximation based on the power system operation and control. In order to obtain accurate angle solution in the BD bus, this work builds nonlinear real power flow balance equations for the BD bus. In other words, the approach assumes that the BD bus is considered to be a PV bus in AC region with 1 p.u. voltage magnitude. All line power flows which are larger than 1 MW are tabulated in Table 1. In order to compare the power flow errors, the mean absolute percentage error (MAPE) of the active line power flow are calculated by taking AC results as a reference: MAPE = From To PAC − Ptest × 100% PAC Real power flow (MW) AC DC (9) MAPE (%) Hybrid DC Hybrid 1 2 159.6 147.9 157.9 7.4 1.1 1 5 75.3 71.1 74.5 5.6 1.1 2 3 75.9 70.1 75.5 7.7 0.6 2 4 55.7 55.2 55.2 0.8 0.9 2 5 41.1 40.9 40.4 0.4 1.6 3 4 -21.1 -24.1 -21.5 14.5 1.9 4 5 -60.0 -62.3 -60.6 3.9 1.0 4 7 28.0 29.6 27.9 5.8 0.6 IV. CASE STUDY 4 9 16.2 17.2 16.1 6.2 0.5 A case study has been conducted on an IEEE 14-bus system, which has 2 generators, 3 synchronous condensers and 20 transmission lines. The test system is slightly modified by removing the reactive power limit of the generators and synchronous condensers. Fig. 3 shows a one-line diagram of the test case system. For the hybrid approach, buses 1, 4, 5 and 14 are selected for the AC region and, according to the network configuration, buses 2, 3, 6, 7, 9 and 13 are the BD bus and the remaining are in the DC region. 5 6 44.1 45.2 42.2 2.3 4.5 6 11 7.4 6.3 6.0 14.3 18.3 6 12 7.8 7.5 7.3 3.3 6.1 6 13 17.8 17.0 17.6 4.2 0.8 7 9 28.0 29.0 27.9 3.4 0.6 9 10 5.2 6.2 4.5 18.3 13.9 9 14 9.4 9.9 9.9 5.0 5.1 10 11 -3.8 -2.8 -3.3 25.8 13.8 13 14 5.6 5.0 5.2 11.9 8.6 7.8 4.5 Average Table 1. Line power flow test results The average MAPE of all lines is 7.8% for the DC power flow model and 4.5% for the hybrid approach. It is noted that the mean error in the AC region is 2.4%. This is a 70% reduction compared to the result from the DC model. Therefore, this approach can give results in the areas of interest similar to the approach with AC model in entire system. V. CONCLUSION This paper explores the development of power flow algorithms for combining an accurate AC model with an approximate DC model. This approach results in an improved power-flow algorithm which has a fast solution without sacrificing accuracy in areas of interest. The test results show the hybrid approach is more accurate than using the DC model. This approach can be utilized in a variety of power system applications and could be a basis of further investigations for the study of power system analysis. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] B. Stott, "Review of load-flow calculation methods," Proceedings of the IEEE, vol. 62, pp. 916-929, 1974. T. J. Overbye, Xu Cheng and Yan Sun, "A comparison of the AC and DC power flow models for LMP calculations," in System Sciences, 2004. Proceedings of the 37th Annual Hawaii International Conference on, 2004, pp. 9 pp. W. F. Tinney and C. E. Hart, "Power Flow Solution by Newton's Method," Power Apparatus and Systems, IEEE Transactions on, vol. PAS-86, pp. 1449-1460, 1967. B. Stott and O. Alsac, "Fast Decoupled Load Flow," Power Apparatus and Systems, IEEE Transactions on, vol. PAS-93, pp. 859-869, 1974. R. Kaye and F. Wu, "Analysis of linearized decoupled power flow approximations for steady-state security assessment," Circuits and Systems, IEEE Transactions on, vol. 31, pp. 623-636, 1984. B. Stott, J. Jardim and O. Alsac, "DC Power Flow Revisited," Power Systems, IEEE Transactions on, vol. 24, pp. 1290-1300, 2009. S. Lu, N. Zhou, N. P. Kumar, N. Samaan and B. B. Chakrabarti, "Improved DC power flow method based on empirical knowledge of the system," in Transmission and Distribution Conference and Exposition, 2010 IEEE PES, 2010, pp. 1-6. K. Purchala, L. Meeus, D. Van Dommelen and R. Belmans, "Usefulness of DC power flow for active power flow analysis," in Power Engineering Society General Meeting, 2005. IEEE, 2005, pp. 454-459 Vol. 1.