Precise Algorithm for Nonlinear Elements in Large-Scale Real-Time Simulator. Olivier TREMBLAY*, Martin FECTEAU, Pascal PRUD HOMME Hydro-Québec CANADA

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1 CIGRÉ CIGRÉ Canada Conference 21, rue d Artois, F PARIS Hilton Montréal Bonaventure http : // Montréal, Québec, September 24-26, 2012 Precise Algorithm for Nonlinear Elements in Large-Scale Real-Time Simulator Olivier TREMBLAY*, Martin FECTEAU, Pascal PRUD HOMME Hydro-Québec CANADA SUMMARY This paper presents an iterative nodal solver for nonlinearities implemented in Hydro- Québec s real-time simulation software. The proposed algorithm, based on fixed-point iteration, combined with the latest supercomputer technology allow precise simulation of nonlinear models with an extra simulation cost per additional iteration of less than 25%. Simulation results for part of the Gaspésie transmission system, where precise nonlinear modeling primary serves to represent overvoltage accurately, show faster than real-time execution for tasks containing many dozens of nodes and hundreds of elements. Moreover, due to numerous nonlinear elements (surge arresters and magnetic saturation of power transformers), the iteration method is required to get results without numerical instability. KEYWORDS Simulation - Real-time - Fixed-point iteration - Magnetic saturation - Surge arrester * Corresponding author: tremblay.olivier@ireq.ca 1

2 1. INTRODUCTION Today s electromagnetic transient simulations require increasingly sophisticated tools. Because of more demanding requirements in terms of the level of detail in modeling, it is essential to provide a simulation engine able to support, with good accuracy, common nonlinear elements. These elements play a significant role in the propagation of overvoltage and overcurrent transients. Offline simulators, such as the Electromagnetic Transient Program (EMTP), already implement precise simulation of such commonly occurring nonlinearities as magnetic saturation in transformers, surge arresters and natural switching devices [1] [2]. The Newton-Raphson technique is widely used in offline simulators to solve the nonlinear system due to its fast convergence. This advantage compensates for the large computational effort required to update the entire linear system as well as the Jacobian matrix [3] [4]. For real-time simulators, it is not an easy task to represent adequately nonlinear elements due to the computational burden. Hydro-Quebec s real-time simulator (Hypersim) is a large-scale multiprocessor simulator used for power system studies and for the development, validation, tuning and commissioning of control systems [5]. The computational effort is automatically spread across available processors using the natural propagation delay of the transmission lines. As a result, the large system impedance matrix is divided into several smaller submatrices which can be solved in parallel by many processors without introducing any error, thus drastically improving simulation speed [6]. For computational load reasons, the network equation solver of Hypersim uses pseudo-nonlinear models. This approach is valid for many cases but actual power system studies today involve more and more nonlinear components, which may introduce instability or reduce simulation accuracy. Supercomputers now have sufficient processing power to provide the larger computational effort required by a real-time iteration engine for nonlinearities. In order to exploit entirely this new computational capability, Hypersim s simulation engine required an overhaul. This paper presents the implementation of a real-time iterative algorithm for nonlinearities. The following section revisits the nonlinear modeling approach in Hypersim. The new real-time iterative algorithm, based on the fixedpoint iteration, and the new iterative nodal solver are then presented. The subsequent section studies the algorithm s performance and precision for a simple case study. The last section gives the results of a real-time simulation of part of the Gaspésie transmission system, which is very sensitive to any numerical imprecision. 2. BACKGROUND TO THE NONLINEAR MODEL Hypersim uses nodal representation to simulate network elements. By using the trapezoidal integration technique (an A-stable numerical method), the dynamic elements are represented by a resistance in parallel with a current source, representing its history values. At each time step, the simulation engine calculates the node voltage with the following equation: V = Y 1 I where Y, I and V are respectively the admittance matrix, the network sources and history injection, and lastly, the corresponding node voltage. The nonlinear elements are incorporated into the linear nodal solver by using a pseudononlinear approximation. Figure 1 shows the representation of a nonlinear resistance (surge arrester) by piecewise linear segments. B( G( Ihist( G( G ( = 1/ R( Vo ( B( = R( I = G( V B( km km + Figure 1: Surge arrester representation ( 2 L( )) G( = T / seg 1 Ihist λkm k 1) λo ( + G( seg L( I = G( seg ) V Ihist ( seg ) km km + ( ( ) ) Vkm( 1) ( = k Figure 2: Nonlinear inductance representation 2

3 Each segment is represented by a slope (corresponding to the resistance value R) and an open-circuit voltage V o. The equivalent admittance G and current injection B are then deduced from the slope equation. In the case of nonlinear inductance (Figure 2), the actual admittance G and the history value Ihist are obtained from the slope equation of the segment. The pseudo-nonlinear approach is fairly accurate and requires no additional computation when the element s operation point is kept inside a linear segment since the admittance contribution is kept constant. Changing network conditions may bring the operation point outside the boundaries of a segment, resulting in a change of admittance contribution to the network. A local algorithm is then used to find the new segment of the elements in order to deduce the new admittance value and the new current injection needed by the nodal solver. In the previous version of Hypersim, the updated values were only taken into account at the next simulation time step. For small system changes, this introduces a minor numerical imprecision affecting only the following time step. However, for severe perturbations requiring many segment changes (e.g., a fault or line energization), the error can cause non-negligible numerical error, possibly leading to instability. The major contribution of this paper is described in the next section, which presents an optimized iterative engine. 3. REAL-TIME ITERATIVE ALGORITHM A. Fixed-point iteration The fixed-point iteration, also called the x = g(x) method, is useful to obtain the root of a function f(x). This iteration technique, which is the basis of many important iteration theories, can be easily used to find the solution of linear and nonlinear network equations. The following steps describe the solution trajectory, using Figure 3: 1) Node voltages are calculated using the past admittance and the current injection of the nonlinear element. 2) With the voltage Vkm calculated in 1), the corresponding nonlinear segment (related to the nonlinear current Ikm) is obtained. 3) The new segment calculated in 2) changes the admittance contribution and the current injection, giving a new node voltage solution. 4) By using the new voltage Vkm calculated in 3), the corresponding segment current is obtained. 5) The final solution is found by updating the admittance contribution and the current injection of the nodal solver, in order to obtain the exact solution. Figure 3: Fixed-point iteration solution 3

4 Since the nonlinear equation is always monotonically increasing (natural logarithm forms), the solution occurs between the first two intermediate solutions, thus guaranteeing convergence of the iterative algorithm. B. Hypersim s new iterative nodal solver The actual solver recalculates node voltages for the entire substation at every segment change of a nonlinear element. Before the simulation starts, the admittance matrix is reordered to move down elements which may change their admittance contribution during the simulation (e.g., nonlinear elements and switches). This minimizes the time spent for re-factorization. Secondly, LDU decomposition and a forward-backward substitution are used to obtain the solution of node voltage V = Y 1 I during the simulation [7]. Begin time step Update Isrc Figure 4 shows the new iterative engine implemented in Hypersim. At every time step, the contribution of sources to Inode is determined before calculating the node voltage. A local algorithm is then used, for each nonlinear element, to find the actual segment corresponding to the node voltage. If a segment changes, the current injection and the new admittance contribution are updated before re-solving the node voltage. This iterative process is done as long as required. When the solution converges to the exact node voltage, element histories are updated in order to calculate the current injection Inode for the next time step. V=Y^-1.Inode Calculate nonlinear segment Segment changed? No Update history LDU = Y Update Inode Yes Furthermore, the iterative solution can be enabled or disabled for each nonlinear element. This granularity makes it possible to concentrate processing resources on key elements while non-critical ones are solved with the standard algorithm. In addition, the maximum number of iterations can be limited in order to meet the real-time computational constraint. 4. ALGORITHM PERFORMANCE Update Inode End time step Figure 4: Iterative engine flowchart Figure 5 shows the single-line representation of an equivalent network connected to a linear transformer feeding a load protected by a surge arrester. The parameters, presented at Table 1, are chosen in order to show the importance of the iteration method when nonlinear models are simulated. Figure 5: Single-line diagram for case study Parameters Source Load Xfo Arrester Values Amplitude: 245kV rms L-L Short-circuit level: 10,000 MVA X/R: 10 P = 30 MW, Q = 30 Mvar Vnom = 230/120 kv L-L Pnom = 300 MVA R = [0.0005, ] p.u. L = [0.05, 0.05] p.u. Rm = 500 p.u., Lm = 666 p.u. Typical parameters for a 120-kV ABB PEXLIM Q arrester Table 1: Case study parameters At t = s, a single-phase-to-ground fault is applied at the secondary side of the transformer (phase a). Figure 6 shows the simulation results (for phase c) when the fault is applied. At that moment, the full voltage is instantaneously applied on the surge arrester, forcing the current to 4

5 increase immediately. The number of maximum iterations is successively changed from 1 (no iteration) to 4 and the results are compared to those generated by EMTP-RV. With only two iterations, the results improve significantly in accuracy: the current calculated at the first iteration (8.83 A) drops to 3.65 A. At the third iteration, the error decreases with a current value of 2.37 A. Finally, with four iterations, the present algorithm and EMTP-RV both give the same solution with a current of 2.45 A. It is clear that accuracy improves as the number of iterations increases, confirming the ability of the fixed-point iteration algorithm to approach the solution at each intermediate step. In a real-time context, the number of iterations could be limited to meet computational constraints (for tasks containing many elements), knowing that the results are more accurate than without the iterative method. Figure 6: Simulation results for case study The case study is simulated in real-time on the latest SGI server technology, Altix UV. This high-end system is a shared-memory supercomputer system based on the Intel Xeon E7 family of processors and has sufficient processing power to provide the greater computational effort required by a real-time iteration engine for nonlinearities. The maximum computation time is about 1.5 µs (for two iterations) and increases by µs (25%) for each additional iteration. 5. SIMULATION RESULTS In order to demonstrate the efficiency of the new algorithm for a network integration study, it is applied below to a part of the Hydro-Québec Gaspésie transmission system, where numerical oscillations caused by the lack of iteration of surge arrester models produce major instabilities. The Gaspésie transmission system, presented in detail in [8], is characterized by series compensation, weak loading, numerous wind power plants, numerous large unloaded power transformers and long transmission lines. The ongoing study requires a very high level of detail with respect to the transmission system and wind power plants, resulting in a large number of nonlinear elements. This system of over 300 busbars (3-phase) contains a large number of nonlinear transformers and surge arresters on every high- and medium-voltage bus. A simulation time step of 24 µs is required to represent the natural propagation delay of the shortest transmission line (7 km). Figure 7 shows a section of the power system fed by an equivalent 315-kV source at Rivière-du-Loup (RDL). Given the transmission lines in this model, it is possible to divide the network computational burden among four processors. Previously, due to the real-time simulation constraint for this study, most nonlinear elements (e.g., surge arresters and magnetic saturation) were ignored, making the analysis of some phenomenon impossible. To complete the study, a high level of detail is required for nonlinear elements in order to analyze adequately overvoltage and overcurrent transients. For this reason, simulation is performed by activating the iteration for all nonlinear elements. 5

6 Figure 7: Gaspésie transmission system section Figure 8 shows the results for the phase b voltage at Bus 1, part of the CPU 2 task, when a 3-phase fault (lasting 3 cycles) is applied at the source at 0.2 s. Figure 8(a) shows that, without the iteration method for nonlinear elements (standard Hypersim surge arrester model), a fault applied under steadystate conditions leads to numerical instability. Figure 8(b) shows the general behavior when the iteration engine is enabled for nonlinear elements. It is clear that the accuracy of the results increases significantly and an overvoltage of 1.3 p.u. can now be observed. By zooming at the beginning of the fault (Figure 8(c)), it is possible to note the simulation transients and observe the slight improvement in accuracy when the number of iterations increases from 2 to 10. Although it is difficult to quantify the precision gain, testing shows that three iterations are sufficient to obtain good results for this network. Figure 8: Real-time transient voltage at Bus 1: (a) without iteration engine, (b) with iteration enabled, (c) zoom at the beginning of the fault with 2 (blue) and 10 (red) iterations. 6

7 Table 2 shows the calculation load for the four processors used in the simulation with the corresponding execution time and the number of iterations used. It can be observed that the heaviest calculation load (CPU 2) is simulated with a highest computation time of 16.9 µs for 2 iterations. This result means CPU 2 needs only 2 iterations to converge during the overall simulation. For CPU 4 (containing nonlinear elements changing abruptly operation points), the resulting execution time is 14.1 µs for 10 iterations. Note that for CPU 4, it is possible to limit the number of iteration to 3, for an execution time of 7.5 µs, without changing significantly the simulation results. In all case, the execution times are far lower than the 24 µs real-time constraint. CPU NB Nodes Transformers Passives Sources Nonlinear Execution time µs (2 iterations) µs (2 iterations) µs (2 iterations) µs (3 iterations) 14.1 µs (10 iterations) Table 2: CPU calculation load 6. CONCLUSION This paper showed the efficiency of the new iterative engine implemented in the real-time simulator Hypersim. The new nodal iterative solver based on fixed-point iteration was presented. A simple case study demonstrated that for an increase of less than 25% in calculation time, it is possible to eliminate numerical oscillations completely while increasing precision considerably. The increased calculation load is offset by the enhanced performance of the new SGI-UV processing units. Furthermore, an optimization of the real-time code generator now makes it possible to concentrate the calculation burden on the iterative elements only. A real-life network, based on part of the Gaspésie transmission system, was simulated in real time. With the new iteration engine, it is now possible to simulate this network including all nonlinear models without numerical instabilities, increasing the reliability and precision of the results. The computational time is well below the real-time constraint imposed by the calculation step even for tasks requiring many iterations to converge to the exact solution. The Hypersim simulator is now equipped with a simulation engine achieving a precision comparable to an offline simulator, allowing Hydro-Québec to perform system studies and to commission control systems on networks containing ever more complex nonlinear systems. 7. BIBLIOGRAPHY [1] H.W. Dommel. Nonlinear and Time-Varying Elements in Digital Simulation of Electromagnetic Transients (IEEE Transactions on Power Apparatus and Systems, Vol. PAS-90, No.6, pp , Nov. 1971). [2] J. Mahseredjian, L. Dube, Ming Zou, S. Dennetiere, G. Joos. Simultaneous solution of control system equations in EMTP (IEEE Transactions on Power Systems, Vol. 21, No. 1, pp , Feb. 2006). [3] Y. Chen, V. Dinavahi. An Iterative Real-Time Nonlinear Electromagnetic Transient Solver on FPGA (IEEE Transactions on Industrial Electronics, Vol. 58, No. 6, pp , June 2011). [4] B. Asghari, V. Dinavahi. Real-Time Nonlinear Transient Simulation Based on Optimized Transmission Line Modeling (IEEE Transactions on Power Systems, Vol. 26, No. 2, pp , May 2011). [5] V. Q. Do, J.-C. Soumagne, G. Sybille, G. Turmel, P. Giroux, G. Cloutier, S. Poulin. Hypersim, an Integrated Real-Time Simulator for Power Networks and Control Systems (ICDS 99, Vasteras, Sweden, May 1999, pp.1-6). [6] D. Paré, G. Turmel, J.-C. Soumagne, V. A. Do, S. Casoria, M. Bissonnette, B. Marcoux, D. McNabb. Validation tests of the Hypersim digital real time simulator with a large AC-DC network (Proceedings of the International Conference on Power Systems Transients, New Orleans, LA, Sept. 28 to Oct. 2, 2003, pp ). 7

8 [7] V.Q. Do, D. McCallum, P. Giroux, B. De Kelper. A backward-forward interpolation technique for a precise modelling of power electronic in HYPERSIM (Proceedings of the International Conference on Power Systems Transients, IPST 2001, Rio de Janeiro, Brazil, June 2001). [8] R. Gagnon, M. Fecteau, P. Prud Homme, E. Lemieux, G. Turmel, D. Paré, F. Duong. Hydro- Québec Strategy to Evaluate Electrical Transients Following Wind Power Plant Integration in the Gaspésie Transmissions System (IEEE Transactions on Sustainable Energy, Vol. PP, No. 2, pp. 1-10, July 2012). 8