M-Arctan estimator based on the trust-region method Yacine Hassa¨ ıne Benoˆ ıt Delourme Pierre Hausheer French transmission system operator (RTE) RTE RTE Yacine.Hassaine@rte-france.com Benoit.Delourme@rte-france.com Patrick Panciatici Eric Walter RTE L2S (CNRS-SUPELEC-U.Paris Sud) Patrick.panciatici@rte-france.com Walter@lss.supelec.fr Abstract - In this paper a new approach is proposed to increase the robustness of the classical L2-norm state estima- tion, to achieve this task a new formulation of the Levemberg Marquardt algorithm based on the trust-region method is applied to a new M-estimator, denoted M-Arctan. Results obtained on IEEE network up to 300 buses are presented Keywords - State estimation, robustness, trust region method, M-estimator. 1 Introduction Electrical power problems provides challenging ap- plications to mathematics in general, and state estimation in particular. In the early seventies, Schweppe [14] in- troduced static state estimation in the context of power network analysis. L 2 -norm estimation combined to the Gauss-Newton algorithm was the approach mainly used. This contribution is also related to static state estimation. The state vector is thus assumed invariant ([12], [11] and [6]) ; several cost functions were studied : L. Mili et al. [10] used Huber’s M-estimator, R. Baldick et al. [1] used a non-quadratic cost function closely related to Hu- ber’s M-estimator and M.K C ¸ elik et al. [3] proposed a weighted L 1 norm estimator using constrained optimiza- tion. In the most general case, the state vector may consist of a mixture of real and Boolean variables [7]. Here, the state vector is supposed to be real valued. In this paper we propose a new algorithm to increase the robustness of the classical L 2 state estimator based on the Gauss-Newton algorithm. To achieve this task, two types of robustness are defined. (i) Statistical robustness, aims at decreasing the influence of outliers on the estima- tion quality [8]. (ii) Numerical robustness, aims at guaran- teeing convergence of the algorithm in ill conditioned en- vironment or in the presence of outliers. The outliers may correspond to bad measurements, topology errors or erro- neous characteristics of branches. The non linear model is given by z = h(x)+ ǫ, (1) where z is the m-dimensional measurement vector, x is the n-dimensional state vector, h is a nonlinear vector function and ǫ is the m-dimensional vector of measure- ment errors. The L 2 estimator is known to be very sen- sitive to outliers in contrast with the M-estimators and the Gauss-Newton algorithm applied to L 2 estimation reaches its limits when the optimum is ill conditioned or singular [13]. In this paper an examples illustrating the problem is given in a context of maximum power transfer capacity. To increase statistical robustness, a new strictly convex M-estimator, denoted M-Arctan, is introduced. To improve numerical robustness, a trust-region algorithm is used [5] with a new approach to calculating the Lagrange multiplier µ. The algorithm obtained is applied to the M-Arctan estimator to obtain an estimation that is both statistically and numerically robust. The paper is organized as follow. Section 2 introduces the problem of ill conditioning and details the special case of maximum power transfer capacity. Section 3 propose the new M-estimator denoted M-Arctan. Section 4 details the Levemberg Marquardt algorithm based on the trust- region method. Finally the approach proposed is tested in Section 5. 2 Problems of ill conditioned optimum Several problems may cause the divergence of state estimation based on a Newtonian algorithm. In this sec- tion, one of them namely the problem of maximum power transfer capacity is studied. Consider a network with two busses connected by a branch with a known admittance y. Assume that a power flow measurement and a phase reference on one of the two busses are available. Assume further that voltage magnitude is fixed in the two busses. The objective of this example is to estimate the unknown phase θ. Under these conditions, one can write T = v 2 y sin(θ)+ ǫ, (2) where ǫ is a white noise, T is the active measurement, v is the know voltage magnitude and θ is the unknown phase. To estimate θ, we minimize the quadratic cost function J (θ)=(T − v 2 y sin(θ)) 2 , (3) The minimizer  θ of (3) satisfies the stationarity condition dJ dθ =0, (4) so −v 2 y cos(  θ)(T − v 2 y sin(  θ)) = 0, (5) 15th PSCC, Liege, 22-26 August 2005 Session 26, Paper 2, Page 1