State estimation in distribution grids

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1 State estimation in distribution grids Ratmir Gelagaev, Pieter Vermeyen, Johan Driesen ELECTA ESAT, Katholieke Universiteit Leuven, Belgium, Abstract Problems and techniques for estimating the state of an observable distribution grid are investigated. A distribution grid is observable if the state of the grid can be fully determined. For the simulations, the modified 34-bus IEEE test feeder is used. The measurements needed for the state estimation are generated by the ladder iterative technique. Two methods for the state estimation are analyzed: Weighted Least Squares and Extended Kalman Filter. Both estimators try to find the most probable state based on the available measurements. The result is that the Kalman filter mostly needs less iterations and calculation time. The disadvantage of the Kalman filter is that it needs some foreknowlegde about the state. I. INTRODUCTION Due to the liberalization of the energy market and increased environmental awareness there is a tendency in the direction of distributed generation. In centralized distribution, the power flows in one fixed direction namely from large central generation units to the consumers. When there are many small generators used in the distribution grid, the direction in which the power flows cannot be unambiguous determined. This will be the case in the state of normal working as well as under fault conditions what can have many safety issues. A possible solution can be found in the state estimation of the grid. State estimation is broadly used in the transmission systems, but for distribution systems, which are radial in nature, this is not the case. The result of the state estimation is the most probable state of the system based on the quantities that are measured. The state of a distribution grid, like the state of a transmission system, is defined by the values of the voltages V and the phase angles θ at every node of the grid. If these values are known then the value and the direction of the power in each branch of the grid can be calculated. Placing measurements is very expensive. Therefore is it important to determine the minimal number of measurements to be placed so that the system becomes fully observable. Besides this there should be some redundant measurements so that the bad data coming from the meters can be detected and identified []-[5]. If there are not enough measurements it is not possible to estimate the full state of the grid. In this case the grid is called unobservable. The only way to restore the observability is by placing extra measurements. Another method is to use the available measurements and partly estimate the state of the grid. Here is the unobservable grid divided in small observable grids from which the states are separately estimated. However in what follows it is assumed that there are enough measurements so that the grid is fully observable. II. SIMULATION GRID AND MEASUREMENTS In Fig. is the modified IEEE-distribution grid [8] shown which is used for the simulations. Important remarks are that this grid is the one phase equivalent of the three phase IEEE grid and the state estimation is done in steady state. For the two other phase the state estimation can be done separately in the same way as shown later. Each arrow at a node represents a user consuming a certain complex power S. There are two generators coupled at nodes 5 and 3. These generators can be seen as negative loads. In case of induction generators there is an active power P injected in the grid and a reactive power Q taken from the grid. By using power electronics or capacitors it is possible to regulate the reactive power Q. Besides the generators there are also two capacitor banks coupled to the grid. The capacitor banks reduce the current flows through the distribution system by compensating the reactive load locally. Hence they reduce system losses and improve the voltage profile. This is why there are two condensators at nodes 9 and 7. ~ Fig.. S S3 S4 S6 S7 S S5 S9 9 S S S Cap9 S5 S3 S4 S3 3 4 ~ S4 S S9 Gen5 The simulated IEEE-distribution grid 5 S6 S9 S8 S S S5 S S8 Cap7 S6 S S3 S3 S ~ Gen3 To estimate the state of a system there are some quantities that should be measured. In case of a distribution grid these can be the active and reactive power, current or voltage measurements. Because every measurement has an error there will be contradictions in the measurements [9]. In state estimation these measurement errors are filtered to obtain optimal estimates. Because of the delays in the communication it is not possible to do all measurements simultaneously. Therefore a certain time delay between the measurements is tolerated. This is justified by the slow changes of the state of the distribution grid under normal circumstances. In practice only the line impedances and the rated voltage V rated V rated = 49ej 3 in the understation are known. Here is 4376e j V for one phase. The values S33 S /8/$5. 8 IEEE

2 of the current and voltage at each node, the active and reactive power in every branch have to be measured. For the simulation offline it is impossible to tell what these values are. Otherwise if the line impedances, rated voltage V rated and the complex power S i at each node are known then the values of the current, voltage, active and reactive power can be calculated using the so called ladder iterative technique [7]. Each iteration of the ladder iterative technique exists of two steps: forward sweep and backward sweep. For the modified IEEE-distribution grid the calculation is done as follows: Forward sweep: Assume rated voltage V rated at the end node voltages (V 5, V, V 4, V 8, V, V 4, V 9, V 3 and V 34 ). Start with ( the) node 3 and compute the node current S3 I 3 = V Apply the Kirchhoff s current law to 3 determine the current flowing from node 3 toward node 3: I (3 3) = I 3. Compute with this current the voltage V 3 = V 3 + Z (3 3) I (3 3). Node 3 is a junction node. ( Select) node 34 and compute the node current I 34 = S34 V Apply the Kirchhoff s current law to determine the current flowing from node 33 toward node I (33 34) = I 34. Compute with this current the voltage V 33 = V 34 + Z (33 34) I (34 34). ( ) S33 Compute with this voltage the current I 33 = V 33 Apply the Kirchhoff s current law to determine the current flowing from node 3 toward node 33 I (3 33) = I (33 34) + I 33. Compute with this current the voltage V 3 = V 33 + Z (3 33) I (3 33). This will be referred to as the most recent voltage at node 3. Compute with( the ) most recent voltage at node 3 the S3 current I 3 = V Apply the Kirchhoff s current law 3 to determine the current I (3 3) = I (3 3) +I (3 33) + I Using the current I ( ) compute the voltage V = V + Z ( ) I ( ). At the end of the forward sweep the magnitude of the compute voltage V is compared to the magnitude of the rated voltage V rated : Error = V rated V If the error is less than a specified tolerance, the solution has been achieved. A typical tolerance is. per unit what for 4376V equals to 4, 376V. If the error is greater than this tolerance, the backward sweep begins. The backward sweep begins at the node with the rated voltage V rated = 4376e j V and all currents from the forward sweep. Backward sweep: Start with node and V = V rated. Compute the voltage V = V Z ( ) I ( ). Compute the voltage V 3 = V Z ( 3) I ( 3).... Compute the voltage V 34 = V 33 Z (33 34) I (33 34). After the backward sweep the first iteration is completed. At this point the forward sweep will be repeated, only this time starting with the new voltage at end nodes. These steps will be repeated until the error is less than the specified tolerance. At the understation the voltage is mostly taken 5% bigger than the rated voltage. The computed voltage, active and reactive power at every node are shown in Fig.. It should be noted that the actual sequence of the nodes is not the same as in the Fig.. Actual voltage [V] Active/Reactive power [Watt/Var] Fig...6 x Voltage Nominal voltage x Active power consumption Reactief vermogen verbruik Voltage, active and reactive power in nodes III. STATE ESTIMATION WITH WEIGHTED LEAST SQUARES The Weighted Least Squares (WLS) method tries to determine the most probable state of the system given the quantities that are measured [9]. The assumptions that are made by the WLS method are that: The measurements are normal distributed. The mean value of the measurement errors is zero: E(e i )=for i =,...,m with m the number of the measurements. The measurement errors are independent: E[e i e j ]=. To determine the most probable state, the WLS method minimizes the sum of the weighted squares of the residuals: m min W ii ri () i= s.t. z i = h i (x)+r i for i =,...,m () where r i is a normal distributed random variable noise term and h i (x) is a nonlinear function which expresses the measured quantity z i in terms of the state variable x. In case of distribution grid, z i consists of voltage, current, active and reactive measurements. The weights W ii represent the certainty on the measurements. The certainty of a measurement is represented by the inverse of its error variance σ. As can be seen from the equations () and () the optimal solution is found by computing the state variable x which minimizes the weighted squares of the residuals.

3 3 The solution of the equation () is found iteratively. The objective function that is being minimized can be written in the matrix form as follows: m J(x) = (z i h i (x)) /R ii i= = [z h(x)] T R [z h(x)] (3) where R is a covariance matrix that plays the role of the weights (R = W ): σ σ Cov(e) =E[e i e j ]=R =... (4) σm The minimum of the equation (3) can be found with the following algorithm [9]: [G(x k )] x k+ = H T (x k )R [z h(x k )] (5) where, x k is the state variable in k-th iteration where one of the phase angles is chosen as a reference (slack). Mostly it is the phase angle at the understation. In this case it is node. The state vector of a distribution grid with n nodes has n elements. x T =[θ θ 3...θ n V V...V n ] (6) H(x k ) is the jacobian which has the number of rows equal to the number of measurements and the number of columns equal to the number of the states minus one because one phase angle is taken as slack H(x k )= P inj P flow Q inj Q flow I mag P inj P flow Q inj Q flow I mag mag and G(x k )=H T (x k )R H(x k ) is the so called gain matrix. IV. STATE ESTIMATION WITH EXTENDED KALMAN FILTER Like the WLS method the Kalman Filter also determines the most probable state of the system. Kalman Filters represent the belief bel(x k ) of the state at time step k by the mean μ k and the covariance Σ k []. One of the conditions of the Kalman Filter is that the measurements can be expressed as linear function of the state. However a distribution grid is described by nonlinear equations what means that the measurements cannot be expressed as linear function of the state. Therefore the Kalman Filter cannot be used for the state estimation of the grid and the nonlinear or the so-called Extended Kalman Filter (EKF) should be used. The EKF defines the state (7) transition function and the relation between the states and the measurements by means of the nonlinear functions g and h respectively: x k = g(u k,x k )+ε k (8) z k = h(x k )+δ k (9) where ε k is the uncertainty introduced by the state transition and δ k is the measurement noise. Their means are zero and the covariances are Q n n and R m m respectively. EKF works with the linearization of the nonlinear functions g and h around a working point. Accordingly like the Kalman Filter, the EKF represents the belief bel(x k ) at time k by mean μ k and covariance Σ k. The goal of the EKF is thus approximating the exact belief bel(x k ) through a Gaussian distribution completely described by its mean and covariance. The calculation of μ k and Σ k is done in two steps: - Prediction - Correction μ k = g(u k,μ k ) () Σ k = G(u k,μ k )Σ k G T (u k,μ k )+Q () K k = Σ k H T (μ k )[H(μ k )Σ k H T (μ k )+R] () μ k = μ k + K k [z k h(μ k )] (3) Σ k = [I K k H(μ k )]Σ k (4) where the nonlinear function g(u k,x k ) is linearized around mean μ k, G(u k,μ k )= g(u k,x k ) x k xk =μ k and Q is the covariance of the uncertainty introduced by the state noise. The matrices H, h and R are the same as for the WLS method. In the prediction step (equations ()-()) the belief bel(x k ), represented by μ k and Σ k, is predicted without incorporating the measurement z k. In the correction step (equations ()- (4)) the belief bel(x k ) is subsequently transformed into the desired belief by incorporating the measurement z k. The variable K k, computed in equation () is called Kalman gain. It specifies the degree to which the measurement is incorporated into the new state estimate. In general for every iteration in the EKF there is a new measurement used in the correction step to correct the state. However the state of a distribution grid changes very slowly. Therefore it can be assumed that the mean does not change during the estimation process. Because of this the equation () becomes μ k = μ k (5) From this it follows that the function g in equation () projects the state μ k to the state μ k. This means that it is approximated by a linear function with the unit slope. Thus the matrix G(u k,μ k )= g(u k,x k ) x k becomes an xk =μ k identity matrix I. From equations () and (5) it also follows that the function g is independent from u k. Finally the EKF becomes: - Prediction x k = x k (6) Σ k = Σ k + Q (7)

4 4 - Correction K k = Σ k H T (x k )[H(x k )Σ k H T (x k )+R] (8) x k = x k + K k [z k h(x k )] (9) Σ k = [I K k H(x k )]Σ k () The EKF can be used instead of the WLS method if there is a foreknowledge about the state. This foreknowledge is incorporated by the covariance matrix Q. V. SIMULATION RESULTS Consider the modified IEEE distribution grid in Fig.. This grid is simulated for both estimation methods: WLS and EKF. As initial state x all phase angles and voltages are taken equal to the rated values: x =[θ...θ n,v...v n ]=[..., ] As said earlier the phase angle at the node is taken as reference. The convergence criterion is that the difference between the successive voltage values is less than V and the difference between the successive phase angle values is less than. degrees: Δx k+ = [Δθ,k+...Δθ n,k+, ΔV,k+...ΔV n,k+ ] < [....,...] () A. Weighted Least Squares Assume that only the active and reactive power flow measurement are done in every branch. The results are shown in Fig. 3 and 4. In the top graphs of Fig. 3 and Fig. 4 the actual values of respectively the voltage and phase angle (full line) are shown with their estimations (dot line). Error [V] Maximum error [V] Voltage [V] Fig x 4 Actual voltage ( ) and estimated voltage ( ) Voltage results The errors between the actual and estimated values are shown in the middle graphs of Fig. 3 and Fig. 4. The average error on the voltage is 6V. This large error is due to the fact that there should at least one voltage reference. If there is no voltage measurement then the estimation will completely depend on the choice of the initial state. Even for a good choice of the initial state it is not sure that the algorithm will give good results. Error [degrees] Maximum error [degrees] Fig. 4. Phase angle [degrees] Actual phase angle ( ) and estimated phase angle ( ) Phase angle results The largest error is at node where there is a large voltage dip. This is the case for the estimated voltages as well as for estimated phase angles. In the lower graphs of Fig. 3 and Fig. 4 is the maximum error shown for each iteration. The maximum error does not necessary reduce per iteration. By incorporating just one voltage measurement the error on estimated voltages is drastically reduced. The results with one voltage measurement at node 7 are shown in Fig. 5 and Fig. 6. In the middle graph of Fig. 5 can be seen that the maximum error is drastically reduced namely from 3V to 3V.In practice it is better to use the voltage measurement at the understation because this one is known and equal to the rated value V rated. There have been done many tests with redundant Voltage [V] Error [V] Maximum error [V] Fig x Actual voltage ( ) and estimated voltage ( ) 3 Voltage results with an extra voltage measurement current, voltage and power measurements. The best results for the estimated voltages were achieved with redundant voltage measurements. The maximum error on voltage was further reduced but the overall error was increased. This is because the errors of redundant voltages are strongly correlated []- [5]. The error on phase angles is not reduced by redundant voltage measurements. This can be improved by using the current measurements. A too severe convergence criterion or a bad choice of the initial state can lead to divergence of the WLS method. It is recommended to use the rated values for the initial state.

5 5 Maximum error [degrees] Phase angle [degrees] Error [degrees] Fig Actual phase angle ( ) and estimated phase angle ( ).3 3 Phase angle results with an extra voltage measurement A voltage dip means that there is a big consumption there. The state estimator gives bad results in these places. If there is a foreknowledge about the high loads and thus big consumptions, then it is better to put the redundant measurements around those places to improve the estimation. The more measurement are used the longer it takes to calculate the state. The calculation time increases at least quadratically with the number of measurements or states. This can be seen in the matrix G(x k ) = H T (x k )R H(x k ) where there is a quadratic product of jacobian and the number of measurements is equal to the number of rows in jacobian and the number of states equal to the number of columns. B. Extended Kalman Filter In EKF as in Kalman Filter it is important to make a good trade off between the covariance matrices Q n n and R m m. The matrix Q n n tells something about how much the EKF trusts the estimated states while R m m tells something about how much the EKF trusts the measurements. If the value of Q n n is bigger than the value of R m m then it is assumed that the state is more noisy than the measurements. In this case the EKF trusts more the measurements than the estimated states. This is the most obvious choice. The uncertainty Σ k on the states is taken large in begin. Through the iterations the value of this matrix should reduce. If only the active and reactive flow measurements are done in every branch then, like for the WLS method, the EKF will give biased result of the voltages. Like for the WLS method the maximum error was at the node with the voltage dip. However the error was less than in the WLS method namely 85V instead of 3V. Assume that only the active and reactive power flow measurement are done in every branch. The results are shown in Fig. 7 and 8. This error can be further reduced by incorporating just one voltage measurement. Again the most obvious choice is the rated voltage at the understation. Voltage [V] Error [V] Maximum error [V] Fig. 7. Maximum error [degrees] Phase angle [degrees] Error [degrees] Fig x Actual voltage ( ) and estimated voltage ( ) Voltage results Actual phase angle ( ) and estimated phase angle ( ).3 3 Phase angle results The tests indicated that the error on the voltage estimation reduces when redundant voltage measurements are used. The error on the phase angles were best reduced with the redundant current measurements. If the matrix Q is taken zero then the EKF often diverges. This is obvious as the EKF fully trusts the estimated states and does not consider the measurements. A too severe convergence criterion or a bad initial state can also lead to divergence. As in the WLS method, the calculation time increases with the increasing number of measurements or states. Each iteration of the Kalman Filter algorithm is lower bounded by approximately O(k.4 ) where k is the dimension of the measurement vector z. Mostly the EKF needed less iterations than the WLS method. Comparing the Fig. 5 and Fig. 6 with respectively the Fig. 7 and Fig. 8 it can be seen that the WLS method and the EKF give comparable results. The WLS method is actually a special case of the EKF with the covariance matrix Q equal to zero. This means that the WLS method completely trusts the measurements and don t consider the state noise. If there is a

6 6 foreknowledge about the state noise then it can be incorporated in the EKF by the covariance matrix Q. VI. CONCLUSION For the state estimation has been the modified 34-bus IEEE test feeder used. The measurements needed for the state estimation were generated by the ladder iterative technique. Two methods for the state estimation in distribution grids have been analyzed: Weighted Least Squares and Extended Kalman Filter. Both methods try to determine the most probable state of a distribution grid based on the quantities that are measured. If there is no reference voltage measurement then the state estimation fully depends on the choice of the initial state. The most obvious choice for the reference voltage is the rated voltage at the understation because this one is known. A too severe convergence criterion or a bad choice of the initial state can lead to divergence. The number of rows and columns of the jacobian matrix is equal to the number of the measurements and states respectively. In both investigated algorithms there is a product of two jacobian matrices which means that the calculation time increases at least quadraticly with the number of measurements or states. For many simulations the Extended Kalman Filter needed less iterations than the Weighted Least Squares methods. But a disadvantage of the Extended Kalman Filter is that it needs a foreknowledge about the state and thus a trade off between the state covariance matrix Q and the measurement covariance matrix R. [4] J.L. Marinho, P.A. Machado and C. Bongers, On the Use of Line Current Measurement for Reliable State Estimation in Electric Power Systems, 979 PICA Conference. [5] IEC standard voltages, IEC 638 Ed. 6.,, p.. REFERENCES [] A. Monticelli and A. Garcia, Reliable Bad Data Processing for Real- Time State Estimation, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-, No.3, 983, pp [] E. Handschin, F.C. Schweppe, J. Kohlas and A. Fiechter, Bad Data Analysis for Power System State Estimation, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-94, No., 975, pp [3] Nian-de Xian, Shi-ying Wang and Er-keng Yu, A New Approach for Detection and Identification of Multiple Bad Data in Power System State Estimation, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-, No., 98, pp [4] A. Garcia, A. Monticelli and P. Abreu, Fast Decoupled State Estimation and Bad Data Processing, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-98, September 979, pp [5] K.A. Clements and P.W. Davis, Multiple Bad Data Detectability and Identification: a Geometric Approach, IEEE Transactions on Power Apparatus and Systems, Vol. PWRD-, No.3, July 986. [6] F.C. Schweppe and E.J. Handschin, Static state estimation in electric power systems, IEEE Proc., Vol.6, pp.97-98, July 974. [7] Distribution System Modeling and Analysis, William H. Kersting, New Mexico State University, Las Cruces, New Mexico, p [8] testfeeders.html [9] Power System State Estimation, Ali Abur, Antonio Gómez Expósito [] Thrun, Burgard Wolfram, Fox Dieter Probabilistic robotics, Cambridge, 5. [] G. Peter and J.H. Wilkinson, The Least-squares Problem and Pseudoinverses, The Computer Journal, Vol.3, No.4, August 97, pp [] Practical State Estimation for Eletric Distribution Networks, Roy Hoffman, PSCE 6 IEEE, pp [3] H.L. Fuller and T.A. Hughes, State Estimation for Power Systems with Mixed Measurements. IEEE/PES Summer Meeting Paper C , Anaheim, 974.

State Estimation and Power Flow Analysis of Power Systems

JOURNAL OF COMPUTERS, VOL. 7, NO. 3, MARCH 01 685 State Estimation and Power Flow Analysis of Power Systems Jiaxiong Chen University of Kentucky, Lexington, Kentucky 40508 U.S.A. Email: jch@g.uky.edu Yuan