Real Time Distribution System State Estimation Based On Interior Point Method

This approach for a three-phase Distribution System State Estimator (DSSE) considers all the analog measurements including voltage and current measurements, active and reactive power measurements, as well as historical load-information. The estimation of load information is based on an interior point optimization. It reduces the estimation problem to load groups which are located in so called measurement areas. By grouping the loads with the same weighting factors, the number of state variables for the optimization is minimized. The power mismatches at the boundary between neighboring areas are eliminated by equality constraints. This significantly reduces size of the estimation problem. The proposal is designed for radial as well as for networks with some meshes.


Abstract
This approach for a three-phase Distribution System State Estimator (DSSE) considers all the analog measurements including voltage and current measurements, active and reactive power measurements, as well as historical load-information. The estimation of load information is based on an interior point optimization. It reduces the estimation problem to load groups which are located in so called measurement areas. By grouping the loads with the same weighting factors, the number of state variables for the optimization is minimized. The power mismatches at the boundary between neighboring areas are eliminated by equality constraints. This significantly reduces size of the estimation problem. The proposal is designed for radial as well as for networks with some meshes. NOMENCLATURE NLG total number of load groups. NLGK number of load groups in area k. NPQ total number of power measurements in system including the ones converted from currents. NI total number of current measurements. NMK number of P-/Q-pairs converted from currents in area k.

INTRODUCTION
From last several decades state estimation has been successfully established for the calculation of transmission systems. Since [1] proposed the estimation of the state variables of energy system in 1970, state estimation has been intensively improved. Using a redundant set of active and reactive power, current and voltage measurements, it has become possible to calculate observable parts. As long as the local redundancy is sufficient, even large measurement errors can be identified ( [3], [4], [5]). However, the advantages of the approaches used in transmission systems cannot be directly transferred to distribution systems, due to the completely different electrical properties in distribution systems such as: radial or tree-like topology, large R/X ratio for cables, small number of real-time measurements, mostly current-instead of power measurements, unbalanced loading, unsymmetrical construction. The missing redundancy of real time measurements is a more serious problem. Thus, different models for distribution systems are required. This problem has been addressed in many publications. For the optimal placing of measurements in a network in order to get it observable, several methods are proposed ([6]- [9]). For the loads of the network, additional assumptions are required. Generally, their time behavior is defined by historical data, e.g. by load curves, or with typifying loads, which have a similar behavior. But, in [17] it is pointed out, that this historical information does normally not match the real time, and additional treatment is required for the load curve values. In [15], [16], a load flow approach is proposed, where the loads are scaled in such a way, that the loads and measurements fit. This approach has some disadvantages, especially, when there are some loops in the network. Also, the use of current measurements is not completely satisfying. Approach is based on estimation of the system's state, using a WLSestimator ( [10], [11], [20]- [22]). It is mainly designed for radial systems and mostly based on branch currents as state variables. In this paper, a new algorithm is presented for an improved estimation of the loads. The optimization problem can be significantly reduced by defining so called measurement areas, which are characterized by measurements on the boundary branches to the neighboring measurement areas. For each of areas, loads groups are defined and then adapted by an estimation of scaling factors. A power mismatch at the boundaries of these areas is eliminated by equality constraints. Thus, this new approach is a combination of a load flow based scaling and an interior point optimization. It is also applicable for the calculation of distribution networks with some meshes.

DESCRIPTION OF THE IDEA
Typically, the structure of a distribution system is based on radial or tree like feeders, probably with loops, which are often operated as open rings. In these networks, measurements are mainly located at the feeder-heads, either as voltage measurements, active and reactive power measurements or as current measurements. Figure1 represents a small network with an openly operated loop. Both radial feeders have some measurements of different types. Using these types, sets of measurement areas are defined. The load information in these measurement areas is to be scaled in such a way, that the network areas are balanced with respect to the injected power and the network losses, the measured information is matched, the power losses of these areas are correctly considered and power mismatch at the boundary branches between two measurement areas is eliminated. The neighboring measurement areas are coupled via the flows between these areas (Figure2). The flow leaving one area is injected to another. This is used for an improved scaling based on weighted least absolute value estimation. With these couplings between neighboring measurement areas, an iterative process can be started with • a load flow calculation for the complete network • an estimation which adapts the loads and eliminates the power mismatch at the boundaries. When using different weighting for different quality of information, the load information as well as measured information or pseudo measurements can be treated in the same way. This ensures, that the real measurements are matched (large weighting), whereas pseudo measurements or load curve values allow a larger discrepancy between the input and the calculated value (low weighting). The optimization uses the load information as state variable. The structure and the admittances of measurement areas are not directly required; it only needs the sum of their scaled loads in the different iteration steps, the losses and the loading of the boundary branches. Voltages and currents are indirectly used after a power flow calculation has been executed. Defining groups of loads which have the same weighting reduces the effort for the estimation significantly in each iteration step. Such a load group consists of loads, which have the same quality of information, thus, they are assigned the same weighting. Consequently, their values are scaled with the same factors. In figure 3 the flow chart of the proposal is displayed.

CONVERSION OF CURRENT MEASUREMENTS
The scaling process is pretty simple for areas, which have complete sets of P-and Q-measurements. The loads are scaled in such a way, that they match the sum of the injected and measured sum of the power at the boundary. The more critical workflow is given for injections into a measurement area, which is only based on current measurements. Current measurements cannot be directly used for the scaling of the load information, and conversion of current information to a complex power becomes necessary. For each measurement area, the conversion of current measurements to P-and Q-pseudo measurements is executed individually. For each area with only one current measurement at its boundary, following objective is minimized using the constraints In the next estimation step, the scaled values k i P ,P i LG and LG are used as new starting point. The conversion of current measurements starts with the measurement areas where is only one current measurement to be converted. It is executed "bottom up", starting with the area which only has one current measurement which was not converted jet. For feeder 2 in figure1, measurements area 6 is considered after 7 and 8, because it has 2 branches with current measurements. Converting the current measurements of area 7 and 8, the number of the current measurements in area 6, which are not converted, is reduced. On this way, the number of non converted current measurements is reduced until all of them are converted.

STATE ESTIMATION
As the conversion of currents to P-and Q-pseudo measurements is done individually for each measurement, existing couplings between these areas is not considered. They are considered in the next estimation step, which takes into account the complete network. This step forces the flow at the boundaries of neighboring areas to match. The estimation problem is defined as a minimization of the objective J described by the equation (3). The first term of equation (3) considers, that the differences between measured and estimated power information is minimized, the second term is responsible for the minimization of the differences in the load information. The third term considers the differences between measured and estimated currents. Additionally the following equality constrains must be fulfilled: • Sum of estimated P for real and pseudo measurements and P contributed by internal generators covers active power losses (equation (4)) • Sum of estimated Q for real and pseudo measurements and Q contributed by generators and capacitors covers reactive power losses (equation (5) • For each area with pseudo measurements converted from current measurement, the estimated P C E , Q C E , I C E and V C must fulfill equation (6) [ ] with the equality constraints (6) and the inequality constraints same as in equation (2).

STATE ESTIMATION WITH INTERIOR POINT OPTIMIZATION
An introduction in interior point methods is given in [22]. The estimation approach, described here, is mostly based on this presentation. The problem for both current to power conversion and the estimation for the whole system can be defined as the following minimization problem: M are i-th component of a n dimensional vector f measurement and estimated state, w i is i-th component of a n dimensional vector of weighting factors, g(x), h(x) are m dimensional vector of equality and inequality constraints, and h � , h p dimensional vectors with lower and upper limits for inequality constraints. The given optimization problem is transformed while replacing inequality constraints by corresponding equality constrains using non-negative slack vectors: where s,z are p dimensional vectors of slack variables, and s,z≥0. The non-negativity conditions are satisfied while introducing logarithmic barrier terms in the problem formulation: where μ k is barrier parameter with μ k >0, decreasing to zero. The entire problem is solved using Lagrange multipliers combined with Merhotra's Predictor-Corrector Method: where y=(s,z,π,v,x,λ) T and π,v is p dimensional vectors of Lagrange multipliers. The Mehrotra's Predictor-Corrector Method reduces the number of iterations until convergence while including additional information into standard Newton representation. The third term of the right hand side of (11) is the extended part. It is based on the incorporation of the new point y k+1 =y k +Δy into Newton representation as shown in [22]. where S is a p by p matrix with vector s at the diagonal, Z is a p by p matrix with vector z at the diagonal, e is identity matrix, v � is sum of vectors v and π. The desired solution is split into a predictor-and a corrector-step. In the predictor-step, the system is solved for the pure Newton direction (also called affine-scaling direction), i.e. without barrier parameter and delta terms in the right hand side of (11) [22]. The found solution is used to estimate the barrier parameter dynamically and to approximate delta terms from right hand side. For updating the update barrier parameter, the step length parameters are first calculated for the pure Newton direction: { }   (13) where ρ af k is complementarity gap for affine scaling direction. The barrier parameter is finally calculated from: Where ρ k =�s k � T π k +�z k � T v � k . In the corrector step, the full Newton direction Δ is calculated. Missing barrier parameter value and delta terms are approximated using results from previous predictor step. Both predictor and corrector steps use the same matrix factorization.

HANDLING OF MESHED TOPOLOGY
When there are two or more current measurements on boundary branches to one measurement area, the conversion of current measurements to pseudo power flow information is not uniquely possible. In many cases, this problem is simply solved by converting all the current measurements leading to an area with only one nonconverted current measurement exists. The number of nonconverted currents is decreased for the neighboring area. This is always successful for radial or tree-like networks. In general, this is not given for networks with some meshes or a parallel injection. The impedances inside the loop or the coupling branches are to be considered in the estimation process. Each branch of such an area with several remaining current information is defined to be a measurement area of its own, having a defined active and reactive virtual information at each of its terminals. These measurements are called virtual, because they are not derived from a real current measurement. The complex power at both sides of a branch, which leads from i to j, is given by (16) From the preceding load flow calculation, this power information is extracted and used like low weighted measured information. Please note, that the calculation of these virtual power measurements is only necessary for the sides of the branches which is connected to a node inside the meshed network part. No calculation is executed for branch sides, where real power measurements are available. Fig. 5 display the newly created measurement areas and the corresponding virtual measurements. In following, the newly created areas, which contain no branches, are called virtual node areas. The objective in equation (3)- (6), which are used for the optimization process, is to be extended by the additional virtual information according to The virtual node areas must fulfill the first Kirchhoff -law.  (19) The virtual power measurements are considered as real power measurements during estimation, but with a low weighting. They have their origin in the load flow calculation, which precedes the estimation.

TEST CASES
In the following, test cases for a modified IEEE34 network [24] are discussed. Fig. 6 shows the schema for this network. The network is extended by an injection transformer and additional buses. There are 6 P-and Qmeasurement pairs at injection transformer and at the feeder head for each phase and 19 current magnitude measurements downstream the feeder head.  Figure 6. Modified IEEE34 network with used measurements (marked by "X"). The weighting factors for all the real measurements are equally set to 900, the weighting factors for load data are set to 0.5. The test scenario mirrors the situation in a distribution system: low number of real measurements; more current measurements available than power measurements; real measurements are more reliable than pseudo measurements at loads. Measurement values are obtained from power flow engine while the load input data are multiplied with a factor of 1.6. The inequality constraints for load scaling factors are derived from load setup and set to the range [1,2]. That means, that calculated load scaling factors should be less than 2, but higher than 1. Table1 summarizes the results for given real measurements. The deviation between measured and calculated data is below 0.5%, the maximum is 2.38% for some measurements at phase C. The average deviation is 0.67%. Compared with the results in [25], where a WLS minimization was used, the deviations between the input information and the calculated results are significantly reduced by use of interior point optimization. This approach delivers more accurate results.  Table 2 shows the deviation between input data and results for loads. Load scaling factors, which express the ratio between estimated and original load data, are in a range [1.24, 1.98]. As the measurements have been multipied with a factor of 1.6 for this calculation, this is an excellent result. All the loads where scaled by some factor around this multiplier. As expected the mismatch between real measurements and load data is shifted mainly to load data which have a poor accuracy.

CONCLUSION
This paper presents a new approach for a distribution system state estimation. This approach is a combination of power flow based load scaling and a WLAV-state estimation with equality and inequality constraints. For the estimation process itself, the set of state variables are sets of load information, which are used as input for a load flow calculation. With different weightings to the available information, this approach allows to model the accuracy or the available information. This and the use of inequality constraints increase the accuracy of the results. Further investigations are intended with respect to the detection and identification of errors in the measurements and load data.