Enhancement of Distribution System State Estimation Using Pruned Physics-Aware Neural Networks
EEnhancement of Distribution System StateEstimation Using Pruned Physics-Aware NeuralNetworks
Minh-Quan Tran
Department of Electrical EngineeringEindhoven University of Technology
Eindhoven, The [email protected]
Ahmed S. Zamzam
National Renewable Energy Laboratory
Golden, CO, United [email protected]
Phuong H. Nguyen
Department of Electrical Engineering
Eindhoven University of TechnologyEindhoven, The [email protected]
Abstract —Realizing complete observability in the three-phasedistribution system remains a challenge that hinders the imple-mentation of classic state estimation algorithms. In this paper,a new method, called the pruned physics-aware neural network(P2N2), is developed to improve the voltage estimation accuracyin the distribution system. The method relies on the physicalgrid topology, which is used to design the connections betweendifferent hidden layers of a neural network model. To verifythe proposed method, a numerical simulation based on one-year smart meter data of load consumptions for three-phasepower flow is developed to generate the measurement and voltagestate data. The IEEE 123-node system is selected as the testnetwork to benchmark the proposed algorithm against the classicweighted least squares (WLS). Numerical results show that P2N2outperforms WLS in terms of data redundancy and estimationaccuracy.
Index Terms —Distribution system state estimation, physics-aware neural network, phasor measurement unit.
I. I
NTRODUCTION
State estimation is an important function for grid monitoringand control. The traditional weighted least squares (WLS) isoften used to estimate the system state (e.g, voltage magnitude,voltage angle). Different from transmission systems, distri-bution systems are nominally unobservable [1], [2]. Causedby the scarcity of measurement devices, WLS is no longerapplicable in a more extensive distribution system because thesignularity of the the gain matrix hinders the solvability forthe state variables [3].A practical solution of the unobservable grid is to usepseudo measurements, which are forecasted from historical
The authors would like to acknowledge the financial support for this workfrom the Enabling flexibility for future distribution grid project FlexiGrid (ECfunding number: 864048). This work was authored in part by the NationalRenewable Energy Laboratory (NREL), operated by Alliance for SustainableEnergy, LLC, for the U.S. Department of Energy (DOE) under Contract No.DE-AC36-08GO28308. The work of A. S. Zamzam was supported by theLaboratory Directed Research and Development (LDRD) Program at NREL.The views expressed in the article do not necessarily represent the viewsof the DOE or the U.S. Government. The U.S. Government retains and thepublisher, by accepting the article for publication, acknowledges that the U.S.Government retains a nonexclusive, paid-up, irrevocable, worldwide licenseto publish or reproduce the published form of this work, or allow others todo so, for U.S. Government purposes. data or calculated by interpolating observed measurementsdata. In distribution systems, pseudo measurements can beobtained from smart meter data, distributed energy resourcegeneration based on the forecasting model of photovoltaic(PV) irradiance or wind speed. In [4], a game theoretic-baseddata-driven technique is studied with the purpose of generatingpseudo measurements in distribution system state estimation(DSSE). The parallel machine learning model is developedto learn load patterns and then to generate accurate activepower pseudo measurements. For the same purpose, in [5],a frequency-based clustering algorithm is implemented, whichdetermines the load patterns and estimates the daily energyconsumption. On the other hand, a probabilistic data-drivenmethod is used to generate time-series pseudo measurementsfor unmeasured PV systems [6].Besides exploiting pseudo measurements from abundantdata to improve the grid monitoring, distribution system op-erators (DSOs) will benefit from methods that can predict thesystem state with limited sensing. An estimation method witha combination of forecasting and the state estimation modelwas proposed in [7]–[10]. These methods proposed data-drivenmodels, which rely on minimum mean squares estimationand Bayesian estimation. The advantage is that these meth-ods do not require observability or redundant measurements.Recently, the authors in [10] proposed a deep learning-basedBayesian state estimation approach for unobservable distribu-tion grids. The data-driven techniques present a very promisingsolution to improve grid observability in distribution systems.Motivated by these approaches, we propose a data-drivenstate estimation with limited sensing to solve the problemDSOs are facing. In [11], a method called the physics-awareneural network (PAWNN) model was proposed. The idea is toembed the physical connection of the distribution system intothe neural network model; however, the connection betweenconsecutive layers in the model is kept the same, which leadsto possible unnecessary connections. To this end, this paperproposes the pruned physics-aware neural network (P2N2).A graphic summary of the proposed approach is shownin Fig. 1. First, Monte Carlo simulations are set up with a r X i v : . [ ee ss . S Y ] F e b ig. 1. The methodology. ISTRIBUTION S YSTEM S TATE E STIMATION
In this section, the three-phase estimator is presented inPart A, which is a rectangular voltage-based DSSE. In PartB, different types of measurements in the distribution systemare discussed in detail.
A. Three-Phase Estimator
The state estimation is a well-known method that aims toestimate the system’s state variables from measurements basedon the the mathematical relations between system states andmeasurement points. The synthesizing function can be writtenas follows: z = h ( x ) + e (1)where z is a measurement vector obtained from grid measure-ments and pseudo measurements; h ( x ) is a vector functionfrom state variables, x , to measurements, z ; e is a measurementnoise vector that is assumed to be independent zero-meanGaussian variables. The covariance of the measurement noiseis denoted by R . In general, the objective of the WLS methodis minimizing the sum of the square of the residuals: J ( x ) = [ z − h ( x )] T R − [ z − h ( x )] . (2)The Gauss-Newton method is usually used to obtain thesolution. The iterative process stops when the number ofiterations is higher than the limited value or when the changesin residual are less than the limited tolerance (usually as 10E-5or 10E-7). x k = x k − + G (cid:0) x k (cid:1) − H (cid:0) x k (cid:1) T R − [ z − h ( x )] (3)where H ( x ) and G ( x ) are: H ( x ) = ∂h ( x ) ∂x (4) G (cid:0) x k (cid:1) = H (cid:0) x k (cid:1) T R − H (cid:0) x k (cid:1) (5)Different from transmission systems, distribution networksare highly unbalanced systems. This leads to signularity of the gain matrix, G (cid:0) x k (cid:1) , and hence the single-phase stateestimation model used for transmission state estimation isoften not applicable for DSSE. In this work, the three-phasestate estimator from [12] is used, which is based on therectangular voltage. The state variables of the network arerepresented by three-phase rectangular form (i.e, the real partand imaginary part) at every node. B. The Used Measurements
Because distribution systems are highly unobservable, theapplication of the WLS algorithm needs additional pseudomeasurements to remedy the low-observability issue. Themeasurements used in this work are:1) Phasor measurement units (PMUs): this is the three-phase synchronized measurement. Normally, it is locatednear the step-down transformer, which is used to mea-sure the voltage phasor at the node and current phasorof the connected branches. With PMUs, the maximumerror is 1% for the magnitude and 10E-2 rad for thephase angle.2) Smart meters (SMs): these measurements are installedat the household (customer measurements). The powerconsumption of customers is obtained normally every15 minutes. With SMs, the maximum error is 2% forpower measurement.3) Pseudo measurements: the historical data are used at thebuses where no measurement device is installed. Thethree-phase active and reactive power can be obtainedas pseudo measurements. The maximum error of thepseudo measurement could be up to 50% for active andreactive power absorbed from loads.4) Zero injection buses: the buses without any loads or gen-erators connected are considered zero injection buses.The active and reactive power injection measured atthese buses is zero, with maximum error equal to0.001%.III. P
RUNED P HYSICS -A WARE N EURAL N ETWORK
In this section, the proposed method of P2N2 is discussed indetail. The background of the partitioning of the DSSE basedon the PMU location is explained in Part A. An example ofa 6-bus system is presented. We show the way we design theP2N2 based on the physical distribution grid. Then, in Part B,we present the model validation.
A. Layer Design-Based Physical Model
As mentioned earlier, the PMU is a three-phase synchro-nized measurement of the real-time measured value withvery high accuracy. Considering this advantage of the PMU
Fig. 2. Example of 6-bus system with PMU located at Bus 4.ig. 3. Vertex-cut partitioning example with PMU located at Bus 4. (a)Partition 1 with buses 1, 2, 3, and 4. (b) Partition 2 with buses 4 and 5.Partition 4 with buses 4 and 6.Fig. 4. The designed connection between layers for 6-bus system. (a) theconnection between the input layer and Layer 2. (b) the connection betweenLayers 2 and 3. (c) the connection between Layer 3 and the output layer. measurement, the estimated voltage at a specified bus doesnot require the information of all available measurements inthe network. This means that with an accurate measurementat a specified bus, other measurements behind this bus canbe neglected. This separability property was proven in [11].As an example, Fig. 2 shows a simple 6-bus system with aPMU installed at Bus 4. Applying the concept of vertex-cut,the system can be divided into three different partitions, asshown in Fig. 3.1) Partition 1 in Fig. 3 (a): buses 1, 2, 3 and 4.2) Partition 2 in Fig. 3 (b): buses 4 and 5.3) Partition 3 in Fig. 3 (c): buses 4 and 6.The PAWNN model is designed with multiple layers, whichare built based on the physical connections of the distributionnetwork. The required number of layers is the maximumdiameter of each partition. In this case, the number of hiddenlayers is 3 because the maximum diameter of all partitions is3. Then, the connections between layers are designed based onthe physical connection of the network. This is the idea behindthe physics-aware technique, which prunes the connectionsthat are not present in the physical network. Fig. 4 showsthe result of the designed connections between layers for the6-bus system. Fig. 4 (a) shows exactly the structure of thenetwork admittance matrix of the 6-bus network.Let the input layer of the PAWNN model be denoted by x ,and y is the output layer of PAWNN. The output vector, y, Fig. 5. The graph-pruned neural network model. (a) The structure of PAWNN.(b) The designed structure of P2N2. represents the voltage at the buses of the network. Let k ( i ) denoted the intermediate output at i - th layer. Then, we have: k i +1 = σ i ( W i k i ) (6)where σ i is a point-wise nonlinearity; and w has a size of N x N weight matrix, with N the number of output y . Thematrix W is designed the same as the connection shown in Fig.4 (a). The ( i, j ) element in the matrix W is pruned if nodes i and j are not connected. Then, the structure of PAWNN isshown in Fig. 5 (a); however, this structure leads to possibleunnecessary connections. Partitions 2, and 3 have the samediameter of 2, meaning that we can get the voltage valuesof buses 5 and 6 after Layer 2. To this end, the P2N2 isproposed to reduce unnecessary connections between layers.As shown in Fig. 4 (b), the connection between Layer 2 andLayer 3—four unnecessary connections of (4,5), (4,6), (5,4),and (6,4)—are zeroed out. Similarly, the new structure of theconnection between Layer 3 and the output layer is shown inFig. 4 (c). Then, three different weight matrices are used forthe P2N2 model, which is shown in Fig. 5 (b). Therefore, theoutput of the P2N2 can be written as: y i = (cid:40) σ ( W σ ( W σ ( W x + b ) + b ) + b ) if i =1,2,3,4 σ ( W σ ( W x + b ) + b ) if i =5,6 (7)where b , b , b are the bias vectors of the P2N2 model. B. Model Validation
In this work, TensorFlow [13] was used to train the model.The data were divided into 90% training and 10% testing. Themodel was trained based on the ADAM optimizer [14], andthe optimization function is formulated as follows: min { b t ,W t } Tt =1 (cid:88) j (cid:107) v j − g T ( z j ; { b t , W t } Tt =1 ) (cid:107) (8)where v j and z j are the true state and measurement in the j - th training sample, respectively. g T is the j - th mapping realizedby the T-layer of the model parameterized by { b t , W t } Tt =1 . Thenetwork structures are imposed on the P2N2 model; hence, thenumber of neurons in each layer is proportional to the number ig. 6. The IEEE 123-node test system of buses. Finally, we used the average estimation to calculatethe accuracy of each algorithm as follows: ν = 1 N N (cid:88) i =1 (cid:107) ˆ v i − v itrue (cid:107) (9)where ˆ v i is the estimated voltage.IV. S IMULATION AND R ESULTS
In this section, the test case and the simulation results arepresented. First, the IEEE 123-node test system is described.Then, the methodologies explained in sections II and III areapplied. Three different scenarios were carried out to assessthe performance of the proposed model.
A. Test Network
In this work, the IEEE 123-node test system is used, asshown in the grid topology in Fig. 6. The IEEE 123-nodesystem is a radial distribution grid with single-phase loads andtwo-phase loads; thus, the grid is a highly unbalanced network.The grid has four different voltage regulators and differentvoltage levels. The detailed grid parameters are available in[15]. There are four switches (13-152, 60-160, 97-197, and 18-135), which have been modified as connection buses. Further,voltage regulators are excluded in this work. Generally, thesemodifications are common for this kind of study [12] withoutaffecting the generality of the study. The DSSE algorithm isbuilt in the MATLAB environment, and the OpenDSS is usedfor the power flow calculation. In addition, we assumed thatthe system has two PMUs, one each installed at Bus 149and Bus 60. Then, the whole system can be split into twopartitions. The first area in the dashed line has one PMU,at Bus 149. By exporting the maximum diameter of possiblepartitions in this area, the diameter is 14. Similarly, in thesecond area, the diameter is 11. Then, the neural networkmodel is designed with 14 layers.
B. Simulation Scenarios
To perform the behavior of the estimator, one-year time-series collected data of the SM are used with 35,040 data
Fig. 7. The simulation and model evaluation process.Fig. 8. Estimated voltage magnitude at Phase A in Scenario 1. points. Then, M= 35,040 possible operation conditions (nor-mally, a data set of 10,000 is sufficient to ensure the qualityof the results) is fed into the power flow model as the loadconsumption. By extracting from each power flow simulation,the measurements and the true voltage magnitude values at thebuses are collected. Hence, we have 35,040 sets of measure-ments (z) for the WLS and 35,040 sets of measurements (z)and system states (voltage magnitude, v true ) for the P2N2.In addition, 90% of the 35,040 sets of data is used fortraining, and the rest is used for testing. The process of thesimulation and model evaluation is shown in Fig. 7. Then, theperformance of each algorithm is calculated using (9).This whole process is tested with three different scenarios:1) The algorithms are tested with a large number of mea-surements.2) We kept the same amount of measurements, and in-creased the error of the pseudo measurements from 30%to 50%.3) Limited measurements are used for this scenario, i.e, 14pseudo measurements are removed.In the first scenario, the network has 2 voltage measure-ments and 2 current injection measurements at Bus 149and Bus 60. Further, 118 pseudo measurements are used,which consist of 85 load power measurements and 33 zeroinjection measurements. As an example, we present only thevoltage magnitudes at Phase A of all the buses. Fig. 8 depictsthe estimated voltage magnitudes in the first scenario. Theresults show the robustness of the WLS in case of redundantmeasurements. However, the PAWNN and P2N2 also show thehigh accuracy of the estimated voltage magnitudes. To showthe performance of the proposed method, we increased theerror of the pseudo measurements from 30% to 50% whilekeeping the same number of measurements. As shown in Fig.9, the PAWNN and P2N2 show a better result when comparedwith the WLS. This means that the neural network model witha large set of training data can provide reliable estimation ig. 9. Estimated voltage magnitude at Phase A in Scenario 2.Fig. 10. Estimated voltage magnitude at Phase A in Scenario 3. performance.The third scenario is carried out with a limited number ofmeasurements, and 14 load power measurements are neglected(compared with the first scenario). In this case, the networkis unobservable because of the limited number of measure-ments, and thus, the WLS cannot obtain estimates for thevoltage magnitudes. However, the PAWNN and P2N2-basedneural network show effectiveness even with the unobservabledistribution system. Further, the average estimation errors ofdifferent scenarios are shown in Table II. It shows the accuracyof the WLS in the case of the observable distribution networkwith noiseless measurements. However, the better estimationresult is achieved by PAWNN and P2N2 in the case ofhigher noise from the measurements or when the network isunobservable. Table II summarizes the estimation time of eachtime step; it shows the robustness of the proposed P2N2 andit is nearly 160 times faster than the WLS.V. C ONCLUSIONS
This paper proposed a data-driven state estimation methodfor the distribution system. The model was designed based onthe physical connections of the distribution network, whichpruned the unnecessary connections between layers. One-yearsmart meter data were used to generate the training data set byperforming the power flow analysis. Then, the set of 35,040data points were collected for the training and testing phase.Three different scenarios were carried out in the IEEE 123-node test network to show the performance of the proposedmethod. Numerical results show the efficacy of P2N2 in termsof reliable performance under different observability scenarios.
TABLE IT HE A VERAGE E STIMATION A CCURACY
Scenario WLS PAWNN P2N2 HE E STIMATION T IME OF E ACH T IME S TEP
Scenario WLS PAWNN P2N2
Compared with WLS, the proposed method achieves betterestimation accuracy in low-observability scenarios. Also, theproposed P2N2 approach can achieve almost the same per-formance as PAWNN while having a significant reduction inthe number of parameters, thus reducing the training effort.Several extensions are possible to improve the method in thefuture. For example, system parameters can be exploited todesign more efficient learning models.A
CKNOWLEDGMENT
The authors acknowledge the financial support for this workfrom the Enabling flexibility for future distribution grid projectFlexiGrid, https://flexigrid.org/.R
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