Observability of power systems based on fast pseudorank
Contents
1. In order to come up with a solution for a non observable system various strategies can be followed One option is to reset all non observable states such that some manually entered values or historic data is used for these states An alternative option is to use so called pseudo measurements for non observable states A pseudo measurement basically is a measurement with a very poor accuracy These pseudo measurements force the algorithm to converge In order to improve the quality of the result observability analysis and state estimation should be run in a loop Here at the end of each state estimation the measurement devices undergo a so called bad data detection the error of every measurement device can be estimated by evaluating the dif ference between calculated and measured quantity Extremely distorted measurements i e the estimated error is much larger than the standard deviation of the measurement device are not considered in the subsequent iterations The process is repeated until no bad measurements are detected any more Since observability analysis is performed very often in this framework we have the requirement that the observability analysis should serve as an accurate and fast preprocessing step for the state estimation III ALGORITHMIC APPROACH FOR OBSERVABILITY ANALYSIS Formally speaking the problem of analyzing the observ ability of a network can be described as follows Assume that we are given an2. 3 Algorithm to determine linear dependencies of two sparse vectors In this example while sweeping over the two vectors v and v2 the constant ratio A 2 is detected up to row index 3 In the following step the pointers p and p will be advanced to indices 6 and 5 respectively For these indices do not coincide the sweep will stop and return that the vectors are linearly independent Proof The vectors are linearly dependent if there exists some A such that v Avg The corresponding algorithm sweeps over the non zero elements of the sparse vectors v and Vg as depicted in Figure 3 We use two pointers p and p that simultaneously proceed through the entries from left to right If the pointers point to elements with equal row index the ratio of the corresponding values determines A The algorithm stops if at some step during the simultaneous sweep either 1 the coordinate index of the two elements differs or 2 the coordinate index coincides but the corresponding values have a ratio different than A In these cases the vectors are linearly independent Otherwise 1 e if the sweep of the pointers simultaneously reaches the end of both vectors the vectors are linearly dependent E This simple computation can be used to determine the pseudorank of a given matrix A Proposition 1 The pseudorank of a sparse n x r matrix A can be determined by the algorithm in Figure 5 An upper bound on the worst case running time o
3. to provide consistent load flow results for an entire power system based on real time measurements manually entered data and the network model The problem has been studied in great detail over the last years see 3 5 6 and the references therein for a comprehensive overview A general input instance for a state estimation algorithm includes a set of measurements located in the network and a selection of states that should be estimated using the given measures Observability analysis is a fundamental part of state esti mation A necessary requirement for an observable system is that the number of available measurements is equal or larger than the number of estimated variables But it can also happen that only parts of the network are observable and some other parts of the system are not observable even if the total number of measurements is sufficient Hence it is not only important that there are enough measurements but also that they are well distributed in the network The entire network is said to be observable if all states can be estimated based on the given In principal a measurement could be any measured value in the network branch power flows current magnitudes voltages etc Similarly one might be interested in estimating any quantity such as power flows at injections and consumers tap positions of transformers or shunts etc Markus Poller DIgSILENT GmbH Heinrich Hertz Str 9 D 72810 Gomaringen Germany E
4. transformer taps of 2 and 3 winding transformers and all active and reactive power flows at each consumer and at each injection We combined the various settings to obtain two further test scenarios see Fig 6 Scenario II uses all 4397 measurements to estimate the restricted set of 164 states whereas Scenario III combines both the full set of 4397 measurements and the full set of 813 states to be estimated In each of the scenarios our pseudorank calculation which is performed in a small fraction of a second considerably outperform the calculation of the rank which takes up to a few seconds see Fig 6 The test were performed on a HP workstation wx8000 with two 3 2 GHz HPTC Xeon CPUs VII CONCLUSION We demonstrated a novel approach for observability analysis in large scale networks The core part of the presented algo rithm is to perform fast rank calculations on a corresponding Sparse sensitivity matrix which is drawn from the location of the measurements and the states to be estimated Our imple mentation of the corresponding algorithm which intrinsically uses the notion of the pseudorank yields very encouraging results for practical settings The presented approach has the advantage of being extremely flexible since it is possible to take into account any measured quantity and any possible state to be estimated In addition a very detailed observability report redundancy of measurements detection of indi
5. Observability of power systems based on fast pseudorank calculation of sparse sensitivity matrices Jochen Alber DIgSILENT GmbH Heinrich Hertz Str 9 D 72810 Gomaringen Germany Email j alber digsilent de Abstract This paper describes a novel approach for the observability analysis in state estimation of large scale power sys tems We draw a one to one correspondence of the observability of a network to the rank of a corresponding sensitivity matrix This general framework is not purely based on topological aspects but takes into account all electrical quantities of the network and turns out to be very generic and flexible In order to solve the observability problem a novel algorithm for very fast pseudorank calculations on sparse matrices is developed This approach allows on the one hand to identify equivalence classes of redundant measurements On the other hand the algorithm can detect all observable islands and group unobservable states according to their observability deficiency Our algorithm bares high potential in coping with unobservable areas a method is described which incorporates a minimum number of pseudo measurements to yield observability The performance of the algorithm is tested on real world network with data gained from an underlying ABB MicroSCADA system and compared to common rank calculation with singular value decompositions on sparse matrices I INTRODUCTION State estimation is the task
6. actical example is discussed in Section VI II COMPONENTS OF A STATE ESTIMATOR Before presenting our algorithmic approach for the observ ability analysis we will have a brief glance at the complete framework of state estimation We summarize the various components of a State Estimator following our implemen tation in the power system analysis software DIgSILENT PowerFactory 7 1 Plausibility Check 2 Observability Analysis 3 State Estimation Non Linear Optimization Figure illustrates the algorithmic interaction of the different components In a first phase the Plausibility Check is sought to detect and separate out all measurements with some apparent error in order to avoid any heavy distortion of the estimated network state due to completely wrong measurements Various test criteria can be thought of such as checking for large measured branch flows on open ended branches checking for consistent measured power flow directions at each side of the branch elements checking for reasonably measured node sums etc In a second phase the network is checked for its Observ ability Roughly speaking a region of the network is called observable if the measurements in the system provide enough non redundant information to estimate the state of that part of the network Finally the State Estimation itself evaluates the state of the entire power system Mathematically speaking the objective is to minimize the weighted sq
7. amPower In the presented test scenarios the pseudorank calculation clearly outperforms an ordinary rank calculation using singular value decomposition on sparse matrices Furthermore our approach allows for a very detailed ob servability report e We do not only identify for each state whether it is observable or not We also subdivides all unobserv able states into so called equivalence classes Each equivalence class has the property that it is observable as a group even though its members 1 e the single states cannot be observed e We additionally determine redundant and non redundant measurements Moreover we subdivides all redundant measurements according to their information content for the system s observability status In this sense we are able to calculate a redundance level which then indicates how much reserve the network measurements provides This helps the system analyst to precisely identify weakly measured areas in the network The paper is organized as follows In a first introductory section we classify the role of observability analysis in the general framework of state estimation In the two subsequent sections III and IV we provide the theoretical part for our new algorithmic approach to compute the system s observability namely sensitivity matrices and fast rank calculations on sparse matrices The results are summarized in Section V Finally the algorithm s behavior on a pr
8. ed pseudorank which formally is an upper bound on the rank However in most of the sensitivity matrices drawn from practical settings this pseudorank turns out to be equal to the rank More formally we give the following definition Definition 3 Let A be an n x r matrix Assume that A has a pair of linearly dependent rows Removing from A one of these rows is called an elimination of pairwise linearly dependent rows Similarly one defines an elimination of p w linearly dependent columns to be the removal of all but one columns among a set of pairwise linearly dependent columns Let A be the resulting n x r matrix after successively applying eliminations of p w linearly dependent rows and columns until no further such elimination is possible Define prank A min n r to be the pseudorank of A It is easy to see that prank A is well defined i e inde pendent of the elimination scheme Moreover it holds that prank A gt rank A For an algorithmic approach of computing the pseudorank it is necessary to implement a fast check which determines whether two sparse vectors are linearly dependent Lemma 1 t can be checked in time O n whether two given sparse vectors v and v s are linearly dependent Here n min n n2 and n is the number of stored non zeros of the vectors Vj p2 Fig
9. eld observability VI PERFORMANCE TESTS ON REAL WORLD DATA The presented algorithms were implemented as part of the state estimator package in the power system analysis software DIgSILENT Power Factory 7 We tested the implementation on the Namibian network as operated by NamPower The modeled network consists of 1400 nodes It uses approximately 200 lines 140 2 winding transformers and 41 3 winding transformers A total of around 370 loads are modeled in the network Our implementation in the software package PowerFac tory 7 is fully integrated to NamPower s ABB MicroSCADA central master system which collects over a communication network the measured values sent by the installed Remote Telemetry Units RTUs This allows at any time to directly populate the measurement models with the actually measured values For test purposes we randomly used the NamPower system s snapshot taken on August 18th 2004 at 5pm A Test scenario I A total of 435 measurements are physically installed in the network Among which we have 161 active power measure ments P meas 121 reactive power measurements Q meas 11 current magnitude measurements I meas and 142 voltage magnitude measurements V meas In the given model we selected a total of 164 states to be estimated More precisely we were interested in 79 active power states and 79 reactive power states of selected loads and generator injections respectively In addition 6 ta
10. f this pseudorank calculation is given by O 4 n r where is the maximum number of non zero elements in a row or a column Consider the algorithm in Figure 5 It very closely follows the definition of the pseudorank see Def 3 i e it suc cessively eliminates all pairwise linearly dependent rows and columns of the matrix until no further elimination is possible We remark that our algorithm behaves very well in practice i e for sensitivity matrices drawn from real world scenarios Moreover we observed that our algorithm outperforms a full rank calculation on sparse matrices which uses singular value decomposition by far see Section VI This is due to the fact that the sub procedure Eliminate see Fig 4 and hence the overall algorithm is extremely fast in the following two opposing situations a If many rows columns are linearly dependent this results in a fast reduction of the matrix which implies that the outer loop in the sub procedure Eliminate is rarely called b If on the other hand most rows columns are linearly procedure Eliminate Z 21 for 1 1 to m do if z not eliminated then for 7 i 1 to m do if z and z are linearly dependent then eliminate zj od en a es fi od end Fig 4 Subroutine Eliminate using the check for pairwise linear depen dence as described in Lemma 1 procedure PseudoRank A N begin do Eliminate row r0W Eliminate c
11. flow located at both sides of the transformer and at Load 1 respectively and two voltage magnitude measurements located at the two terminals are distributed in the network For simplicity the sensitivity matrix was discretized to the range 1 0 1 Clearly the matrix has full rank 1 e all five states can be observed In this simple setting the two P and two Q measurements at Load 1 and at Trafo Term 1 respectively and the additional V measurement at Term 2 would have sufficed for observability of the network From the outline above we get the following core statement Claim 1 A given set of r states X in a network is observable by a set of n measured values M if and only if the n x r sensitivity matrix SX has rank r The sensitivity matrix can be numerically computed by infinitesimal disturbances of the corresponding states The computation basically relies on a backsubstitution of a modi fied Jacobian matrix which needs to be extended by setpoints for each estimated state For an algorithmic treatment of rank calculations and linear dependencies it is essential that the matrices and vectors are discrete Hence let D R N be some discretizing function We denote by D sens x and D sensx m the coordinatewisely discretized sensitivity vectors Similarly D S is the discretized sensitivity matrix Our algorithm was tested using various kinds of such functions ranging from a very c
12. in state estimation IEEE Transactions on Power Systems 19 2 699 706 2004 5 A Monticelli State Estimation in Electric Power Systems A Generalized Approach Kluwer Academic Publishers 1999 6 A Monticelli Electric power system state estimation Proceedings of the IEEE 88 2 262 282 2000 7 DIgSILENT GmbH D gSILENT PowerFactory V13 User Manual 2005
13. mail m poeller digsilent de measurements If a network is not observable it is still useful to determine the islands in the network that are observable Three main approaches for observability analysis are dis tinguished in the literature topological numerical and hybrid methods see 5 We present a concept which is closely related to the numeri cal approach since it also takes into consideration all electrical parameters of the network However we purely focus on the network s sensitivities of all measured values with respect to the estimated states This framework turns out to be extremely generic and flexible It intrinsically allows to incorporate all sorts of measurements including e g current magnitude measurements see 1 2 and a wide range of estimated states such as tap positions of transformers and shunts see 4 where such issues were dealt with individually The core part for our approach is to compute the rank of a corresponding sensitivity matrix Since rank calculations on large systems in general are time consuming we propose the notion of pseudorank which yields an upper bound on the rank We demonstrate a simple algorithm to compute the pseudorank of a given matrix This computation scheme may highly profit from the matrix sparsity and is proven to be extremely efficient The performance of the pseudorank calcu lation is exhibited exemplarily on the Namibian network model as operated by N
14. nd we explicitly find all individual non observable states and can group them according to their lack of deficiency A Equivalence classes for redundant measurements If we simply label each group of p w linearly dependent rows during our pseudorank computation then we get the more fine grained information on the redundance of measurements If from each group we eliminate all but one measurements the observability status of the system will not change In this sense the cardinality of each group indicates the degree of redundance of the measurements inside that redundance class B Equivalence classes for non observable states Similarly we may group during the pseudorank calcu lation all p w linearly dependent columns The system is observable if each such group is a singleton Otherwise the system is unobservable If in case of unobservability from each group with more than one members all but one corre sponding states were neglected to be estimated the system would be observable In this sense the states of each group are observable as a whole but not individually In order to reestablish full observability of the whole system so called pseudo measurements can be introduced at the location of each non observable state This corresponds to inserting a unit row vector in the sensitivity matrix Processing in this way at all unobservable states a minimum number of pseudo measurements is incorporated to yi
15. oarse to a very fine grained discretization As a very simple example for the construction of the sensitivity matrix consider the network in Figure 2 with the corresponding 1 0 1 discretized matrix D S IV FAST RANK CALCULATION ON SPARSE MATRICES The sensitivity matrix turns out to be sparse in most practi cal settings Clearly the sparsity of the discretized sensitivity matrix D S depends on both the discretizing function D and the problem setting i e the network topology as well as the location of both measurements and states to be estimated We now turn our attention to fast rank calculations for sparse matrices as required by Claim 1 in order to check observability of the network It is important to note that determining the rank of a discrete matrix can be done using the singular value decomposition or the Gaussian elimination scheme which requires O n running time For sparse ma trices one encounters the problem of generating fill ins along the elimination However since the sensitivity matrix ain practical setting may be large we are interested in much faster computations The key observation is the following It turns out that in practical settings instead of determining the rank of the sensitivity matrix it is sufficient to only check for pairwise linearly dependent rows and columns of the sensitivity matrix Hence instead of determining the rank of the sensitivity ma trix we compute a so call
16. ol col while further rows and columns are eliminated A n x r submatrix of A after elimination return min n r end Fig 5 Computing the pseudorank of an n x r matrix A The algorithm uses the subroutine Eliminate as specified in Fig 4 independent the check for pairwise linear dependence see Lemma 1 will profit since the sweep over the two vectors will be aborted very soon This implies that the inner loop in the sub procedure Eliminate is extremely fast V DETAILED OBSERVABILITY ANALYSIS Putting our results together we have a fast test to determine the observability of a network with n measured values M and r states X 1 determine the n x r sensitivity matrix S of M with respect to X see Definition 2 2 choose an appropriate discretizing function D R N 3 compute the pseudorank prank D S of the dis cretized matrix D S see Proposition 1 4 if prank D S lt X the network is not observable Under the assumption that prank D S rank D S which is most often the case in practical settings the inverse also holds true if prank D S X the network is observable Besides determining the system s observability our ap proach even offers a much more detailed description of the system s observability status On the one hand the algorithm is able to determine redundant measurements and group them into redundance classes On the other ha
17. ordered set of n measured quantities M M1 Mn in the network and in addition an ordered set of r states X x1 2 The question is whether the states can be observed by the given measurements Definition 1 For a given state x define the sensitivity vector with respect to the measured quantities M to be om om M 1 n sens 2 hs Ca ar an Here Uui denotes the sensitivity of m with respect to x Similarly let om om sensx m x m a an denote the sensitivity for a measured value m with respect to the state set X The following two basic observations are the core part of our algorithmic approach e Redundance of measurements 1 If for two distinct measured values mM Mmi E M the sensitivity vectors sens x m and sens x mj are linearly dependent then m and m carry the same information content in order to distinguish between the states X This implies that in terms of the system s observability status one of the two measurements is redundant 2 Generally speaking each subset mj mi of M has non redundant measurements if and only if the corresponding sensitivity vectors sensx mMj Sens x m are linearly indepen dent e Observability of individual states 1 On the other hand if for two distinct states i Zi X the sensitivity vectors sens x and sens x are linearly dependent then the given set of meas
18. p positions of 2 and 3 winding transformers were included in the set of states to be estimated see Scenario I in the table of Fig 6 The given dimensions resulted in a corresponding 435 x 164 sensitivity matrix The continuous matrix was discretized to the range 100 100 We detected high sparsity only 3 09 non zero elements Our method computes the pseudorank in 2ms whereas an ordinary rank computation performed by MATLAB is more than 20 times slower Even if we use MATLAB to perform sparse rank calculations based on a singular value decomposition for sparse matrices the pseudorank calculation is 16 times faster see the last three rows of the table in Fig 6 In addition our investigations revealed 184 redundant mea surements among which we found 39 P meas 23 Q meas 10 I meas and 112 V meas The redundant measurements could be grouped into 62 redundance classes The highest level of redundance detected for such a class was 6 B Test scenarios II and III This speed up pays off in particular for larger systems In order to obtain sensitivity matrices of higher dimensions we fully supplied the network with further artificial measure ments we additionally introduced P Q and I measurements at each side of any branch and V measurements at each node This resulted in a total of 4397 measurements Moreover we increased the number of states to be estimated to 813 In this setting we estimated all possible
19. trix We gain speed ups by a factor ranging from 5 up to 20 define the c pseudorank c prank A of a matrix A It then holds for c gt co gt 2 that rank A oo prank A lt lt cy prank A lt co prank A lt lt 2 prank A prank A In other words the higher the value of c the more accurate the observability analysis will be On the other hand the running time for computing c prank A will increase for higher values of c The task then is to find a suitable trade off between accuracy and running time The question remains whether there are like for the case c 2 also fast algorithms to compute the c prank A on sparse matrices for any fixed c which outperform the computation of rank A ACKNOWLEDGMENT The authors would like to thank NamPower Namibia for providing the network specifications of their model REFERENCES 1 A Abur and A G Exposito Detecting multiple solutions in state estimation in the presence of current magnitude measurements IEEE Transactions on Power Systems 12 1 370 375 1997 2 A G Exposito an A Abur Generalized observability analysis and measurement Classification IEEE Transactions on Power Systems 13 3 1090 1095 1998 3 J J Graininger and W D Stevenson PowerSystem Analysis McGraw Hill Series in Electricacl and Computer Engineering 1994 4 G N Korres P J Katsikas and G C Contaxis Transformer tap setting observability
20. uare sum of all deviations between calculated and measured branch flows and bus bar voltages whilst fulfilling all load flow equations This can be expressed with a weighted square sum of all deviations between calculated calVal and measured meaVal branch flows and bus bar voltages iu Swi calVal x meaVal i 1 The state vector x contains all voltage magnitudes voltage angles and also all variables to be estimated such as active and reactive power injections at all bus bars Because more accurate measurements should have a higher influence to the final results than less accurate measurements every measurement error is weighted with a weighting fac tor w to the standard deviation of the corresponding mea surement device Plausibility Check Eliminate Errornous Measurements Observability Analysis Repair Unobservability Still Unobservable Observable State Estimation non linear Optimization Eliminate Bad Measurements Bad Data Detection No Bad Measurements Exists OK Fig 1 Scheme of the PowerFactory state estimator algorithm Our implementation uses an iterative Newton approach to solve the minimization problem based on Lagrange multipli ers Here all load flow equations are formulated as equality constraints for the optimization problem If the observability analysis in the previous step indicates that the entire power system is observable convergence in general is guaranteed
21. urements M does not allow to distinguish between the states x and x In other words the states x and x are only observable as a group but not individually 2 Again a subset of states 7 7 of X is indi vidually observable if and only if the corresponding sensitivity vectors sens x sens x are linearly independent Summarizing these ideas we gain information about the observability of the network by considering the following matrix Definition 2 For given sets of measured values M and states X we call eons Ss E e E sens zi ocx sens 7 the sensitivity matrix of M with respect to X External Grid Besg 4 DigSILENT Term 1 E Load 1 L1 P Q Load 2 L2 P Q estimated states measurements P L1 Q LI Tap P L2 Q L2 V Term 1 0 0 0 0 0 P Trafo Term 1 1 0 0 1 0 Q Trafo Term 1 0 1 0 0 1 P Trafo Term 2 1 0 0 1 0 Q Trafo Term 2 0 1 0 0 1 V Term 2 0 0 1 0 0 P Load 1 1 0 0 0 0 Q Load 1 0 1 0 0 0 Fig 2 Trivial example network together with its sensitivity matrix to illustrate the construction according to Def 2 Here five states of the network are to be observed by eight measurements The estimated states are the active and reactive powers in loads Load and Load 2 and the tap position Tap of the transformer In total three measurements for active power flow three measurements for reactive power
22. vidual unobservable states can be derived from our computations Further investigations on this topic might focus on an extension of our notion of the pseudorank To get a closer match to the rank of a matrix it could be of interest not only to take into consideration the pairwise linear dependencies of rows and columns of the sensitivity matrix but investigate the linear dependencies of any fixed number c of row and column vectors In this sense we could analogously to the case c 2 test scenarios Scen I Scen II Scen III no of measurements 435 4397 4397 P meas 161 1070 1070 Q meas 121 1070 1070 I meas 11 1070 1070 V meas 142 1187 1187 no of estimated states 164 164 813 P states 79 79 307 Q states 79 79 307 Tap pos states 6 6 199 sensitivity matrix size 435x164 4397x164 4397x813 sparsity 3 09 2 74 2 81 rank full MATLAB 41 ms 422 ms 6437 ms rank sparse MATLAB 32 ms 218 ms 1344 ms pseudorank Prop 1 2 ms 31 ms 250 ms Fig 6 Test results on the network model as operated by NamPower Namibia Three different scenarios with varying dimension of the sensitivity matrix were investigated The last three rows compare the running times of rank calculations of the sensitivity matrix by MATLAB full matrix mode and sparse matrix mode with our pseudorank algorithm Prop 1 In each of the scenarios our algorithm to compute the pseudorank considerably outperforms the computation of the rank of the sensitivity ma
Download Pdf Manuals
Related Search