Structured Model Reduction Of Power Systems
2012 American Control ConferenceFairmont Queen Elizabeth, Montréal, CanadaJune 27-June 29, 2012Structured Model Reduction of Power SystemsChristopher Sturk, Luigi Vanfretti, Federico Milano, and Henrik SandbergAbstract— This paper shows how structured model orderreduction can be applied to power systems. For power systemsdivided into a study area and an external area, the proposedalgorithm can be used to reduce the external area to a low orderlinear system, while retaining the nonlinear description of thestudy system. The reduction of the external area is done in sucha way that the study system is affected as little as possible.It is shown that a lower model order can be attained wheninformation about the study system is taken into consideration,than if the external system is reduced independently of it.Index Terms— Model reduction of power systems, internalsystems, structured model reductionI. I NTRODUCTIONPower systems today are colossal networks of interconnected power apparatus often spanning large geographicalareas. With the growing trend in facilitating additional interconnections to neighbouring systems, the size and complexity of these networks will likely continue to increase,and thus, bring challenges for planning, operations andcontrol of these networks. Hence, reduced-order models ofpower systems are desirable for many applications and studies, particularly for fast and cost-efficient stability analysis.Research on power system model reduction is extensive,and some methodologies focusing on specific applicationshave been implemented in software for automated modelreduction. Generally speaking, model reduction of largepower networks, typically known as Power System DynamicEquivalencing [1], has the main aim of providing a systemequivalent model able to reproduce the aggregated steadystate [2] and dynamic characteristics of the full-order network [3], while at the same time being compatible with theavailable computation tools for power system analysis [4],[5].Model reduction techniques are often developed for specific applications such as: steady-state power flow for security assessment [6], small-signal stability analysis [7], transient stability [8], [9], network equivalents for electromagnetic transient studies [10], and control design applications[11]. In these applications the power system is divided intoC. Sturk and H. Sandberg are with KTH Royal Institute of Technology,ACCESS Linnaeus Centre, School of Electrical Engineering, AutomaticControl Lab, Stockholm, Sweden. E-mails: {chsturk, hsan}@kth.se. C.Sturk and H. Sandberg are supported in part by the Swedish ResearchCouncil under Grant 2007-6350 and Grant 2009-4565, and the SwedishFoundation for Strategic Research under the project ICT-Psi.L. Vanfretti is with KTH Royal Institute of Technology, School of Electrical Engineering, Electric Power Systems Division, Stockholm, Sweden.E-mail: luigiv@kth.se. L. Vanfretti is supported by the STandUP for Energycollaboration initiative and the KTH School of Electrical Engineering.F. Milano is with the Department of Electrical Engineering, Universityof Castilla - La Mancha, 13071 - Ciudad Real, Spain. E-mail: Federico.Milano@uclm.es.978-1-4577-1096-4/12/ 26.00 2012 AACCan internal (or study system) and an external system. Theinternal or study system is usually set by the boundaries ofthe part of the system that a specific utility owns or it maybe the area in which a particular power market is defined,usually this portion of the network is electrically close.However, it would be advantageous to arbitrarily set the areacorresponding to the study system without any restrictions,which could aid in the design of power plant controllersand coordination of system protections. The external systemis usually electrically distant, and in practice it is used todenote the service area of other utilities. The internal systemis usually represented in detail while a reduced model is usedfor the external system.From these approaches, there is particular interest in thosethat can preserve network structures and can be used forsmall signal analysis, transient stability analysis, and controldesign. Coherency based methodologies [1], [9], [12], [13]have been well accepted by the power engineering community and automated software for their application exist[4]. These methods start by identifying coherency in thegenerators of the power system [14], and in a second stageto proceed with the dynamic reduction of the system. Thedynamic reduction process itself is carried out by aggregatingthe network [15] and aggregating the generators [16], [13].At a later stage, it may even be possible to aggregate excitation controllers [17]. Although coherency methods havebeen accepted as the most reliable for power system dynamicequivalencing, the major drawback is that it may not alwaysbe possible to reduce specific parts of the power network,the nature of coherency is to cluster generator groups whichimposes the areas in which the network can be divided. Toovercome this limitation, some approaches that are capableof retaining a part of the network have been proposed [18],[19]With the recent developments in power systems an increased interest has been seen from the automatic controlcommunity. Among other things it has been shown howmodel reduction algorithms popular in control can be applicable. These algorithms typically have a strong theoreticalfoundation and they are also very general in the sense thatthey are not targeted to a particular application. This makesthem a good candidate for the reduction of power systemscomposed not only of synchronous machines but also of forinstance renewable energy sources.In [20] it is shown how Krylov subspace model reductioncan be applied to the external area before reconnectionwith the study area. Principal component analysis (PCA) isanother popular model reduction algorithm. In particular itis well-suited for reduction of nonlinear systems. In [21] it2276
was used to project the states of the external system onto asubspace of lower dimension.What we propose in this paper is to apply structured modelorder reduction to power systems with an external and astudy area. The idea is to reduce the external system whiletrying to minimize the effect it will have on the study area.This is the main objective of structured model reduction,namely to reduce models locally while ensuring a smallglobal model error.Model reduction where various structural constraints aretaken into account (”structured model reduction”) has beenconsidered in several papers. For example, in [22] frequencyweighted model reduction problems are considered, and in[23] controller reduction is considered. More general interconnection structures have been considered in, for example,[24], [25], [26].The structure of the paper is as follows: In Section IIwe formulate the problem we want to solve. Section IIIsummarizes the theory of structured model order reductionand in Section IV it is shown how this theory can beapplied to power systems. Section V introduces the WSCC3-machine, 9-bus system to which the proposed model reduction algorithm is applied.II. P ROBLEM F ORMULATIONWe will assume that the power system we want to reducecan be divided into one part with system variables of interestto us, called the study area, and another part called theexternal area, that is only of interest in terms of its effecton the system variables in the study area (Fig. 1). This ismotivated by the fact that one often only focuses on parts ofthe power system. Under this assumption the aim is to reducethe external area to a linear time-invariant (LTI) system oflower order, while retaining the nonlinear description of thestudy area. This is relevant since it allows for a physicalinterpretation of the reduced power system, which is helpfulif one wants to simulate changes to the study area in termsof for instance transmission line failures or for the purposeof nonlinear control design.V1study ,studyd1Tie-Line 1V1ext ,ext1StudyArea V2study ,studyd2Tie-Line 2V2ext ,ext2Vnstudy ,studydnTie-Line nVnext ,extnExternalEAreaFig. 1. The power system is divided into a study area and an external area,which is to be reduced.The interface between the systems is defined by their ntie-lines and buses. Given that the network with N buses hasan admittance matrix Y G jB, every bus satisfies thetwo power equationsN P̃i Vi2 Gii Vi Vj Bij sin(θi θj )j 1;j i N Vi Vj Gij cos(θi θj )j 1;j iQ̃i Vi2 Bii N Vi Vj Gij sin(θi θj )j 1;j i N Vi Vj Bij cos(θi θj )(1)j 1;j iwhere P̃i and Q̃i are the injected real and reactive powerrespectively of bus i. This means that in order to have theequations (1) well-defined for the buses of the external area,it is required that it has the voltage magnitudes V istudy andphases θistudy that are adjacent to it as input signals. Andhaving the external area output the voltage magnitudes V iextand phases θiext will ensure that the study area has all therequired bus variables available to make (1) well-defined forall its buses that are adjacent to the external area. The inputand output signals are analogously defined for the study area.Given this interface the external area can be linearizedaround a steady-state and reduced after which it is reconnected to the study area. The objective is to do the modelreduction so that the dynamics of the nonlinear study areais affected as little as possible when replacing the nonlineardescription of the external area with the reduced order linearmodel.III. M ODEL R EDUCTION M ETHODStructured model order reduction is a model reductiontechnique that can be applied to systems composed of subsystems that are interconnected with some network dynamics.The idea is to reduce the model order of the subsystems whileretaining the interconnection structure and keeping the globalmodel error small. The general system (Fig. 2) to whichstructured model reduction can be applied is composed of qsubsystems described by the transfer function matrix 0G1 (s)AG BG .(2)G(s) :.CG DG0Gq (s)with transfer functions G k (s) Cpk mk and an interconnecting network AN BN,1 BN,2E(s) F (s) : CN,1 DEN (s) DF .H(s) K(s)DKCN,2 DH(3)We want to find a reduced order system that approximatesNthe mapping from u N1 to y1 defined by the lower linearfractional transformationFl (N, G)2277 E(s) F (s) (I G(s)K(s))A B :C D 1G(s)H(s)(4)
y1NyN (s )N2uG (s )uGN2Thus if the original state vector has the structure,u1Nx [xTN xT1 . xTq ]T ,then the transformed system will have the states x̄ definedbyT x̄ x,yGFig. 2. The interconnected system. G is the system that should be reducedNand N is the interconnecting network. uN1 and y1 are external input andoutput respectively.with the constraint that the interconnecting network N isretained, i.e. the objective is to find the reduced order systemĜ such that Fl (N, G) Fl N, Ĝ (5)is made as small as possible andĜ {F (s) : F (s) diag {F1 (s), ., Fq (s)}}where Fk (s) Cpk mk , k 1, ., q.Finding the optimal minimum to (5) is very difficult, sinceit is a nonconvex optimization problem. We will thereforehave to resort to suboptimal methods, which yield solutionssatisfying the constraints while trying to minimize the normof the model error. The model order reduction algorithm usedin this paper is based on the idea of balanced truncation,see for example [22] and [27]. To enforce the structuralconstraints we use a generalization of balanced truncationas described in [28], [29], [25], [26]. The notation used hereclosely follows the one used in [25] and [26].It uses the reachability and observability Gramians P andQ given by the Lyapunov equationsAP P AT BB T 0,AT Q QA C T C 0, (6)with the matrices A, B, C defined by (4). It is helpful to usea partition with the blocks Q N , PN for the interconnectingnetwork that is not reduced and with the blocks Q k , Pk forsubsystem k that should be reduced separately, but in a wayso that the closed-loop dynamics is retained. Q1 . . . Q1qQN QN G . (7).Q , QG . QTN G QGTQ1q · · · QqP PNPNT GPN GPG P1 ., PG .TP1q . . . P1q. . · · · PqP̄k Q̄k Σk diag {σk,1 , ., σk,nk } ,σk,1 . σk,nk 0,λj (Pk Qk ) λj (P̄k Q̄k ).whereT diag(TN , T1 , ., Tq ), TN RnN nN , Tk Rnk nk ,and nN and nk are the order of system N and G k respectively. A nonsingular matrix T defining the balancingcoordinate transformation can always be found given thatthe system is stable, see Corollary 7.7 in [30].Having made a coordinate transformation, either truncation or singular perturbation is used to calculate the reducedorder systems Ĝi of the subsystems in G. To this end thestructured Hankel singular values can guide the choice ofwhich states to retain as explained below.The strength of the structured model reduction algorithmis that the block-diagonal elements of the Gramians definedby (6) tell us how reachable and observable the states ofthe subsystems are when we control the global input signalNuN1 and observe the global output signal y 1 (Fig. 2). Moreprecisely, the following holds [25]:If all states excepts the ones in subsystem k are zero at timezero, then y1N (t) 2[0, ] xk (0)T Qk xk (0).(9)(11)and assuming that all states of the interconnected system arezero at t and that we would like to control the statesof subsystem k to the specific state x k (0) x k , while thestates xN (0) and xi (0), i k are free variables, then theminimum control signal satisfiesmin2 T 1 uN1 (t) [ ,0] (xk ) Pk xk ,uN1 L2 ( ,0)x(0) X kXk {x : x has structure (10) and xk x k }.(12)The structured Hankel singular values (9) together with(11) and (12) indicate which states that are most importantto retain; the larger they are, the more controllable andobservable will they be through the exogenous input u N1 andoutput y1N . However if it is desired to have an upper boundon the model error one has to resort to LMIs [26].IV. S TRUCTURED M ODEL R EDUCTION OF P OWERS YSTEMS(8)The method balances the subsystems G k (s) by the coordinate transformation x k Tk x̄k that makes the transformedGramians Q̄k TkT Qk Tk and P̄k Tk 1 Pk Tk T subsystembalanced, which means thatσk,j (10)The structured model reduction algorithm accounted forin Section III is based on the notion of dividing the systeminto the subsystems N and G. This makes it suitable forapplication to power systems with one study area, which wewant to retain a non-linear description of and one externalarea which we want to reduce. We will henceforth assumethat the system G only consists of one subsystem that wewant to reduce, i.e. with the notation introduced in SectionIII, q 1. Although we make this assumption, there is2278
nothing preventing G from being composed of more thanone subsystem, something that could be of interest in largepower systems with several areas of particular interest. Wenow propose the following four-step algorithm:1. Defining the modelA general power system will be described by differentialalgebraic equations (DAE) of the formẋ f (x, xalg , u)0 g(x, xalg , u).(13)The states x will be describing the generators, controllers,etc., whereas the algebraic variables x alg will be the voltagesand phases of the buses as well as algebraic variablesdescribing the generators, controllers, etc. The signal u willbe an exogenous input to the power system. It could forinstance describe time-varying loads. Divide this system intoa study area, denoted N and an external area denoted G. Ageneral system with this structure can be described by theDAEsGẋG f G (xG , xGalg , u )(14)G0 g G (xG , xGalg , u )andNNẋN f N (xN , xNalg , u1 , u2 )NNNNN0 g (x , xalg , u1 , u2 )(15)The variables u G and uN2 are the voltage magnitudes andphases of the buses at the tie-line as described in Section IIand uN1 is the same exogenous input as in (13), (Fig. 2).2. LinearizingIn order to apply the structured model reduction algorithmdescribed in Section III it is first necessary to linearizeboth the study area and the external area. By solving thepower flow problem, the steady-state of the power systemis acquired around which the linearization is done. Thelinearization of (14, 15) will take the form G G G G xA11 AGB1ẋ12 uGGGx0AGAB2Galg2122 N N N xA11 ANẋ12 NxN0ANA 21 N 22 N alg N B11 B12u1 .NNB21B22uN2and 1NNN NẋN (AN11 A12 A22 A21 )x N u1NN N 1NN N 1 (B11 A12 A22 B21 ) (B12 A12 A22 B22 )uN2 N 1y10 AN22MNy2N N 0 Iu1N NN NN N·(A21 x B21 u1 B22 u2 ) uN0 02(17)We can note that the system N has one input signalNuN1 and one output signal y 1 apart from the input/outputpair that defines its interconnection with the externalarea G. These are the exogenous inputs and the globaloutputs and they will be elaborated upon in Subsection IV-A.3. Model reductionWith G and N on the form (16) and (17) the state spaceequations for the interconnected system can readily be foundfor which the reachability and observability Gramians canbe calculated with (6). Selecting the submatrices P G andQG from the matrices (8) and (7) a change of coordinatesfor the system G can be found and guided by the structuredHankel singular values (9) the model order can be selected.4. Nonlinear modelWith the system G being reduced toĜ(s) AĜCĜBĜDĜit can be reconnect to the non-linear description of the studyarea yielding the reduced interconnected systemẋĜ AĜ xĜ BĜ uĜĜĜuN2 yĜ CĜ x DĜ uNNẋN f N (xN , xNalg , u1 , u2 )NNNNN0 g (x , xalg , u1 , u2 )uĜ y2N MN xNalg .A. Algorithm preferencesHaving linearized the system, we want to find the reducedsystem Ĝ that makes (5) as small as possible. The outcomeof the model reduction will of course be dependent on theWhat makes these systems easy to work with is that the choice of exogenous input signals u N1 and output signalsalgNandxcanbesolvedforalgebraic variables x algy.Intheexamplethatwillbeusedto demonstrate theGN1N 1algorithmtheinputsignalsuwerechosenas the loads ofGG GG G1xGalg A22 (A21 x B2 u )the buses 5 and 6 (Fig. 3). This choice could be of interestN 1N NN NN Nif one wants to design a power system stabilizer (PSS) atxNalg A22 (A21 x B21 u1 B22 u2 ).the generator connected to bus 1 that attenuates oscillationsIf the matrices M G and M N select which algebraic variables following load variations. However, the choice should bethe two subsystems output, i.e. the tie-line voltage magniapplication dependent.tudes and phases, the DAEs can be recast into the followingThe most natural choice for the output variables y 1N isordinary differential equationsto choose them as the voltage magnitudes and phases ofG G 1 GGGG G 1 G G the buses of the external area at the tie-lines. This makes aẋG (AG11 A12 A22 A21 )x (B1 A12 A22 B2 )ugood choice considering how we interconnect the two areas.G 1GG Gy G M G A22(AG(16) Assuming that the reduced order external system exhibited21 x B2 u )2279
the same voltage amplitudes and phases on the buses at thetie-lines as the full external system, then it would follow thatthe trajectories of the study system of both models wouldbe identical, given that they had the same initial condition.Therefore if we can make the tie-line variables of the reducedsystem and the full system be close to each other, then thetwo study systems should have similar trajectories which isthe objective of the model reduction.connected to buses 2 and 3 respectively as well as theremaining buses. The synchronous machines are modeled asfourth order systems with connected AVRs of the standardIEEE model 1. The DAEs for each machine are Ωb (ω 1) (pm pe D(ω 1)) /M ′ ′ eq (xd x′d )id vf /Td0 ′ ′ ed (xq x′q )iq /Tq0vq ra iq e′q x′d id ′ ed 0 B. Required assumptionsThe structured model reduction algorithm, which is basedon balanced truncation, require that the interconnected system is asymptotically stable, since
Research on power system model reduction is extensive, and some methodologies focusing on speciÞc applications have been implemented in software for automated model reduction. Generally speaking, model reduction of large power networks, typically known as Power System Dynamic Equivalencing [1], has the main aim of providing a system
model reduction of power systems Problem and results Model reduction of nonlinear large-scale power system Clustering, linearization, and reduction of external area Application of structured balanced truncation Reference. Sturk, Vanfretti, Chompoobutrgool, Sandberg: "Coherency-Independent Structured Model Reduction of Power Systems".
2. Reduction: (i) Reduction of aldehydes and ketones to primary or secondary alcohol using sodium borohydride or lithium aluminum hydride. (ii) Reduction of aldehydes or ketones to hydrocarbons using Clemmenson reduction or Wolff-Kishner reduction Clemmensen reduction Wolff-Kishner reduction 3. Oxidation: Aldehydes can be easily oxidized to carboxylic acids using nitric acid, potassium
Figure: R. Podemore: Coherency in Power Systems 3 / 1 Model Reduction, Identification, and Distributed Optimization of Power Systems . Identification of Dynamic Reduced-Order Models of Power Systems Power System Model Reduction A Case Study– NPCC 48 Machine Model G 34 G 48 G 46 G 45 43 G 7 G 6 G 4 G 5 G 10 36 97 92 33 91 96 95 93 94 111 .
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employed including chemical reduction using hydrazine or NaBH 4 [14], high-temperature annealing reduction [15], hydrothermal reduction using supercritical water [16], green chemistry method [17], and photocatalytic reduction using semiconductors [18-21]. Among them, the photocatalytic reduction is an environment-friendly and
Model Reduction of Power System Dynamics using a Constrained Convex-optimization Method Sanjana Vijayshankar, Maziar S. Hemati, Andrew Lamperski, Sairaj Dhople Abstract—This paper discusses methods for model reduction of power system dynamics. Dynamical models for realistic power-systems can very easily contain several thousands of states.
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