Measurement-based Coherency Identification And Aggregation .
1Measurement-based Coherency Identificationand Aggregation for Power SystemsShaobu Wang; Shuai Lu, Member, IEEE; Guang Lin, Member, IEEE; and Ning Zhou, Senior Member, IEEE1Abstract —Model reduction techniques are often applied tolarge-scale complex power systems to increase simulation performance. The bottleneck of existing methods to get a high reduction ratio lies in: ① Coherency identification is static and conservative. Some coherent generators are not detected when systemtopology or operating point changes. ②Solitary generators outsideany coherency group are not aggregated regardless of their importance. To overcome the first problem, a measurement-basedonline coherency identification method was used in this paper. Byanalyzing post-fault trajectories measured by phasor measurement units (PMUs), coherency generators were identified throughprincipal component analysis. The method can track conherencygroups with time-varying system topology and operating points.To address the second problem, sensitivity analysis was employedinto model reduction in this paper. The sensitivity of tie-line powerflows against injected active power of external system generatorswas derived. Those generators having minimal impacts on tie-linepower flows were replaced with negative impedances. Case studiesshow that the proposed method can handle well these solitarygenerators and the reduction ratio can be enhanced. Future workwill include generalization of the sensitivity method.Index Terms — Power systems, Coherency identification,Generator aggregation, Model reduction, Phasor measurementunit, Sensitivity analysis.WI. INTRODUCTIONITH the increasing interconnection of regional powersystems, the size of modern power systems is becominglarger and larger. As a result, the number of state variables(dimension) undergoes explosive growth. This presents a severe challenge to stability analysis and control of modern powersystems (high computational cost in time domain simulation).In order to overcome this problem (to accelerate time domainsimulation), three primary approaches have been developed.The first approach relies on parallel or distributed algorithms toaccelerate the process of numerical integration of power systemdynamic equations so that the trajectory can be obtained in realtime or faster than real time [1]-[5]. The second approach is todevelop advanced algorithms that can be used to simulatepower system dynamics with larger time steps that can beFinacial support of this work was provided by Advanced ScientificComputing Research (ASCR) program of U.S. Department of Energy (DOE)Office of Science. The Pacific Northwest National Laboratory is operated byBattelle for the U.S. DOE under Contract DE-AC05-76RL01830.Shaobu Wang, Shuai Lu, Guang Lin, and Ning Zhou are with PacificNorthwest National Laboratory and can be contacted by email at (shaobu.wang@pnnl.gov, shuai.lu@pnnl.gov, guang.lin@pnnl.gov , ning.zhou@pnnl.gov).978-1-4673-2729-9/12/ 31.00 2012 IEEEdozens of times the time steps of traditional numerical integration methods, so as to accelerate time domain simulation [6].The third approach relies on model reduction to get a simplifiedpower system model that has the similar dynamics with the fullorder system [7]-[19]. This paper will discuss the third approach.Model reduction has been the focus of research for decades[7]-[19]. Model reduction consists of coherency generatoridentification and aggregation. In fact, performance of modelreduction mainly depends on coherency generators identification. For example, if the coherency group identification is basedon an operating point of power systems, then the model reduction performance cannot be guaranteed when large disturbancestake place or the operating point changes, because the oscillations or dynamics of generators depend on the operating point.Existing coherency group identification methods can bemainly classified into three types. The first type relies on analyzing linearized models of power systems around an operatingpoint [12]-[13]. Generally speaking, these kinds of methodsdepend on the operating point where the model is linearized.For example, when grid topology changes following largedisturbances with lines tripping, the identification results obtained before the faults may not hold well for post-fault powersystems. The second type identifies coherency groups by analyzing the results from offline simulation [14]. These methodsare reliable and can get a higher reduction ratio; but becausethey require traversing all possible scenarios, the computationalcost of these methods is high. The third type of coherencyidentification method relies on online measurements andcomputation [15]-[16]. These methods depend on advancedhardware (e. g., PMUs which was reported to measure voltages,rotor angles, etc. [20]). Along with popularization of PMUs anddevelopment of communication technique [21] in modernpower systems, advances of these kinds of identificationmethods emerge gradually.According to the above discussion, challenges of powersystem model reduction can be summarized as follows. ①Selfadaptive or robust coherency identification methods should bedeveloped to handle changes of system topology and operatingpoints. ② Higher reduction ratios are desired to handlelarge-scale power systems with large number of state variables.Existing methods are relatively conservative. Some coherencygenerators may not be detected and aggregated, especiallywhen power systems undergo large disturbances with systemtopology changed (e.g., line tripping).
2In this paper, first a PMU-based online coherency identification method is presented for model reduction. This methodidentifies coherency groups by analyzing measured oscillationsof state variables. Generators, whose rotor angles oscillate atsimilar modes, are considered as a coherency group. Then asensitivity analysis-based model reduction method is proposedin this paper. In external systems, there are some generatorswhose active power has little impact on tie-line power (lowsensitivity). Our reduction method ignores these generators’dynamics by replacing them with static models (e.g., idealcurrent sources or negative impedances). As a result, a highreduction ratio is obtained in this paper.The remaining parts of this paper are organized as follows.Section II presents a measurement-based coherency identification method. In Section III, a coherency generators aggregationmethod is introduced. In Section IV, a sensitivity analysis-based reduction method is proposed to enhance reductionratios. In Section V, a case study is presented to illustrate application of the method. Finally, some conclusions are drawn inSection VI.II. MEASUREMENT-BASED ONLINE COHERENCYIDENTIFICATIONMeasurement-based coherency identification methods haveattracted more and more attention, especially since WAMS(Wide Area Measurement Systems) emerged in the 1990s.Because online analysis or computation is required for measurement-based coherency identification, a simple and effectivealgorithm plays the key role in measurement-based coherencyidentification. In this paper, we adopt principal componentanalysis (PCA) [15], [24] as an online analysis tool to identifycoherency groups. The PCA is comparatively simple and withlow computational cost. This method can be summarized asfollows.The basic idea of PCA is to decompose a set of coupledvariables to a set of decomposed variables called principalcomponents. These decomposed variables are a linear combination of the original variables and are arranged in decreasingorder of their weights in the original variable oscillations (e. g.,the first principal component accounts for the largest proportionin the oscillations in the original variables). Now we illustratehow to apply the above method to measurement data and coherency identification.Suppose Y [ y1 , y2 ,L , ym ]T are m power system measurements (for examples, y can be rotor angles or velocities).Here y1, y2, , ym are n-dimensional column vectors (timeseries); Y is a m n matrix. Using singular value decompositionalgorithm Y UDV’, we can get three matrices: U, D and V. Andthen let T UD [T1, T2, , Tn] where T is an m n matrix; T1, T2, , Tn are m-dimensional column vectors. The m elements of Tireflect the proportion of the ith mode in the m measurements.The decoupled variables are arranged in decreasing order oftheir weights in original variable oscillations; therefore, the firstfew columns of T reflect the weights of the first few principalmodes in the measured oscillations. As a result, if we chose thefirst two columns of T as coordinates, a score plot can be obtained in a two-dimensional space; if we chose the first threecolumns of T as coordinates, then we can get another score plotin a three- dimensional space. The measurements from a coherency group have similar T-coordinates and cluster together.Therefore, the coherent generation groups can be identified byforming clusters. The detailed description about the method canbe found in [15], [24].III. MODEL REDUCTION--COHERENCY GENERATORSAGGREGATIONThere are several methods available for coherency generatoraggregation, for example: inertial aggregation [9], slow coherency aggregation [9], terminal bus methods [17]-[18], synchrony-based methods [19], [22]-[23], etc. Each method has itsown advantages and disadvantages. In this paper, we adopt theinertial aggregation method, which is straightforward and effective. The method can be summarized as follows.Suppose Generator i and Generator j belong to the samecoherency group. The inertial aggregation method aggregatescoherent generators at generator internal nodes, which are illustrated in Fig. 1. In this method, first the machine internalnode voltages are calculated, and then these nodes are connected to a common bus ‘p’ by introducing appropriate transformers and phase shifters to guarantee the original power flowis preserved precisely. Finally, extend the network at node ‘p’by introducing two more additional nodes with reactance x’dqand -x’dq as shown in Fig. 2. Here x’dq is the transient reactanceof the equivalent machine. The detailed steps for the aboveaggregation are given as follows: Step 1: Calculate the machine internal node voltagesWith the known bus injections Pg jQg, the machine currentinjection phasors into bus ‘i’ can be computed as:I%i conj[( Pgi jQgi ) / V%i ] .After that, the internal voltage phasor of the machine can beobtained by E% i' V%i jxdi' I%i . The internal voltage of generator jcan be computed in a similar way. Step 2: Construct the common bus ‘p’The voltage of the common bus ‘p’ is a linear combination ofindividual generator internal voltage, and can be obtained byusing an inertial weighted average of the individual generatorinternal voltage. Step 3: Connect buses ‘i’ and ‘j’ to bus ‘p’ with new linesThe new lines are introduced to connect buses ‘i’ and ‘j’ tobus ‘p’ as shown in Fig. 2. The impedance of the line betweenbus ‘p’ and bus i is equal to x’di, and its charging is equal to 0.The tap ratio of the transformer on the line is V%p / E% i' , and itsphase shift angle is arg(V%p / E% i' ) . The parameters of the linefrom bus ‘j’ to bus ‘p’ can be computed in a similar way. Step 4: Aggregate coherency generatorThe transient reactance and the inertia of the equivalentmachine can be computed by: x’dq 1/(1/ x’di 1/ x’dj), Hq Hi
3Hj. Here the parameters of machines ‘i’ and ‘j’ are on a commonMVA base. Step 5: Create bus ‘q’Because there are two (or more) buses connected to bus ‘p’,it is unreasonable using it (bus ‘p’) to represent a generatorinternal node in power systems. An equivalent bus ‘r’ havingthe same voltage as bus ‘p’ can be obtained by extending bus‘p’ to a bus ‘q’ with a line of impedance x’dq, and then to a bus‘r’ with a line of impedance -x’dq. With the equivalent nodes,bus ‘q’ can represent the internal bus, and bus ‘r’ can representthe terminal bus of the equivalent generator. The voltage phasorat bus ‘q’ is computed by: V%p and the power transfer to buses‘i’ and ‘j’. Step 6: Modify the property of buses ‘i’, ‘j’ and ‘q’After the aggregation, bus ‘i’, bus ‘j’, and bus ‘q’ are nolonger generator terminal buses, and these buses should bemodified into PQ buses.E% i'E% 'jjxdi'jxdj'V%iV%jFig. 1. Two coherency Generatorsr'jxdqppE% i'E% Fig. 2. Diagram for Inertial AggregationIV. ENHANCING REDUCTION RATIO THROUGH SENSITIVITYANALYSISA. Tie-Line Power Flow Sensitivity against Generator RotorAnglesIn power systems, external system’s dynamics couple withinternal system’s dynamics through tie-lines that connect theinternal area and the external area (shown in Fig. 3). Moreover,active power couples mainly with voltage angles. Therefore, thetie-line power flow sensitivity against voltage angles (rotorangles) can be reflected fully by the sensitivity against theinjected active power of generator terminal buses. As a result,we can use the latter to represent the former. Now, using Fig. 3,we show how to calculate the tie-line power flow sensitivityagainst the injected active power.V j θ jVi θiPijVk θ kFig. 3. Power Flow on Tie-LineHere Rij and Xij are the resistance and reactance of the tie-line,respectively. This active power sensitivity against the injectedpower of generator ‘k’ can be written as Pij Pij Vi Pij V j Pij θ i Pij θ j, (2)Ts Pk Vi Pk V j Pk θi Pk θ j Pkwhere Pk is the injected power of generator ‘k’, and Pij Rij 2Vi V j cos(θi θ j ) X ijV j sin(θ i θ j ) 22V RX iijij Pij RijVi cos(θi θ j ) X ijVi sin(θi θ j ) VRij 2 X ij 2j, (3) Pij RijViV j sin(θi θ j ) X ijViV j cos(θ i θ j ) θ Rij 2 X ij 2 i Pij RijViV j sin(θ i θ j ) X ijViV j cos(θ i θ j ) Rij 2 X ij 2 θ jand Vi / Pk , V j / Pk , θi / Pk , θ j / Pk can be computed asfollows.From power flow distribution sensitivity, we have Δθ ΔP ΔQ J 1 , VD 2 ΔV q' jxdqV%pE% rIn Fig. 3, the active power on the tie-line (from bus ‘i’ to bus‘j’) can be formulated as:Rij Vi 2 ViV j cos(θi θ j ) X ij ViV j sin(θ i θ j ) Pij . (1)Rij 2 X ij 2(4)where ΔP1 ΔQ1 Δθ1 ΔV1 Δθ ΔP ΔQ ΔV ΔP 2 , ΔQ 2 , Δθ 2 , ΔV 2 , M M M M ΔPn 1 ΔQm Δθ n 1 ΔVm VD 21 diag(V1 , V2 ,LVm ) .According to (4), we have Δθ 1 ΔP (5) 1 J , ΔQ VD 2 ΔV which give the values of Vi / Pk , V j / Pk , θi / Pk , θ j / Pkin – J -1. Substituting these values and (3) into (2), we can getthe sensitivity value. We show how to enhance the reductionratio using this sensitivity value in the following subsection.Remark 1: This sensitivity is computed using post-faultsystem, so the change of system topology (parameters) can bereflected in these values.B. Enhancing Reduction Ratios through Sensitivity AnalysisUsually, the number of external generators is very large; buta few do not have significant impact on tie-line power flow.Sometimes these generators stand alone (do not belong to anycoherency group); traditional reduction methods hardly handlethem.The sensitivity analysis discussed in the previous subsectionreflects the degree of generator impact on the tie-line powerflow, and a lower sensitivity means that the generator has less
4static impact on the tie-line power flow. With this idea, we canreplace these generators by connecting a negative impedance tothe terminal bus to generate the active and reactive power,which is illustrated in Fig. 4. Here Zk Vk/conj[(Pk jQk)/Vk]. Asa result, the dynamics of generator ‘k’ is ignored and less statevariables are left in the reduced system. We test if the sensitivity,based on static power flow, provides good guidance to reducethe dynamic model.V j θ jVi θiSijVk θ kFig. 4. Enhancing Reduction Ratios Using Sensitivity AnalysisThe sensitivity in (2) is derived under unit active powerperturbation. In order to quantify the impact of substituting thenegative impedance for the generator on the tie-line powerflow, the metric shown in (6) need be introduced. Here Pij andPk are defined in (2).S k Ts Pk ( Pij Pk ) Pk(6)External AreaFig. 5. Diagram of IEEE 50 Machine System [25]V. CASE STUDIESIn this section, the IEEE 145 bus 50 machine system [25]shown in Fig. 5 is used to test the proposed method. There are16 machines in the internal system, and 34 machines in theexternal system. All of the generators are models using theclassical model. A three-phase metal short-circuit fault is configured on Line 116-136 at bus No. 116 at t 1 second. Thefault lasts for 60 milliseconds, and then the line trips followingthe fault clearance. The measurements are machine rotor anglesthat are used to identify coherency group. The measured datawithin interval 1.2 t 5 (in sec.) are analyzed to determinecoherency groups (the highlighted generator in the internalsystem is the reference machine). Using the PCA methodmentioned in Section II, we chose the first three columns of T ascoordinates, and the Matlab clustering toolbox is used to doaided analysis so that visual clustering results can be obtained.By doing so, we obtained coherency clusters for the rotor angleoscillations in the external area shown in Fig. 6 in which thehorizontal axis represents machine numbers in the externalsystem; and the vertical axis scales distances between groups.Internal Area
5indicates that generators with higher sensitivities have moreimpact on the oscillations.102level 219TABLE ICOHERENCY GROUPS IN EXTERNAL SYSTEM8Distance7Groups No.165Generators No.11,14,261,4,5,12, 17,19,20, 21, 248,18,33,34,352,630,319,107,13, 1522, 2732, 3624level 1321013345678981139765432112122 27 1 5 17 20 19 21 24 4 12 11 26 14 2 6 8 35 34 18 33 25 32 36 3 7 13 15 16 9 10 23 30 31Machine numberFig. 6. Coherency ClusteringIn Fig. 6, different coherency groups can be obtained atdifferent distance levels. For example, at Level 2 shown, wecan get 2 groups: Machine 23, 30 and 31 constitute one group,and the others constitute another group. Similarly, there are 13groups at Level 13 shown in the plot. The less groups result inthe more simplified system. Now we take Level 13 as an example to illustrate the method in this paper. At level 13, thereare 4 solitary machines shown in Table II, and other coherencygroups are given in Table I. As shown by Fig. 5, there are21 tie-lines between the internal and external systems. According to (6), we use the metric defined in (7) to measure theimpact of substituting the negative impedance for generator ‘k’on the power flow in 21 tie-lines.21S k Pk r 1 Pr Pk(7)In (7), Pr is the active power Pij defined in (1); Pk represents theinjected active power of generator terminal bus ‘k’. Then, weget the sensitivity results of those solitary machines shown inTable II. As shown by Table II, Generator 23 and Generator 16have the highest and lowest sensitivities, respectively.Under the same disturbance as mentioned previously, therotor angle oscillations of the closest generator (G29 on Bus116 ) to the fault are given in Fig. 7 in which the blue solid linerepresents the dynamic responses of the original system(full-order system); the black dotted line is the dynamic responses of the reduced model with 13 machines in the externalsystem (without enhancing the reduction ratio); and the reddash-dot line is the dynamic responses of the reduced modelwith 12 machines in the external system (with enhanced reduction ratio by removing G16 which has the lowest sensitivity). It can be seen from Fig. 7 that the dominant oscillationsresponses of reduced models match those of the full model verywell, and substituting the negative impedance for G16 has littleimpact on the oscillations. Now we substitute the negativeimpedance for G23 which has the highest sensitivity, and getthe results shown in Fig.8 under the same disturbance. Bycomparing Fig. 8 and Fig.
The third approach relies on model reduction to get a simplified power system model that has the similar dynamics with the full order system [7]-[19]. This paper will discuss the third ap-proach. Model reduction has been the focus of research for decades [7]-[19]. Model reduction consists of coherency generator identification and aggregation.
Novel Results on Slow Coherency in Consensus and Power Networks Diego Romeres Florian Dorfler Francesco Bullo Abstract—We revisit the classic slow coherency and area aggregation approach to model reduction in power networks. The slow coherency approach is based on identifying sparsely and densely connected areas of a network, within which
SOURCE NODE EXPANSION ALGORITHM FOR COHERENCY BASED ISLANDING OF POWER SYSTEMS by Issah Ibra
proach to model reduction in power networks. The slow coherency approach is based on identifying sparsely and densely connected areas of a network, within which all generators swing coherently. A time-scale separation and singular pertur-bation analysis then results in a reduced low-order system, where coherent areas
Coherency-based power system model reduction [9], [13], [14], [15] have been well accepted by the power engineer-ing community and automated software for their application exists [7], [6]. These methods consider two important stages, first coherency in the generators of the power system [16]is
3 are partial derivatives of the power transfer between machines and terminal buses, K 1 is diagonal K 4 network admittance matrix and nonsingular. the sensitivity matrices Ki can be derived analytically or from numerical perturbations using the Power System Toolbox cJ.H.Chow (RPI-ECSE) Coherency November 1, 2017 9 / 29
coherency control protocols. We consider two locking and two optimistic schemes which operate either under central or distributed control. For coherency control, we investigate so-called on-request and broadcast invalidation schemes, and employ buffer-to-buffer communication to exchange modified pages directly between different nodes.
methods in the model reduction of power systems. Additionally, a connection between the Krylov subspace model reduction and coherency in power systems is proposed, aiming at retaining some physical relationship between the reduced and the original system. Index Terms—Coherency, Krylov subspaces, power system dy-
3.6 BEM as TQM framework . 43 3.6.1 Deming Prize . 44 3.6.2 The Malcolm Baldrige National Quality Award . 45 3.6.3 European Foundation for Quality Management . 48 3.7 Abu Dhabi Award for Excellence in Government Performance . 50 3.7.1 The ADAEP model . 51 3.8 Comparison between the BEMs . 52 3.9 Chapter summary . 54 4 Chapter Four: Framework of the Study .
