An Adaptive Three-bus Power System Equivalent for Estimating Voltage Stability Margin from Synchronized Phasor Measurements

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1 An Adative Three-bus Power System Equivalent for Estimating oltage Stability argin from Synchronized Phasor easurements Fengkai Hu, Kai Sun University of Tennessee Knoxville, TN, USA Alberto Del Rosso, Evangelos Farantatos, Navin Bhatt Electric Power Research Institute Knoxville, TN, USA Abstract This aer utilizes an adative three-bus ower system equivalent for measurement-based voltage stability analysis. With that equivalent identified online, a measurementbased aroach is develoed to estimate real-time voltage stability margin for a load-rich area suorted by remote generation via multile tie lines. Comared with traditional Thevenin equivalent based aroach, this new aroach is able to rovide more accurate voltage stability margin for each individual tie line. This aroach is validated on a three-bus system and the IEEE 39-bus system. Index Terms arameter estimation; hasor measurement unit; Thevenin equivalent; voltage stability monitoring I. INTRODUCTION oltage instability is a major concern for ower systems oeration. Usually, it initiates from a local bus or region but may develo to a wide-area or even system-wide stability roblems. Online oltage Stability Assessment (SA) is a key function in system oerations to hel oerators foresee otential voltage insecurity. Traditional SA is based on a simulation-based aroach. By either ower-flow analysis or time-domain simulation, it emloys ower system models to simulate a list of contingencies on a State Estimator solution that reresents the current oerating condition. However, the simulation based aroach has limitations: it is modeldeendent and it requires a convergent State Estimation solution for the current oerating condition. Different measurement-based SA aroaches have been studied to directly estimate real-time voltage stability margin for a monitored load bus or area []-[9] or redict otential ost-contingency voltage insecurity by means of data mining techniques []. A majority of the measurement-based aroaches are based on Thevenin s Theorem. For instance, local measurements at the monitored buses are used to aroximate the rest of the system as an imedance connected to a voltage source, i.e. the Thevenin equivalent. The ower transferred to the bus reaches its voltage stability limit when that external Thevenin imedance has the same magnitude as the load imedance at the bus []. Based on Thevenin equivalent, the voltage stability or reactive-ower reserve indices can be obtained []-[5]. A modified model with two equivalent voltage sources is studied in [6] to redict the stability limit. Paer [7] alies the Thevenin equivalent based aroach to an actual EH network. Some other works consider load ta changers and over-excitation limiters in their models for better detection of voltage instability [8][9]. The above methods work well on a radially-fed load bus or transmission corridors. EPRI develoed a Thevenin equivalent-based method for load center areas, which requires synchronized measurements on boundary buses [4][5]. As illustrated by Fig., the method merges all boundary buses and tie lines to one fictitious boundary bus connected by one tie line with the external system such that the Thevenin equivalent can be alied. An ongoing roject is demonstrating this method in the real-time environment []. Figure. Load area and its Thevenin equivalent However, the Thevenin equivalent-based aroach cannot rovide voltage stability margin for each individual tie line when the monitored load is fed by multile tie lines. For such a case, transfer limits of various tie lines may be reached at different times, or in other words, voltage instability may start near one of the boundary buses sooner and then rogress to This work is suorted by the Electric Power Research Institute and the US Deartment of Energy (under award DE-OE68) /4/$3. 4 IEEE

2 the others. However, by monitoring the total transfer limit through a single equivalent, the Thevenin equivalent based aroach may not detect the time variability across the interface associated with voltage instability. In this aer, an adative three-bus ower network equivalent is roosed for estimating voltage stability margin for a load-rich area fed by multile tie lines. A real-time voltage stability monitoring method is then develoed based on that new equivalent. It is exlained and demonstrated later that such a three-bus equivalent, if alied to a load-rich area fed by two or more tie lines, is able to estimate the real-time ower transfer limit in terms of voltage stability for each individual tie line if synchronized measurements are available on all boundary buses. This new method is tested on a threebus system and the IEEE 39-bus system. II. A THREE-BUS EQUIALENT Figure. Prosed 3-bus equivalent Figure 3. Strategies for three-bus equivalencing N equivalents As shown in Fig., a three-bus ower network equivalent is roosed to monitor voltage stability for a load-rich area. Its three buses include a voltage source and two interconnected load buses reresenting the load center. The voltage source reresents the external system, whose generators are assumed to be strongly coherent without risking any angular instability. The two load buses reresent either actual or fictitious boundary buses, deending on the requirement of voltage stability monitoring. For examle, if it is required to estimate the transfer limit for each of the N tie lines of a load area, any tie line versus the rest can be studied to create N three-bus equivalents, as illustrated by Fig. 3. Then, voltage stability analyses on all such equivalents rovide comrehensive results on all tie lines. In ractice, usually only one or very few tie lines are most vulnerable to voltage instability, so it is unnecessary to study all N equivalents. Since this equivalent does not model generator AR limit, it focuses on detecting or redicting the saddle-oint bifurcation tye voltage collase on the load side []. However, since the equivalent will be estimated in real time from measurement data, it also has otentials in reflecting significant changes on the generation side, e.g. voltage dros due to a generator limit being met. III. APPROACH FOR OLTAGE STABILITY ARGIN CALCULATION Based on the three-bus equivalent, an aroach for calculating voltage stability limits and margins for N tie lines of a load area is resented in this section. The aroach assumes that time-synchronized voltage hasor data ~ N at boundary buses and current hasor data I ~ I N of tie lines are available. The data may be from synchrohasors, e.g. hasor measurement units (PUs) at 3-6 samles er second or a state estimator at a slower rate, e.g. s to several minutes, deending on the seed requirements for voltage stability monitoring. This aer uses synchrohasor data as an examle. The aroach conducts the following stes: i) Determine the number of three-bus equivalents, deending on how many lines need to be monitored in detail for ower transfer limits and margins. For each equivalent, use measurements to calculate voltage hasor data of and and comlex ower-flow data of S and S on two load buses as indicated by Fig.. For instance, if the st bus is selected vs. the others,,, S and S are calculated by () =, S = I N N N = i i i, = i i i= i= i= I I S I ii) At any time when estimation of stability limits or margins is exected, use the data of,, S and S over a latest time window to estimate the other arameters of the three-bus equivalent including those of the external system, i.e. E,,, and those of the load area, i.e. L, L and T. Details are resented in subsections A and B. iii) Find the maximum limit of the active ower transferred to each of the two load buses, denoted by P max and P max. An exhaustive or heuristic searching algorithm may be emloyed to find the limit. Since the searching sace is not large for the three-bus equivalent, subsection C gives an algorithm for exhaustive searching. A. External System Parameters Estimation Assume that E, are are constant over the time window. Thus, similar to [], a least-square method may be adoted to give estimates for E, are. Note that the Thevenin equivalent has 4 real unknowns while this new equivalent has 6 real unknowns to solve, i.e.: T T [ E E R X R X ] ( H H H r i ) () = () where E r +je i =E, R +jx =, R +jx =, and matrices H and are formed based on measurement data at n time oints of the time window. r,k and i,k resectively denote the real and imaginary arts of bus voltage at the k-th time oint.,k and q,k are resectively the active and reactive owers received by bus at the k-th time oint. Similarly, r,k, i,k,,k and q,k are data of bus.

3 r, i, r, i, H = r i r i i, r, i, r, i r i r,,,, [ ] T r, i, r, i, r, n i, n r, n + i, n (3) = L (4) B. Load Area Parameters Estimation To estimate the load area arameters, at least two time oints (denoted by t a and t b ) of measurement data are needed. The following equations could be obtained: Y Y a Y a b Y b + ( + ( + ( + ( a b b a b a Symbols labelled a or b are linked to the corresonding time t a or t b. For examle, denotes the bus voltage hasor at time t a. Y denotes the admittance of the load connected to bus. Y reresents the transfer admittance between the two load buses, which is assumed constant. Another assumtion is that each load imendence has a constant imedance angle, i.e. constant ower factor. Thus, a a b b a b (5) G / B = G / B, G / B = G / B (6) Equations (5) and (6) actually corresond to real equations, which are solved for real unknowns, i.e. real and imaginary arts of comlex unknowns Y, Y, Y, Y a and Y b. The above constant ower factor assumtion can tolerate reasonably slight changes in the imedance angles over a short time window based on our studies. C. Finding the Power Transfer Limits Based on the current oerating condition, which deends on the estimated L and L, the maximum limits of the active ower transferred to two load buses need to be solved. It is assumed that L and L vary in a zone and then an exhaustive search is conducted to check ower-flow solutions of all meshed reresentative oints in that sace. The goal is to find the maximum ower flows delivered to the two load buses without causing voltage insecurity. Since the dimension of the sace is not high, those oints may have a very high density. Also, when solving the owerflow solution at each oint, the ower injected by the slack bus is limited within a range around its original value to avoid unrealistically large changes at the slack bus. A heuristic algorithm may also be alied to utilize the gradient information from two successive oints during the searching to seed on the rocess. I. CASE STUDIES A three-bus system and the IEEE 39-bus system are used to test the roosed aroach. Simulations are conducted and the simulation results at the boundary buses are treated as synchrohasor data. A. Three-bus System The three-bus system has the same structure as the equivalent shown in Fig. with all arameters given below in er unit: E =.475, =. + j., =.3 + j., L = j.595, L = j.345 Time-domain simulations are conducted to continuously decrease the magnitudes of two load imedances by % every second to simulate a load area with increasing load until two lines meet the maximum ower transfer limits. The uroses of this case are to demonstrate: ) how different the limits of two lines may be, and ) differences between the results of this new aroach and those from a traditional Thevenin equivalent based aroach. oltage(.u.) Active Power(W) Figure 4. P curves of two load buses with tight interconnection oltage(.u.) Figure 5. P curves of two load buses with weak interconnection Two cases are simulated with two different values of the transfer imedance, i.e..3+j.5u and.3+j.5u, which resectively reresent a tight interconnection and a relatively weak interconnection between the two load buses (corresonding to the boundary buses of the load area). Fig. 4 and Fig. 5 give the P curves from simulation results at two load buses. Each curve is about the bus voltage magnitude and the active ower transferred to the bus. When the two load buses are more weakly connected, the Bus P Bus P Bus P Bus P Active Power(W)

4 two P- curves are more different, indicating the need of the estimating stability limits for individual buses. The roosed aroach is erformed every second over a sliding time window of second. For tight and weak interconnections, Fig. 6 gives the active line flows P and P, and total interface flow P +P, and their limits calculated by the new aroach, i.e. P max, P max, and P max (new) =P max +P max. For comarison uroses, the total interface flow from the Thevenin equivalent based aroach is also given as P max (Thevenin) in the figures. Based on the results, it can be observed that when the two interface buses have tighter interconnection, the transfer limits of the two lines are met at the same time [around t=38s in Fig. 6(a)], which means the two buses can be reasonably merged into one bus without losing accuracy. That is the basic assumtion of the traditional Thevenin equivalent based aroach, so the Thevenin equivalent based aroach also estimates the total interface limit to be met almost at the same time as individual lines. As shown by Fig. 6(b), when the two buses are weakly connected, the limits of two lines are met at different time instants, at t=47s and t=6s, resectively. On the other hand, the Thevenin equivalent based aroach estimates that the total interface flow limit is met at around t=37s, i.e. not much different from the tight interconnection case. The results illustrate that if only the total transfer limit for the entire interface of a load area is estimated, detection of voltage instability may be delayed since some tie line may be more stressed and voltage instability may occur there first. For the IEEE 39-bus system, a load area is defined as indicated by Fig. 7. It has three interface buses, i.e. buses 4, 8 and 4. The system can be simlified into a three-bus equivalent system, and utilize the aroach roosed in this aer. The following contingency is simulated to create a voltage instability scenario: Starting from t=s, kee increasing the total load of the area from 898 W at a seed around.3w (with slight randomization) er second to create slow decay in the voltage level of the area. At t=439s, tri the generator on bus 3, i.e. one of the two local generators of the load area. Kee increasing the load of the area at the same seed until voltage collase around t=539s. Fig. 8 shows all bus voltage magnitudes, in which the highlighted curves are those inside the load area. Figure 8. New England system bus voltage magnitude (a) (b) Figure 6. Line flow limits for tight (a) and weak (b) interconnections between two buses B. IEEE 39-bus System Figure 7. IEEE 39-bus system diagram P 34 P m Figure 9. Active ower and voltage magnitude of buses on the boundary Fig. 9 gives the active ower flows and the bus voltage magnitudes of three boundary buses. Bus 4 and bus 4 have close voltage curves, so they can be merged into a single fictitious busamed bus E. The two lines connected, i.e. 5-4 and 3-4, are also merged to an equivalent lineamed line E. Thus, the three-bus equivalent is alied. Fig. gives the active ower flows of line 9-8 and line E and the voltage magnitudes at bus 8 and bus E. Fig. gives the P- curves from the simulation results on the two buses. It shows that two curves have different shaes and their nose oints may be reached at different times in the simulation. Parameters of the external system and load area are estimated at each time ste of.5s using measurements (i.e. from simulation results) on the three boundary buses over the latest 5s time window. Let and P E denote the active owers oltage agnitude (.u.) m8 m4

5 in the line 9-8 and line E, whose real-time values are directly from the measurements. At each time ste, in the lane about and P E, a rectangular region of ±5% around the oint corresonding to their real-time values is considered. Power flow solutions are studied for reresentative oints in the region at a density. The real ower change at the slack bus for each time ste is restricted to % in solving the ower flows. The maxima of and P E, i.e. max and P Emax, among all solved ower flows are identified as the limits of the two lines in terms of voltage stability. Fig. gives the identified active ower limits. Fig. 3 comares the ercentage active ower margins, i.e. (max - )/ % and (P Emax -P E )/P E %. At the beginning, the ercentage margin of is larger than that of line E. Two margins become closer after the generator tri at t=439s. Finally, the voltages at two buses almost collase at the same time. The generator tri has more imact on the voltage stability margin of line E because the tried generator is closed to bus 4 and bus 4. Such information is not available from a traditional Thevenin equivalent based aroach. This new aroach offers more accurate monitoring of voltage stability margins for individual tie lines P E 4 6 oltage agnitude (.u.) Figure. Bus equivalent result me Figure. P curves of two buses in England New system max Figure. Active owers of two lines and their voltage stability limits oltage agnitude (.u.) m8 P 8 P E P E P Emax Active Power argin (%) max P Emax Figure 3. Comarison of the ercentage active ower margins of two lines. CONCLUSION This aer roosed a new three-bus equivalent for realtime estimation of voltage stability margin using synchronized measurements at the boundary buses of a load area. The detailed aroach was comared to the traditional Thevenin equivalent based aroach by case studies. The comarison indicates that the new aroach is able to assess voltage stability limits of a load area served by multile lines more accurately than the traditional aroach. I. REFERENCES [] K. u,. Begovic, D. Novosel, and. Saha, Use of local measurements to estimate voltage-stability margin, IEEE Trans. Power Systems, vol. 4o. 3,. 9 35, Aug [] B. ilosevic and. Begovic, "oltage-stability rotection and control using a wide-area network of hasor measurements," IEEE Trans. Power Systems, vol. 8,. -7, 3. [3] Smon, et al, "Local oltage-stability Index Using Tellegen's Theorem," IEEE Trans. Power Systems, ol., No. 3, Aug. 6. [4] P. hang, L. in, et al, easurement based voltage stability monitoring and control, US Patent # 8,6,667, Feb. [5] K. Sun, P. hang, L. in, easurement-based oltage Stability onitoring and Control for Load Centers, EPRI Technical Reort No. 7798, 9. [6]. Parniani, et al, "oltage Stability Analysis of a ultile-infeed Load Center Using Phasor easurement Data," IEEE PES Power Systems Conference and Exosition, Nov 6 [7] S. Corsi and G. Taranto, A real-time voltage instability identification algorithm based on local hasor measurements, IEEE Trans. Power Systems. vol. 3o. 3,. 7 79, Aug. 8. [8] C. D. ournas and N. G. Sakellaridis, "Tracking aximum Loadability Conditions in Power Systems," in 7 irep Symosium Bulk Power System Dynamics and Control - II. Revitalizing Oerational Reliability, 7. [9]. Glavic and T. an Cutsem, "Wide-Area Detection of oltage Instability from Synchronized Phasor easurements. Part I: Princile," IEEE Trans. Power Systems, vol. 4, , 9. [] R. Diao, K. Sun,. ittal, et al, "Decision Tree-Based Online oltage Security Assessment Using PU easurements", IEEE Trans. Power Systems, vol. 4, , ay 9. [] F. Galvan, A. Abur, K. Sun, et al, "Imlementation of Synchrohasor onitoring at Entergy: Tools, Training and Tribulations", IEEE PES General eeting, 3-6 July, San Diego [] C. Canizares, et al, oltage Stability Assessment: Concets, Practices and Tools, IEEE PES Power System Stability Subcommittee Secial Publication, IEEE, Aug