Optimal PMU Placement S. A. Soman Department of Electrical Engineering Indian Institute of Technology Bombay Dec 2, 2011
PMU Numerical relays as PMU System Observability Control Center Architecture WAMS Architecture Outline PMU Placement Problem Integer Linear Programming Approach PMU Placement on WR Grid Optimal Phasing of PMUs Formulation Enhancements PMU outages and line outages Modelling of zero injection constraints PMU Based State Estimator Hybrid State Estimator
Phasor Measurement Unit GPS receiver One pulse per secind Second of Century Counter Analog Inouts Phase locked oscillator Modem Anti aliasing filters A/D conv. Phasor microprocessor
Phasor Measurement Unit PMU measures synchrophasors in a substation Positive sequence branch currents or phase currents Positive bus voltage or phase voltages V I 3 I 1 In practice, number of channels in a PMU are limited and substations implemented with bay controllers may require multiple PMUs. Therefore, a concept PMU can be more than one physical device. I 2
Numerical Relay as a PMU Distance relay will only read line current and line voltage synchrophasors Line voltage may be different from bus voltage With a numerical relay acting as a PMU, one can observe only a pair of busses with one PMU Which one should be preferred in actual implementation? V bus V line Figure: Simultaneous Sampling Scheme
Numerical Relay as a PMU (contd..) Ideally, all phasor measurements in a substation should be synchrophasor measurements
Power System Observability A power system is observable if its connectivity information, captured usually in SLD, is fully available This requires information in real time about CB status and substation arrangement e.g., 1 1/2 bus bar arrangement etc All bus voltage phasors can be measured or estimated with high level of accuracy With this knowledge, flow on lines can be estimated and many what if simulations can be run in the control center
Control Center Architecture
Limitations Assumes a steady state model of power system An operator is helpless during system dynamics Power swings or electromechanical oscillations are in the range of 0.5-2 Hz. There is no central (control center) level monitoring of them, leave aside control Technology Limitations: RTU measurements are time skewed RTU fetch time may be of the order 1-4 sec. Each RTU uses a different reference phasor and we cannot correlate one reference phasor to other To mask off the reference phasor problem, we use MW and MVAR measurements (VI* computation) The resulting estimator becomes nonlinear
SynchroPhasor State Estimator Architecture 1. computation of synchrophasors & 2. communication to control center
Promise of WAMS for Energy Control Centers Synchrophasors as well as digital information will be updated at least every 5 cycles (10 Hz sampling) Directly angle measurements will be available from PMUs Time snapshots of the system can be built say with latency of 100 msec, in the control center Operator can monitor oscillations This can lead to development of intelligent device control strategies during system dynamics Improve emergency control e.g., smart islanding Simplify system restoration after grid incidences PMU based state estimator will become linear, may be even redundant for small systems
PMU Placement Options Aim: To place PMUs to ascertain system observability Options 1: Place PMUs at all busses in the system 2: Place PMUs at selected busses in the system Option 1 more expensive due to PMU and communication costs. Example 1 2 3 4 I 12 I 43 PMU1 PMU 2 V 2 = V 1 I 12 z 12 V 1 = V 4 I 43 z 43 Other options are busses {1, 2, 3}, {2, 3, 4}, {1, 2, 4}, {2, 3}, {2, 4}, and {1, 3}
Minimum PMU Placement Problem What are the minimum number of PMUs required to make system observable? Where should these PMUs be placed? e.g., {1, 4}, {1, 3},{2, 4}, and {2, 3} We discuss an Integer Linear Programming (ILP) Approach The placement problem should also consider PMU positioning requirement due to other analytic applications e.g., 1. secure back protection scheme for critical transmission lines, 2. necessity to observe big generating complexes, 3. monitoring flows, voltages and angular difference across lines which have a tendency to get congested 4. PMU placement near controllable devices like SVC, TCSC 5. Placement in critical locations which cause blind spots (unobservability) in existing estimators
ILP Formulation OPP : min w 1 x 1 + w 2 x 2 +... + w 14 x 14 subject to bus observability constraints (here onwards weights assumed unity for simplicity): Bus 1 : x 1 + x 2 + x 5 1 Bus 2 : x 1 + x 2 + x 3 + x 4 + x 5 1 Bus 3 : x 2 + x 3 + x 4 1 Bus 4 : x 2 + x 3 + x 4 + x 5 + x 7 + x 9 1 Bus 5 : x 1 + x 2 + x 4 + x 5 1 Bus 6 : x 6 + x 11 + x 12 + x 13 1 Bus 7 : x 4 + x 7 + x 8 + x 9 1 Figure: IEEE 14 Bus System
ILP Formulation Bus 8 : x 7 + x 8 1 Bus 9 : x 4 + x 7 + x 9 + x 10 + x 14 1 Bus 10 : x 9 + x 10 + x 11 1 Bus 11 : x 6 + x 10 + x 11 1 Bus 12 : x 6 + x 12 + x 13 1 Bus 13 : x 6 + x 12 + x 13 + x 14 1 Bus 14 : x 9 + x 13 + x 14 1 Figure: IEEE 14 Bus System
Optimal PMU placement for Western Region
Optimal PMU placement for Western Region Total of 211 busses Minimum No. of PMUs required is 60 Approximately 28 % PMU penetration is required Cannot be placed in one year horizon
Optimal Phasing of PMUs Phasing of PMUs at busses identified by OPP formulation in t years. For IEEE 14 bus system phasing of 2, 6, 7 and 9 over 3 year interval with 1 PMU in 1st year, 2 PMUs in IInd year and 1 in IIIrd year In the first and second year, with 1 and 3 PMUs system will not be observable Define Bus Observability variable u i {0, 1} for each bus Objective is to maximize number of busses made observable in the intermediate stages
IEEE 14 Bus System - Phase I Formulation Objective is to maximize the number of observable busses. max u 1 + u 2 + u 14 Placement of the only PMU in phase-i can be on any one of the four busses, i.e. bus number 2, 6, 7, or 9. Thus, x 2 + x 6 + x 7 + x 9 = 1 All other x i s i.e. x 1, x 3, x 4, x 5, x 8, x 10, x 11, x 12, x 13 and x 14 are set to zero
IEEE 14 Bus System- Phasing Constraints for Ist Year Bus 1 : x 2 u 1 Bus 2 : x 2 u 2 Bus 3 : x 2 u 3 Bus 4 : x 2 + x 7 + x 9 u 4 Bus 5 : x 2 u 5 Bus 6 : x 6 u 6 Bus 7 : x 7 + x 9 u 7 Figure: IEEE 14 Bus System
IEEE 14 Bus System- Phasing Constraints for Ist Year Bus 8 : x 7 u 8 Bus 9 : x 7 + x 9 u 9 Bus 10 : x 9 u 10 Bus 11 : x 6 u 11 Bus 12 : x 6 u 12 Bus 13 : x 6 u 13 Bus 14 : x 9 u 14 Optimal solution is at bus 2 Figure: IEEE 14 Bus System
IEEE 14 Bus System - Phase II Formulation After phase-i, busses 1, 2, 3, 4 and 5 become observable. For maximizing observability for the remaining busses, the objective function is as follows: max u 6 + u 7 + u 14 Two PMUs can be placed in this phase among the three busses 6, 7 or 9 can be modeled as follows:. x 6 + x 7 + x 9 = 2 Bus observability constraints for busses 6 to 14 have to be modelled
IEEE 14 Bus System - Phase II Formulation Bus Observability Constraints Bus 8 : x 7 u 8 Bus 9 : x 7 + x 9 u 9 Bus 10 : x 9 u 10 Bus 11 : x 6 u 11 Bus 12 : x 6 u 12 Bus 13 : x 6 u 13 Bus 14 : x 9 u 14 Solving the ILP formulation leads to the PMU placement on busses 6 and 9 In the last year, PMU will be placed at bus 7
Phasing PMUs in Western Region 60 PMUs to be phased in 5 years. Year Obs. Busses I 97 II 144 III 178 IV 199 V 211 Table: Number of Busses observable at each year Figure: % of Busses observable at each year
PMU Placement for All India Grid 2011-12 Scenario with New Grid and SR are in synchronous operation (data provided by CEA) There are 1699 nodes and 3390 lines OPP leads to minimum of 525 PMUs (31 % PMU penetration) Phasing considered over 5 years Year Obs. Busses I 777 II 1197 III 1444 IV 1594 V 1699
Enhancements Number of PMUs can be further reduced by considering zero injection measurements. Redundancy in PMU placement will be desired to account for PMU outages & contingencies. PMU Outage: At least two PMUs should observe a bus i.e., set RHS to 2. Line Outage: Final set of constraints is a collection of constraint set for each topology. Some busses may not be suitable for PMU placement due to lack of communication facility - Set corresponding decision variables to zero. If there are some busses which must have PMUs (or PMU already exists e.g., due to NR pilot), set corresponding x i s to 1. PMU phasing objective could be to maximize the size of observable subsystem in the phasing horizons so that linear state estimators could be attempted.
Modelling of Zero Injection Busses Figure: Optimal PMU placement for a 4 bus system (a) neglecting zero injection constraints and (b) considering bus-2 as zero injection bus In the best case, the minimum number of PMUs required to observe the system can be further reduced by total number of zero injection busses in the system.
Philosophy to Model Zero Injection Busses 1. Unobservable busses, if any, must belong to the cluster of zero injection busses and busses adjacent to zero injection busses 2. In this cluster at most one bus can be unobservable because: V 1 = y 12V 2 + y 1,m 1 V m 1 y 12 + y 1m Figure: Modeling of zero injection busses
Bus 7 : x 4 + x 7 + x 8 + x 9 u 7 Figure: IEEE 14 Bus System IEEE 14 Bus Illustration Bus 7 is zero injection Bus. Out of the four busses 4, 7, 8 and 9, at least three busses observable. The modified ILP formulation is OPP Z min x 1 + x 2 +... + x 14 subject to constraints Bus 1 : x 1 + x 2 + x 5 1 Bus 2 : x 1 + x 2 + x 3 + x 4 + x 5 1 Bus 3 : x 2 + x 3 + x 4 1 Bus 4 : x 2 + x 3 + x 4 + x 5 + x 7 + x 9 u 4 Bus 5 : x 1 + x 2 + x 4 + x 5 1 Bus 6 : x 6 + x 11 + x 12 + x 13 1
IEEE 14 Bus Illustration Bus 8 : x 7 + x 8 u 8 Bus 9 : x 4 + x 7 + x 9 + x 10 + x 14 u 9 Bus 10 : x 9 + x 10 + x 11 1 Bus 11 : x 6 + x 10 + x 11 1 Bus 12 : x 6 + x 12 + x 13 1 Bus 13 : x 6 + x 12 + x 13 + x 14 1 Bus 14 : x 9 + x 13 + x 14 1 Zero injection : u 4 + u 7 + u 8 + u 9 3 Figure: IEEE 14 Bus System Optimal PMU placement is at busses 2,6, and 9 Optimal solution has one less PMU
Synchrophasor State Estimator 1 I 2 12 I 21 I 13 I 23 V pmu 1 V pmu 2 I pmu 12 I pmu 13 I pmu 23 = 1 3 1 y 12 y 12 y 13 y 13 y 23 y 23 z = MV + e V est = (M H M) 1 M H z V 1 V 2 V 3 + e 1 e 2 e 3 e 4 e 5
Hybrid State Estimator-Motivation Synchrophasor Based State Estimator will be a Linear State Estimator It is not realizable at this point of time More realistic option is to use few PMU inputs in existing state estimators Right now each region has its state estimator With a minimum of one PMU in each region, we can correlate state estimator outputs of different regions and create a state estimator output for the new grid If in SLDC systems, we have at least one PMU, then their output can be integrated with RLDC output. This technology can enable open access, as operator can make system wide ATC calculations
Hybrid State Estimator-Motivation (Contd..) Reference phasor of PMU and conventional state estimator reference may be different; this calls for angle correction in conventional state estimator output to match it with PMU output. With multiple PMUs,due to noise in the conventional measurements (including time skew issues), different PMUs may suggest different corrections in angles.
Modelling Approaches 1. Include PMU measurements like conventional measurements in state estimation formulation and proceed. Consider a line modelled as simple reactance (jx). Now, Hence, I 2 = V 2 1 + V 2 2 2V 1V 2 cos(δ 1 δ 2 ) I p = 1 (V1 2 + V 2 2 2V 1V 2 cos(δ 1 δ 2 )) 2 I p Derivative of current measurements in polar coordinate may become undefined at certain points in iterative process. Hence, computation will have to be done in rectangular coordinates.
Modelling Approach (Contd..) 2 Transform PMU measurements into P and Q flow and voltage magnitude measurements. Then integrate in conventional state estimator. We will need to compute the variance of these transformed measurements. 3 Preprocess the PMU measurements to obtain pseudo-voltage phasor measurements for adjacent busses. Estimate the variance for pseudo measurements. Include these pseudo-measurements in the state estimation formulation like any conventional measurements.
Estimating the Variance of Transformed Measurements Let z be n of original PMU measurements and y is the m vector of transformed measurements (y = g(z)). Then σ 2 yi = n ( gi (z) k=1 z k σ zk ) 2 i = 1, m From the PMU data, we know the maximum measurement uncertainty of the measurements Under assumption of uniform distribution, standard deviation of the of measurement is 1 3 times the maximum measurement uncertainty
Approach-4 Issue of vendor cooperation and costs in integrating PMU measurements in conventional estimator Another alternative is to obtain estimate phasors first from conventional estimator and then write a top layer refinement estimator using PMU data. the voltage estimate of conventional estimator can be parameterized in reference angle variable which is unknown Variance of the estimates of conventional estimator are known from the conventional estimator. Finally, combine pseudo voltage measurements from conventional state estimator with PMU measurements using linear state estimation theory PMU generates many measurements in the same time in the time interval in which RTU will generate one measurement
Refences [?], [?] and [?] References J. Thorp, A. Phadke, and K. Karimi, Real time voltage-phasor measurement for static state estimation, Power Apparatus and Systems, IEEE Transactions on, vol. PAS-104, no. 11, pp. 3098 3106, Nov. 1985. D. Dua, S. Dambhare, R. Gajbhiye, and S. Soman, Optimal multistage scheduling of pmu placement: An ilp approach, Power Delivery, IEEE Transactions on, vol. 23, no. 4, pp. 1812 1820, Oct. 2008. M. Zhou, V. Centeno, J. Thorp, and A. Phadke, An alternative for including phasor measurements in state estimators, Power Systems, IEEE Transactions on, vol. 21, no. 4, pp. 1930 1937, Nov. 2006.
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