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Uncertainty Quantification for Networks with PowerDistribution Applications
Scott Vander Wiel* Russell Bent†
Earl Lawrence* Emily Casleton*
*Statistical Sciences Group†Energy & Infrastructure Analysis Group
Los Alamos National Laboratory
June 6, 2013
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Overview
Robust and reliable electricpower grid is essential forsociety to function
The network is complex,unpredictable, and only partiallyobservable
Grid operators determine theleast cost power generationbased on topology and demands
We estimate topologyprobabilities and optimize powergeneration accounting foruncertainty
Image Credit: U.S. DOE (2006).
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Conventional State Estimation
Estimate topology, voltages andflows given system measurements
Iterate nonlinear least squareswith greedy topology search
– estimate state x by NLS– large residuals indicate a
topology error– modify topology and repeat
Use single best fit to decideoptimal generation
Works well in practice, with a fewnotable failures
– Incorrect estimate of networktopology contributed to 2003northeast blackout
Image Credit: Slobodan Pajic (2007)
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Power Flow Solver
Computational model used bystate estimator
– implements electrical laws– uses physical properties of
grid components
Inputs: Grid topology and nodequantities (complex valuedpower load or voltage)
Outputs: power flowing on thelines, a subset of which areobserved
SINGH et al.: RECURSIVE BAYESIAN APPROACH FOR IDENTIFICATION OF NETWORK CONFIGURATION CHANGES 1331
In (9), is a diagonal matrix whose diagonal elements rep-resent the inverse of the variances corresponding to the errorcomponents in the error vector. Large value of the elements of
can magnify the model errors and cause the accelerationof convergence to a single model. The error vector is givenby
(10)
where is the real measurement vector common to allmodels and is the estimated value of the same realmeasurement vector obtained from the estimated states of theth model in the th iteration.
The algorithm proceeds recursively from an equal initialprobability assigned to each model. In each iteration thenew probabilities are computed according to (9). These newprobabilities are the improvements in the probabilities of theprevious iteration.
If a large number of iterations is considered, the model bankasymptotically converges to a single model. Over the iterationsone model has asymptotic probability equal to one while othershave zero probabilities. However, in practice, for the identifi-cation of the correct network configuration, asymptotic conver-gence to a single model with unit probability is not required.The algorithm can be terminated in few iterations as long as oneof the models attains a significantly higher probability than therest.
The main advantage of the recursive Bayesian approach isthat the identification is naturally constrained so the cases inwhich the state estimation diverges are automatically rejected.The rejection of a poor model is exponential and thus very fast.Furthermore, the algorithm is computationally inexpensive andhence a large number of models can be handled efficiently.
IV. STUDY SYSTEMS
Two 11 kV distribution networks, shown in Fig. 2, are consid-ered. The networks are based on the U.K. Generic DistributionSystem (UKGDS) [9], which was modified in order to demon-strate the concept of the proposed approach.
Network 1 consists of 26 buses, 25 overhead lines, 13 loads,and one distributed generator (DG), and Network 2 has 13 buses,13 overhead lines, and eight loads. All loads are in the rangeof 10 to 140 kW, apart from the load at bus #18 of Network1 which is 930 kW. The generator is fixed at 700 kW at 0.95power factor. The two networks are connected via a normallyopen point which can be closed for maintenance or emergencynetwork reconfiguration. In addition, each network is equippedwith a protective device (recloser, sectionalizer, or fuse), a typ-ical means of improving reliability and service continuity inoverhead lines.
Network parameters are obtained from [9]. The buses arerenumbered so the networks are easier to observe.
A. Model Bank Description
Two types of major contingencies are considered for themodel banks of the two study systems: topological changes
Fig. 2. Test network models.
that are associated with the operation of protective devices andstatus change of normally open points, and injection changesthat are associated with loss of a DG or disconnection of amajor load.
It can be shown that disconnection of small loads has little im-pact on the state estimation function. On the other hand, discon-nection of large loads has significant effect on the accuracy ofthe estimated quantities. Thus, loss of large load has been con-sidered as critical change and hence included in the model bank.Also line outages have not been considered individually as, ineffect, they cause the protective devices to operate, resulting inone of the configurations included in the model banks.
The model banks selected for the two study networks are sum-marized in Table I. The model banks considered in this studyrepresent critical configuration changes that have a detrimentaleffect on the state estimation output. In general, these configu-rations are network specific and a detailed contingency study isrequired for their identification. In other words, the number ofmodels in a model bank can be prevented from being excessiveby utilizing the critical contingencies, operators’ experience anddetailed reliability and risk analysis. However, the speed of theproposed approach allows the use of models banks with large
Image Credit: Modified from Singh, Manitsas, Pal, Strbac (2010)Unclassified4
AC Power Flow Equations
Known Property:admittance Gi ,j , bi ,j
power flow Pi ,j ,Qi ,j
voltage|Vi |, θi
power loadPi ,Qi
node quantities2 given, 2 solved:|Vj |, θjPj ,Qj
node i line i , j node j
Lossless flow equations for real (Pi ,Pi ,j) and reactive (Qi ,Qi ,j) power:
Pi =N∑j=1
|Vi ||Vj |[Gi ,j cos(θi − θj) + bi ,j sin(θi − θj)] =N∑j=1
Pi ,j
Qi =N∑j=1
|Vi ||Vj |[Gi ,j sin(θi − θj)− bi ,j cos(θi − θj)] =N∑j=1
Qi ,j
(i = 1, . . . ,N)
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Model Bank Topology Estimation
Singh, et al. consider estimating thecorrect topology from a finitecollection of possibilities
a bank of models contains allimportant networkconfigurations
estimate probabilities for eachmodel in the bank bycombining
– system measurements– prior information about loads– power flow model
Did not implement Bayes rulecorrectly
SINGH et al.: RECURSIVE BAYESIAN APPROACH FOR IDENTIFICATION OF NETWORK CONFIGURATION CHANGES 1331
In (9), is a diagonal matrix whose diagonal elements rep-resent the inverse of the variances corresponding to the errorcomponents in the error vector. Large value of the elements of
can magnify the model errors and cause the accelerationof convergence to a single model. The error vector is givenby
(10)
where is the real measurement vector common to allmodels and is the estimated value of the same realmeasurement vector obtained from the estimated states of theth model in the th iteration.
The algorithm proceeds recursively from an equal initialprobability assigned to each model. In each iteration thenew probabilities are computed according to (9). These newprobabilities are the improvements in the probabilities of theprevious iteration.
If a large number of iterations is considered, the model bankasymptotically converges to a single model. Over the iterationsone model has asymptotic probability equal to one while othershave zero probabilities. However, in practice, for the identifi-cation of the correct network configuration, asymptotic conver-gence to a single model with unit probability is not required.The algorithm can be terminated in few iterations as long as oneof the models attains a significantly higher probability than therest.
The main advantage of the recursive Bayesian approach isthat the identification is naturally constrained so the cases inwhich the state estimation diverges are automatically rejected.The rejection of a poor model is exponential and thus very fast.Furthermore, the algorithm is computationally inexpensive andhence a large number of models can be handled efficiently.
IV. STUDY SYSTEMS
Two 11 kV distribution networks, shown in Fig. 2, are consid-ered. The networks are based on the U.K. Generic DistributionSystem (UKGDS) [9], which was modified in order to demon-strate the concept of the proposed approach.
Network 1 consists of 26 buses, 25 overhead lines, 13 loads,and one distributed generator (DG), and Network 2 has 13 buses,13 overhead lines, and eight loads. All loads are in the rangeof 10 to 140 kW, apart from the load at bus #18 of Network1 which is 930 kW. The generator is fixed at 700 kW at 0.95power factor. The two networks are connected via a normallyopen point which can be closed for maintenance or emergencynetwork reconfiguration. In addition, each network is equippedwith a protective device (recloser, sectionalizer, or fuse), a typ-ical means of improving reliability and service continuity inoverhead lines.
Network parameters are obtained from [9]. The buses arerenumbered so the networks are easier to observe.
A. Model Bank Description
Two types of major contingencies are considered for themodel banks of the two study systems: topological changes
Fig. 2. Test network models.
that are associated with the operation of protective devices andstatus change of normally open points, and injection changesthat are associated with loss of a DG or disconnection of amajor load.
It can be shown that disconnection of small loads has little im-pact on the state estimation function. On the other hand, discon-nection of large loads has significant effect on the accuracy ofthe estimated quantities. Thus, loss of large load has been con-sidered as critical change and hence included in the model bank.Also line outages have not been considered individually as, ineffect, they cause the protective devices to operate, resulting inone of the configurations included in the model banks.
The model banks selected for the two study networks are sum-marized in Table I. The model banks considered in this studyrepresent critical configuration changes that have a detrimentaleffect on the state estimation output. In general, these configu-rations are network specific and a detailed contingency study isrequired for their identification. In other words, the number ofmodels in a model bank can be prevented from being excessiveby utilizing the critical contingencies, operators’ experience anddetailed reliability and risk analysis. However, the speed of theproposed approach allows the use of models banks with large
B
E
17
18
D
P1 Q1
P2 Q2
Image Credit: Modified from Singh, Manitsas, Pal, Strbac (2010)Unclassified6
Model Bank Topology Estimation
We compute probabilities ontopologies from a vector ofmeasured flows by
propagating priors on nodalloads through each possibletopology
approximating resultantdistributions of power flows
implementing Bayes rule byimportance sampling aroundthe measured flows
Resulting algorithm is fast enoughto be used in real time
Image shows topology estimates andrisk-optimal generation
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Statistical Model Set-Up
Model bank of m network topologies: Ω = ω1, . . . , ωmSystem measurements, Y, are normally distributed with some mean µand standard deviation proportional to the mean
Y ∼ MVN(µ,Σy ), Σy = diag(ρµ)2
The mean is the output from solving the power flow equations for a setof random loads, Z, and a topology, ω
µ = µ(Z, ω)
Loads are also normally distributed with known mean and proportionalerror
Z ∼ MVN(ν,Σz), Σz = diag(βν)2
Goal of analysis is to estimate Pr(ωk |Y = y), ∀k = 1, . . . ,m.
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Offline Precomputation
Large networks imply large dimension of Z and running the power flowsolver in real time becomes computationally prohibitive.
A distribution, ηk , is fit to random outputs from the power flow solverto avoid running the solver in real time:
µ(Z, ωk) ∼ ηk(µ) with Z ∼ MVN(ν,Σz)
Scheme:
1. Draw a large sample of loads Zi (i = 1, . . . ,M).2. Run the solver to compute µi,k = µ(Zi , ωk) (∀i , k).3. For each topology transform µ1,k . . . ,µM,k to approximate normality:
hk(µi,k) ∼ MVN (over i)
Transformations hk are determined through exploratory data analysis
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Example: IEEE Reliability Test System (RTS)
Designed as benchmark forcomparing reliabilitymethodologies
72 nodes
Power observed on the 5interconnects between subsystems
Model bank of 124 topologiesdefined by RTS Task Force
– Normal– All single lines down (except
those monitored)– Some two-line losses
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Example: Normal-Plots of Solver Output µ(Zi , ωk)
Each line assumes one of the 124 model bank topologies.
Real power flow, P Reactive power flow, Q
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Example: Transformation of Reactive Power Solver Output
Transform each measurement,`, to normality
hk,`(µ) = Φ−1[Fk,`(µ)]
where
– Fk,` is a fitted three–parameter Gamma CDF
– Φ is the standard normalCDF
Transformed reactive power, hk(Q)
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Importance Sampling
Reminder: Goal is to compute
Pr(ωk |Y = y) ∝ π(ωk)
∫f (y|µ(ωk ,Z)) πZ (Z)dZ
= π(ωk)
∫f (y|µ) ηk(µ)dµ
where f is the measurement density, MVN with proportional std. dev.
π is a prior on topologies
πZ is the MVN density of Z
ηk is the approximating density of µ(ωk ,Z)
No analytical solution
Brute force Monte Carlo is infeasible because µ ∼ ηk(µ) is rarely closeenough to y to contribute to the integral
⇒ Importance sample µ near y and re-weight to estimate the integral
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Online Real Time Monitoring
1. Obtain a new measurement, y
2. Draw an importance sample of µ near y.
3. Estimate
Pr(ωk |y) ∝ π(ωk)Eηk [f (y|µ)]
as an importance-weighted average for each scenario, ωk
4. Normalize to obtain model bank probabilities.
Result is a vector of probabilities, p(y), for each possible topology
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Simulation Study
For each topology:
1. Generate 1000 simulated observations, y
2. Compute p(y) for 124 toplogies
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Simulation Study
For each topology:
1. Generate 1000 simulated observations, y
2. Compute p(y) for 124 toplogies
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Simulation Study Results
Plot shows
EYk[Pr(ωk |Y)]
verses
max6=k
EYk[Pr(ω`|Y)]
True topology has highestprobability in all but onecase
Ambiguity is often large
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Pr(
true
sce
nario
)
Maximum Pr(incorrect scenario)
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Chance Constrained Optimal Power Flow (CCOPF)
OPF decides how much power to generate and where.
CCOPF makes these decisions with the added constraints that
P(line i , j overloaded) < ε
for all lines and over the uncertain topology
Chance constraints have been implemented for other classes ofuncertainty such as unit commitment and expansion planning toaccount for renewable energy generation
Define setsN = nodes
G = generators
G i = generators associated with node i , (G i ⊂ G )
E = edges (lines)
E i edges connected to node i , (E i ⊂ E )
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DC Power Flow Equations
Known Property:admittance Gi ,j , bi ,j
power flow Pi ,j ,Qi ,j
voltage|Vi |, θi
power loadPi ,Qi
node quantities2 given, 2 solved:|Vj |, θjPj ,Qj
node i line i , j node j
DC power flow equations for real, Pi power at node i ; i = 1, . . . ,N:
Pi =N∑j=1
bi ,j(θi − θj)
Results from multiple simplifying assumptions on the AC equations thatresults in a linear set of equations.
The DC equations are an approximation to the AC equations and areused in the CCOPF due to their simplicity.
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CCOPF formula
The problem in terms of objective function and constraints:
minimize∑k∈G
ckgk +∑i∈N
κiξi
subject to g−k ≤ gk ≤ g+k ∀k ∈ G
0 ≤ ξi ≤ `i ∀i ∈ N∑k∈G i
gk − `i + ξi +∑i,j∈E i
bωi,j(θωi − θωj ) = 0 ∀i ∈ N, ω ∈ Ω
ρωi,j ≥|bωi,j(θωi − θωj )| − qi,j
M∀i , j ∈ E , ω ∈ Ω∑
ω∈Ω
pωρωi,j ≤ ε ∀i , j ∈ E
ρωi,j ∈ 0, 1
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CCOPF: Objective function
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Find a set of gk ; k ∈ G and ξi ; i ∈ N that minimize:
Objective Function:∑k∈G
ckgk +∑i∈N
κiξi
Generator kgk amount generatedg+k maximum generationg−k minimum generationck cost of generation
Node i`i load demandξi amount shedκi cost to shed
Node j
qi ,j capacitybωi ,j(θ
ωi − θωj )flow on (i , j) in ω
ρωi ,j =
1 (i , j) overloaded in ω0 otherwise
CCOPF: Generation constraints
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Find a set of gk ; k ∈ G and ξi ; i ∈ N that minimize:
Objective Function:∑k∈G
ckgk +∑i∈N
κiξi
Constraint 1: g−k ≤ gk ≤ g+k ∀k ∈ G
Constrain the amount generated at each generator to be within plausiblelimits.
Generator kgk amount generatedg+k maximum generationg−k minimum generationck cost of generation
Node i`i load demandξi amount shedκi cost to shed
Node j
qi ,j capacitybωi ,j(θ
ωi − θωj )flow on (i , j) in ω
ρωi ,j =
1 (i , j) overloaded in ω0 otherwise
CCOPF: Load shedding constraints
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Find a set of gk ; k ∈ G and ξi ; i ∈ N that minimize:
Objective Function:∑k∈G
ckgk +∑i∈N
κiξi
Constraint 2: 0 ≤ ξi ≤ `i ∀i ∈ N
Cannot shed more load at a particular node than is demanded.
Generator kgk amount generatedg+k maximum generationg−k minimum generationck cost of generation
Node i`i load demandξi amount shedκi cost to shed
Node j
qi ,j capacitybωi ,j(θ
ωi − θωj )flow on (i , j) in ω
ρωi ,j =
1 (i , j) overloaded in ω0 otherwise
CCOPF: Flow Balance constraints
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Find a set of gk ; k ∈ G and ξi ; i ∈ N that minimize:
Objective Function:∑k∈G
ckgk +∑i∈N
κiξi
Constraint 3:∑k∈G i
gk − `i + ξi +∑i ,j∈E i
bωi ,j(θωi − θωj ) = 0 ∀i ∈ N, ω ∈ Ω
The amount of power generated and passed in must be either consumed,shed, or passed out.
Generator kgk amount generatedg+k maximum generationg−k minimum generationck cost of generation
Node i`i load demandξi amount shedκi cost to shed
Node j
qi ,j capacitybωi ,j(θ
ωi − θωj )flow on (i , j) in ω
ρωi ,j =
1 (i , j) overloaded in ω0 otherwise
CCOPF: Overload detection
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Find a set of gk ; k ∈ G and ξi ; i ∈ N that minimize:
Objective Function:∑k∈G
ckgk +∑i∈N
κiξi
Constraint 4: ρωi ,j ≥|bωi ,j(θωi − θωj )| − qi ,j
M∀i , j ∈ E , ω ∈ Ω
Constraint 6: ρωi ,j ∈ 0, 1If a line is overloaded in scenario ω, set the variable ρωi ,j to 1. Thediscrete-ness of the ρωi ,j makes the problem computationally challenging.
Generator kgk amount generatedg+k maximum generationg−k minimum generationck cost of generation
Node i`i load demandξi amount shedκi cost to shed
Node j
qi ,j capacitybωi ,j(θ
ωi − θωj )flow on (i , j) in ω
ρωi ,j =
1 (i , j) overloaded in ω0 otherwise
CCOPF: Chance constraints
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Find a set of gk ; k ∈ G and ξi ; i ∈ N that minimize:
Objective Function:∑k∈G
ckgk +∑i∈N
κiξi
Constraint 5:∑ω∈Ω
pωρωi ,j ≤ ε ∀i , j ∈ E
Ensures the probability that the flow of power on a line violates its thermallimits is smaller than ε.
Generator kgk amount generatedg+k maximum generationg−k minimum generationck cost of generation
Node i`i load demandξi amount shedκi cost to shed
Node j
qi ,j capacitybωi ,j(θ
ωi − θωj )flow on (i , j) in ω
ρωi ,j =
1 (i , j) overloaded in ω0 otherwise
Algorithms
1. Commercial Mixed Integer Programming Solver
2. Branch and Bound on the Chance Constraints–Recursively branch onenforcing the capacity constraints on one violated chance constraintscenario
3. Constraint Injection on the Chance Constraints–Add chance constraintsincrementally to eliminate constraint violations
4. Disjunctive Programming–Cutting plane algorithm
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Empirical Study
Stressed network with model bank of 340 scenarios
Sample 100 observations from each scenario–perform CCOPF withvarious values of ε and OPF based on most likely topology
ε Violation Probability (%) Average Load Shed
CCOPF OPF CCOPF OPF0 0.00 33.38 143.42 11.77
0.001 0.82 33.38 93.14 11.770.01 2.29 33.38 59 11.770.1 13.53 33.38 21.66 11.77
0.25 26.85 33.38 12.73 11.770.5 38.44 33.38 7.42 11.77
0.75 44.12 33.38 4.83 11.771 100.00 33.38 0 11.77
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Summary
Fast algorithm for capturing uncertainty in topology.
Standard OPF does not recognize uncertainty
Simulations suggests there is more uncertainty than is realized
CCOPF algorithm incorporates uncertainty for robust operation
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Future Work
Fully quantify the uncertainty in state estimation by calculatingposterior load distributions for each topology
[Z | Y = y, ωk ]
– Early work has used Gaussian process emulation techniques– Larger problems will require more scalable algorithms.– Incorporate load uncertainty into OPF
Model bank as a good stepping stone.
– Can be solved for moderately large networks with large models banks.– Can apply the method to the complete set of possible topologies and find
metrics that vary smoothly with the Gaussian approximations.– Goal is to explore the topology space in a structured manner.
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