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System Identification of Reduced‐Order Models of Power Systems from PMU Data
Aranya ChakraborttyNorth Carolina State University
Behzad NabaviNew York Power Authority
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Introduction2
• Power systems dynamic model reduction has seen some 40 years of long and rich research history.
• Model‐based methods:– Need the model of the power system,– Are computationally expensive,– Are based on idealistic assumption about system
structure and clustering,• With the recent increase in number of Phasor
Measurement Units (PMUs) operators are gradually inclining more towards measurement based techniques for aggregation
Dynamic equivalencing for a two-area test system (Chakrabortty et al. 2011)
Dynamic equivalencing for IEEE 39-Bus Model (S. Nabavi et al. 2013)
Areas within WECC System (R. Podmore, 2013)
Proposed Dynamic Model Reduction
1sG
2sG
3sG
4sG
1
spV
2
spV
3
spV
4
spV
1
spI
2
spI
3
spI
4
spI
3
`
Area 1
Area 2 Area 3
Area 4
Remaining Transmission
Network
• Linear reduced-order models give the equivalent reduced-order model in the Kron’s from.
• No information about the equivalent machine parameters such as inertia and active power.
• None of the reduced-order model components have any physical meaning• We next propose an alternative method to identify the equivalent Differential-
Algebraic (DAE) models.
Problem Formulation• Assumptions:
– There are coherent areas, – The area partitioning is given,– The boundary buses of area , denoted by , are equipped with PMUs or
they are observable by PMUs• Objective:
– Finding the equivalent DAE model of a power system using a set of measurements , 1, … , where:
, ∈ , ∈• Proposed Identification Steps
– Finding the equivalent pilot bus voltages and currents– Estimating the equivalent area impedances– Estimating the equivalent generator parameters– Estimating the inter‐area impedances
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1sG
2sG
3sG
4sG
1
spV
2
spV
3
spV
4
spV
1
spI
2
spI
3
spI
4
spI
Equivalent Pilot Bus Voltage and Current Calculation
Step 1: Use to calculate and
≜ ∑ , ≜∑ ∗
∑ ∗
where=∑ ,∉Areak
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Coherent Area k
Coherent Area k
kpV
kpI
PMU
PMU
PMU
Coherent Area k
k
spV
k
spI
Step 1 Step 2
Equivalent Pilot Bus Voltage and Current Calculation
Step 2.1: Decompose ≜ ∠ and ≜ ∠into its slow and fast components using a modal decomposition technique such as Prony:
≅ ≅
≅ ≅
Step 2.2: Form ≜ ∠ and ≜ ∠
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Coherent Area k
Coherent Area k
kpV
kpI
PMU
PMU
PMU
Coherent Area k
k
spV
k
spI
Step 1 Step 2
Equivalent Area Impedance Calculations
• KVL in the equivalent circuit:∠
• For time instance through :
Φ ≜ ⋮
Φ ≜• The estimation of and for area can be posed as the following NLS problem:
min,
Φ ,… ,Φ
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s sk kE
k
sdjx
k k k
s s sp p pV V
k
spIs
kr
the kth equivalent pilot bus
Equivalent Area Generator Parameters
• Solve the following NLS problem
min, ,
| , , , |
• Where1
1∠ . ∗
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s sk kE
k
sdjx
k k k
s s sp p pV V
k
spIs
kr
the kth equivalent pilot bus
Inter‐Area Impedance Calculations• KCL on equivalent pilot buses:
⋯⋮ ⋱ ⋮
⋯
⋯⋮ ⋱ ⋮
⋯
⋯⋮ ⋱ ⋮
⋯
• Estimating by solving:
min
s.t.
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1sG
2sG
3sG
4sG
1
spV
2
spV
3
spV
4
spV
1
spI
2
spI
3
spI
4
spI
A Case Study: IEEE 39‐Bus Model10
.007
4+j.0
268
.003
7+j.0
451
.0100+j.0267
1sG
2sG
3sG
4sG
1
spV
2
spV
3
spV
4
spV
4
4 4
106.6757
1.6541
s
s s
H
D M
3
3
267.234
0
s
s
H
D
2
2 2
82.3485
0.4417
s
s s
H
D M
1
1 1
510.6557
0.3227
s
s s
H
D M
j0.0058 -0.0301 -j 1.1996
j 0.0
815
Predictive Capabilities of the Reduced‐order Model
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More contingencies:
Conclusion
• Coherency‐based dynamic equivalencing has been a useful tool to assist utilities in reducing the running times associated with transient stability studies.
• We propose a new identification method to identify the equivalent dynamic models of large power system in DAE form using PMU measurements.
• Compared to the previous works, the proposed equivalent model provides more details about the equivalent generators such as their inertia, damping, produced active power, and transient impedance.
• One future extension of this method would be to include stochastic dynamic models of renewable generation and loads.
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