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.

9

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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