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Impact of Delays on a Consensus-based Primary Frequency Control Scheme for AC Systems Connected by a Multi-Terminal HVDC Grid J. Dai1

Y. Phulpin1 1 Supélec, 2 FNRS

A. Sarlette2

D. Ernst2

Paris, France

and University of Liège, Belgium

IREP Symposium 2010

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Title: Impact of Delays on a Consensus-based Primary Frequency Control Scheme for AC Systems Connected by a Multi-Terminal HVDC Grid 4 elements: 1

Primary frequency control

2

Multi-terminal HVDC system

3

Consensus-based control scheme

4

Delays

Conclusions

Problem addressed

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Empirical contributions

Problem addressed

Title: Impact of Delays on a Consensus-based Primary Frequency Control Scheme for AC Systems Connected by a Multi-Terminal HVDC Grid 4 elements: 1

Primary frequency control

2

Multi-terminal HVDC system

3

Consensus-based control scheme

4

Delays

Conclusions

Problem addressed

Theoretical contributions

Empirical contributions

Problem addressed

Title: Impact of Delays on a Consensus-based Primary Frequency Control Scheme for AC Systems Connected by a Multi-Terminal HVDC Grid 4 elements: 1

Primary frequency control

2

Multi-terminal HVDC system

3

Consensus-based control scheme

4

Delays

Conclusions

Problem addressed

Theoretical contributions

Empirical contributions

Problem addressed

Title: Impact of Delays on a Consensus-based Primary Frequency Control Scheme for AC Systems Connected by a Multi-Terminal HVDC Grid 4 elements: 1

Primary frequency control

2

Multi-terminal HVDC system

3

Consensus-based control scheme

4

Delays

Conclusions

Problem addressed

Theoretical contributions

Empirical contributions

Problem addressed

Title: Impact of Delays on a Consensus-based Primary Frequency Control Scheme for AC Systems Connected by a Multi-Terminal HVDC Grid 4 elements: 1

Primary frequency control

2

Multi-terminal HVDC system

3

Consensus-based control scheme

4

Delays

Conclusions

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Primary frequency control

Frequency control: Limit frequency variations and restore balance between generation and load demand Primary frequency control: Time scale: a few seconds Local adjustment of generators’ power output Based on locally measured frequency Primary reserves: generators’ power output margin

Larger synchronous area: More generators participating in the primary control Smaller frequency deviations

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Primary frequency control

Frequency control: Limit frequency variations and restore balance between generation and load demand Primary frequency control: Time scale: a few seconds Local adjustment of generators’ power output Based on locally measured frequency Primary reserves: generators’ power output margin

Larger synchronous area: More generators participating in the primary control Smaller frequency deviations

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Primary frequency control

Frequency control: Limit frequency variations and restore balance between generation and load demand Primary frequency control: Time scale: a few seconds Local adjustment of generators’ power output Based on locally measured frequency Primary reserves: generators’ power output margin

Larger synchronous area: More generators participating in the primary control Smaller frequency deviations

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Multi-terminal HVDC system

DC grid P1dc

PNdc

P2dc

AC area 1

AC area N AC area 2

Generally, Pidc are supposed to track pre-determined power settings. Frequencies are independent among AC areas. Primary frequency control is independent from one area to another.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Multi-terminal HVDC system

DC grid P1dc

PNdc

P2dc

AC area 1

AC area N AC area 2

Generally, Pidc are supposed to track pre-determined power settings. Frequencies are independent among AC areas. Primary frequency control is independent from one area to another.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

A control scheme proposed in an earlier work Control objective: Sharing primary frequency reserves among non-synchronous areas by imposing that ∆f1 (t) = . . . = ∆fN (t). dc Control variables: Pidc , . . . , PN−1 N

X dPidc (t) =α bik (∆fi (t) − ∆fk (t)) dt k =1   N X d∆fi (t) d∆fk (t) +β bik − dt dt k =1

i = 1, . . . , N − 1, where ∆fi (t): Frequency deviation of area i from its nominal value. α and β: integral and proportional control gain. bik : coefficient representing the communication graph.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

A control scheme proposed in an earlier work Control objective: Sharing primary frequency reserves among non-synchronous areas by imposing that ∆f1 (t) = . . . = ∆fN (t). dc Control variables: Pidc , . . . , PN−1 N

X dPidc (t) =α bik (∆fi (t) − ∆fk (t)) dt k =1   N X d∆fi (t) d∆fk (t) +β bik − dt dt k =1

i = 1, . . . , N − 1, where ∆fi (t): Frequency deviation of area i from its nominal value. α and β: integral and proportional control gain. bik : coefficient representing the communication graph.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

A control scheme proposed in an earlier work Control objective: Sharing primary frequency reserves among non-synchronous areas by imposing that ∆f1 (t) = . . . = ∆fN (t). dc Control variables: Pidc , . . . , PN−1 N

X dPidc (t) =α bik (∆fi (t) − ∆fk (t)) dt k =1   N X d∆fi (t) d∆fk (t) +β bik − dt dt k =1

i = 1, . . . , N − 1, where ∆fi (t): Frequency deviation of area i from its nominal value. α and β: integral and proportional control gain. bik : coefficient representing the communication graph.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

A control scheme proposed in an earlier work Control objective: Sharing primary frequency reserves among non-synchronous areas by imposing that ∆f1 (t) = . . . = ∆fN (t). dc Control variables: Pidc , . . . , PN−1 N

X dPidc (t) =α bik (∆fi (t) − ∆fk (t)) dt k =1   N X d∆fi (t) d∆fk (t) +β bik − dt dt k =1

i = 1, . . . , N − 1, where ∆fi (t): Frequency deviation of area i from its nominal value. α and β: integral and proportional control gain. bik : coefficient representing the communication graph.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

A control scheme proposed in an earlier work Control objective: Sharing primary frequency reserves among non-synchronous areas by imposing that ∆f1 (t) = . . . = ∆fN (t). dc Control variables: Pidc , . . . , PN−1 N

X dPidc (t) =α bik (∆fi (t) − ∆fk (t)) dt k =1   N X d∆fi (t) d∆fk (t) +β bik − dt dt k =1

i = 1, . . . , N − 1, where ∆fi (t): Frequency deviation of area i from its nominal value. α and β: integral and proportional control gain. bik : coefficient representing the communication graph.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

A control scheme proposed in an earlier work Control objective: Sharing primary frequency reserves among non-synchronous areas by imposing that ∆f1 (t) = . . . = ∆fN (t). dc Control variables: Pidc , . . . , PN−1 N

X dPidc (t) =α bik (∆fi (t) − ∆fk (t)) dt k =1   N X d∆fi (t) d∆fk (t) +β bik − dt dt k =1

i = 1, . . . , N − 1, where ∆fi (t): Frequency deviation of area i from its nominal value. α and β: integral and proportional control gain. bik : coefficient representing the communication graph.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

A control scheme proposed in an earlier work Control objective: Sharing primary frequency reserves among non-synchronous areas by imposing that ∆f1 (t) = . . . = ∆fN (t). dc Control variables: Pidc , . . . , PN−1 N

X dPidc (t) =α bik (∆fi (t) − ∆fk (t)) dt k =1   N X d∆fi (t) d∆fk (t) +β bik − dt dt k =1

i = 1, . . . , N − 1, where ∆fi (t): Frequency deviation of area i from its nominal value. α and β: integral and proportional control gain. bik : coefficient representing the communication graph.

Problem addressed

Theoretical contributions

Empirical contributions

Delay modeling Sources of delays: measurement, transmission, computation, application Assumption: identical regardless of AC areas and communication links Dynamics of the effective power injections N

X dPidc (t) =α bik (∆fi (t − τ ) − ∆fk (t − τ )) dt k =1   N X d∆fi (t − τ ) d∆fk (t − τ ) +β bik − dt dt k =1

where τ : Delay between AC areas.

Conclusions

Problem addressed

Theoretical contributions

Empirical contributions

Delay modeling Sources of delays: measurement, transmission, computation, application Assumption: identical regardless of AC areas and communication links Dynamics of the effective power injections N

X dPidc (t) =α bik (∆fi (t − τ ) − ∆fk (t − τ )) dt k =1   N X d∆fi (t − τ ) d∆fk (t − τ ) +β bik − dt dt k =1

where τ : Delay between AC areas.

Conclusions

Problem addressed

Theoretical contributions

Empirical contributions

Delay modeling Sources of delays: measurement, transmission, computation, application Assumption: identical regardless of AC areas and communication links Dynamics of the effective power injections N

X dPidc (t) =α bik (∆fi (t − τ ) − ∆fk (t − τ )) dt k =1   N X d∆fi (t − τ ) d∆fk (t − τ ) +β bik − dt dt k =1

where τ : Delay between AC areas.

Conclusions

Problem addressed

Theoretical contributions

Empirical contributions

Delay modeling Sources of delays: measurement, transmission, computation, application Assumption: identical regardless of AC areas and communication links Dynamics of the effective power injections N

X dPidc (t) =α bik (∆fi (t − τ ) − ∆fk (t − τ )) dt k =1   N X d∆fi (t − τ ) d∆fk (t − τ ) +β bik − dt dt k =1

where τ : Delay between AC areas.

Conclusions

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Theoretical results on system stability

Assumptions: Constant losses within the DC grid Communication graph of the frequency information access among AC areas: connected, undirected, constant in time. Linearized model

Stability results on the impacts of the delays: Unique equilibrium point: Following a step change in the load of one of the AC areas, there is a unique equilibrium point: ∆f1 = ∆f2 = . . . = ∆fN . Nyquist criterion for the special case where all the AC areas have identical parameters.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Theoretical results on system stability

Assumptions: Constant losses within the DC grid Communication graph of the frequency information access among AC areas: connected, undirected, constant in time. Linearized model

Stability results on the impacts of the delays: Unique equilibrium point: Following a step change in the load of one of the AC areas, there is a unique equilibrium point: ∆f1 = ∆f2 = . . . = ∆fN . Nyquist criterion for the special case where all the AC areas have identical parameters.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Theoretical results on system stability

Assumptions: Constant losses within the DC grid Communication graph of the frequency information access among AC areas: connected, undirected, constant in time. Linearized model

Stability results on the impacts of the delays: Unique equilibrium point: Following a step change in the load of one of the AC areas, there is a unique equilibrium point: ∆f1 = ∆f2 = . . . = ∆fN . Nyquist criterion for the special case where all the AC areas have identical parameters.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Theoretical results on system stability

Assumptions: Constant losses within the DC grid Communication graph of the frequency information access among AC areas: connected, undirected, constant in time. Linearized model

Stability results on the impacts of the delays: Unique equilibrium point: Following a step change in the load of one of the AC areas, there is a unique equilibrium point: ∆f1 = ∆f2 = . . . = ∆fN . Nyquist criterion for the special case where all the AC areas have identical parameters.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Theoretical results on system stability

Assumptions: Constant losses within the DC grid Communication graph of the frequency information access among AC areas: connected, undirected, constant in time. Linearized model

Stability results on the impacts of the delays: Unique equilibrium point: Following a step change in the load of one of the AC areas, there is a unique equilibrium point: ∆f1 = ∆f2 = . . . = ∆fN . Nyquist criterion for the special case where all the AC areas have identical parameters.

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Benchmark system An MT HVDC system with 5 areas: Each area is modeled by an aggregated generator and a load. Communication graph: A circle: an area An edge: a communication channel between the two areas

1

5

2

3

4

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Simulation result: No delays

50.05

τ=0s

1

f (Hz)

50

49.95

49.9 0

10

20 time (s)

30

40

Problem addressed

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Conclusions

Simulation result: τ = 0.35s

50.05 τ=0s τ=0.35s

1

f (Hz)

50

49.95

49.9 0

10

20 time (s)

30

40

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Conclusions

Simulation result: τ = 0.37s

50.05 τ=0s τ=0.35s τ=0.37s

1

f (Hz)

50

49.95

49.9 0

10

20 time (s)

30

40

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Stability criterion

We define the following stability criterion: The system is classified as stable if, 20 seconds after the step change in the load, all the AC areas’ frequency deviations remain within ±50mHz around f e , i.e., |∆fi (t) − ∆f e | ≤ 50mHz, ∀i and ∀t > 22s where f e is the common value to which the frequency deviations of all AC areas converge when no delays is considered.

Problem addressed

Theoretical contributions

Empirical contributions

Stability criterion The ±50mHz band around f e is represented by the two horizontal lines. Unstable when τ = 0.37s. 50.05 τ=0s τ=0.35s τ=0.37s

f1 (Hz)

50

49.95

49.9 0

10

20 time (s)

30

40

Conclusions

Problem addressed

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Empirical contributions

Conclusions

Maximum acceptable delay For certain α, β, there exists a maximum acceptable delay, denoted by τmax , beyond which the system is unstable. Evolutiona of τmax as a function of the controller gains (assuming that α = β):

0

τmax (s)

10

-1

10

-2

10 6 10

7

10 α, β

8

10

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Conclusions

Oscillations of frequencies when τ = 2s

50.1

50.1 f2

f ,f ,f ,f

50.05

1 3 4 5

50

frequency (Hz)

frequency (Hz)

50.05

49.95 49.9 49.85

f (Pdc constant) 2

49.8 49.75 0

10

i

20 time (s)

f1,f3,f4,f5

50 f

49.95

2

49.9 49.85

f (Pdc constant) 2

49.8 30

α = β = 1 × 106

40

49.75 0

10

i

20 time (s)

30

α = β = 1 × 105

40

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Conclusions

Practical issue (impact of delays) in the implementation of a previously proposed control scheme. Theoretical results: Unique equilibrium point for the general case Nyquist criterion for a special case

Simulation results: Frequency oscillations in the presence of delays Relation between the maximum acceptable delay and the controller gains

Perspectives: Theoretical: extention to the general case Practice: benchmark system with more details

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Conclusions

Practical issue (impact of delays) in the implementation of a previously proposed control scheme. Theoretical results: Unique equilibrium point for the general case Nyquist criterion for a special case

Simulation results: Frequency oscillations in the presence of delays Relation between the maximum acceptable delay and the controller gains

Perspectives: Theoretical: extention to the general case Practice: benchmark system with more details

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Conclusions

Practical issue (impact of delays) in the implementation of a previously proposed control scheme. Theoretical results: Unique equilibrium point for the general case Nyquist criterion for a special case

Simulation results: Frequency oscillations in the presence of delays Relation between the maximum acceptable delay and the controller gains

Perspectives: Theoretical: extention to the general case Practice: benchmark system with more details

Problem addressed

Theoretical contributions

Empirical contributions

Conclusions

Conclusions

Practical issue (impact of delays) in the implementation of a previously proposed control scheme. Theoretical results: Unique equilibrium point for the general case Nyquist criterion for a special case

Simulation results: Frequency oscillations in the presence of delays Relation between the maximum acceptable delay and the controller gains

Perspectives: Theoretical: extention to the general case Practice: benchmark system with more details

Impact of Delays on a Consensus-based Primary ...

Frequency Control Scheme for AC Systems Connected by a. Multi-Terminal ... Multi-terminal HVDC system. 3 .... α and β: integral and proportional control gain.

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