Dra

ftSparsity-promoting wide-area control ofpower systems

F. Dorfler, Mihailo Jovanovic, M. Chertkov, and F. Bullo

2013 American Control Conference

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Electro-mechanical oscillations in power systems

• Local oscillations? single generators swing relative to the rest of the grid

? typically damped by Power System Stabilizers (PSSs)

• Inter-area oscillations? groups of generators oscillate relative to each other

? associated with dynamics of power transfers

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Inter-area oscillations

• Blackout of Aug. 10, 1996? resulted from instability of the 0.25Hz mode

western interconnected system: California-Oregon power transfer:

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Slow coherency theory

• WHERE ARE THE INTER-AREA MODES COMING FROM?

? slow coherency theory Chow, Kokotovic, et al. ’78, ’82

RTS 96 power system: linearized swing equation:

220

309

310

120103

209

102102

118

307

302

216

202

time

gene

rato

ran

gles

time

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Conventional control

• Blue layer: generators with transmission lines

• Fully decentralized controller? effective against local oscillations

? ineffective against inter-area oscillations

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Wide-area control• Blue layer: generators with transmission lines

wide-area controller

KEY CHALLENGE:

identification of a signal exchange network

performance vs sparsity

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wide-areacontroller

powernetwork

dynamics

generator

transmission line

wide-area measurements

(e.g. PMUs)

remote control signals

uwac(t)

uloc(t)

uloc(t)

+

+

+

channel andmeasurement noise

local control loops

...

system noise

FACTS

PSS & AVR

⌘(t)

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Outline

¶ SPARSITY-PROMOTING WIDE-AREA CONTROL

? Safeguard against inter-area oscillations

? Performance vs sparsity

· CASE STUDY

? IEEE New England power grid model

¸ SUMMARY AND OUTLOOK

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Case study: IEEE New England Power Grid• MODEL FEATURES

? detailed sub-transient generator models

? exciters

? carefully tuned PSS data

15

512

1110

7

8

9

4

3

1

2

17

18

14

16

19

20

21

24

26

27

28

31

32

34 33

36

38

39 22

35

6

13

30

37

25

29

23

1

10

8

2

3

6

9

4

7

5

F

Fig. 9. The New England test system [10], [11]. The system includes10 synchronous generators and 39 buses. Most of the buses have constantactive and reactive power loads. Coupled swing dynamics of 10 generatorsare studied in the case that a line-to-ground fault occurs at point F near bus16.

test system can be represented by

!i = "i,Hi

#fs"i = !Di"i + Pmi ! GiiE

2i !

10!

j=1,j !=i

EiEj ·

· {Gij cos(!i ! !j) + Bij sin(!i ! !j)},

"##$##%

(11)

where i = 2, . . . , 10. !i is the rotor angle of generator i withrespect to bus 1, and "i the rotor speed deviation of generatori relative to system angular frequency (2#fs = 2# " 60Hz).!1 is constant for the above assumption. The parametersfs, Hi, Pmi, Di, Ei, Gii, Gij , and Bij are in per unitsystem except for Hi and Di in second, and for fs in Helz.The mechanical input power Pmi to generator i and themagnitude Ei of internal voltage in generator i are assumedto be constant for transient stability studies [1], [2]. Hi isthe inertia constant of generator i, Di its damping coefficient,and they are constant. Gii is the internal conductance, andGij + jBij the transfer impedance between generators iand j; They are the parameters which change with networktopology changes. Note that electrical loads in the test systemare modeled as passive impedance [11].

B. Numerical Experiment

Coupled swing dynamics of 10 generators in thetest system are simulated. Ei and the initial condition(!i(0),"i(0) = 0) for generator i are fixed through powerflow calculation. Hi is fixed at the original values in [11].Pmi and constant power loads are assumed to be 50% at theirratings [22]. The damping Di is 0.005 s for all generators.Gii, Gij , and Bij are also based on the original line datain [11] and the power flow calculation. It is assumed thatthe test system is in a steady operating condition at t = 0 s,that a line-to-ground fault occurs at point F near bus 16 att = 1 s!20/(60Hz), and that line 16–17 trips at t = 1 s. Thefault duration is 20 cycles of a 60-Hz sine wave. The faultis simulated by adding a small impedance (10"7j) betweenbus 16 and ground. Fig. 10 shows coupled swings of rotorangle !i in the test system. The figure indicates that all rotorangles start to grow coherently at about 8 s. The coherentgrowing is global instability.

C. Remarks

It was confirmed that the system (11) in the New Eng-land test system shows global instability. A few comments

0 2 4 6 8 10-5

0

5

10

15

!i /

ra

d

10

02

03

04

05

0 2 4 6 8 10-5

0

5

10

15

!i /

ra

d

TIME / s

06

07

08

09

Fig. 10. Coupled swing of phase angle !i in New England test system.The fault duration is 20 cycles of a 60-Hz sine wave. The result is obtainedby numerical integration of eqs. (11).

are provided to discuss whether the instability in Fig. 10occurs in the corresponding real power system. First, theclassical model with constant voltage behind impedance isused for first swing criterion of transient stability [1]. This isbecause second and multi swings may be affected by voltagefluctuations, damping effects, controllers such as AVR, PSS,and governor. Second, the fault durations, which we fixed at20 cycles, are normally less than 10 cycles. Last, the loadcondition used above is different from the original one in[11]. We cannot hence argue that global instability occurs inthe real system. Analysis, however, does show a possibilityof global instability in real power systems.

IV. TOWARDS A CONTROL FOR GLOBAL SWING

INSTABILITY

Global instability is related to the undesirable phenomenonthat should be avoided by control. We introduce a keymechanism for the control problem and discuss controlstrategies for preventing or avoiding the instability.

A. Internal Resonance as Another Mechanism

Inspired by [12], we here describe the global instabilitywith dynamical systems theory close to internal resonance[23], [24]. Consider collective dynamics in the system (5).For the system (5) with small parameters pm and b, the set{(!,") # S1 " R | " = 0} of states in the phase plane iscalled resonant surface [23], and its neighborhood resonantband. The phase plane is decomposed into the two parts:resonant band and high-energy zone outside of it. Here theinitial conditions of local and mode disturbances in Sec. IIindeed exist inside the resonant band. The collective motionbefore the onset of coherent growing is trapped near theresonant band. On the other hand, after the coherent growing,it escapes from the resonant band as shown in Figs. 3(b),4(b), 5, and 8(b) and (c). The trapped motion is almostintegrable and is regarded as a captured state in resonance[23]. At a moment, the integrable motion may be interruptedby small kicks that happen during the resonant band. That is,the so-called release from resonance [23] happens, and thecollective motion crosses the homoclinic orbit in Figs. 3(b),4(b), 5, and 8(b) and (c), and hence it goes away fromthe resonant band. It is therefore said that global instability

!"#$%&'''%()(*%(+,-.,*%/012-3*%)0-4%5677*%899: !"#$%&'

(')$

Authorized licensed use limited to: Univ of Calif Santa Barbara. Downloaded on June 10, 2009 at 14:48 from IEEE Xplore. Restrictions apply.

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Preview of a key result

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

spy(B

K⇤ �)

2 2

15

512

1110

7

8

9

4

3

1

2

17

18

14

16

19

20

21

24

26

27

28

31

32

34 33

36

38

39 22

35

6

13

30

37

25

29

23

1

10

8

2

3

6

9

4

7

5

F

Fig. 9. The New England test system [10], [11]. The system includes10 synchronous generators and 39 buses. Most of the buses have constantactive and reactive power loads. Coupled swing dynamics of 10 generatorsare studied in the case that a line-to-ground fault occurs at point F near bus16.

test system can be represented by

!i = "i,Hi

#fs"i = !Di"i + Pmi ! GiiE

2i !

10!

j=1,j !=i

EiEj ·

· {Gij cos(!i ! !j) + Bij sin(!i ! !j)},

"##$##%

(11)

where i = 2, . . . , 10. !i is the rotor angle of generator i withrespect to bus 1, and "i the rotor speed deviation of generatori relative to system angular frequency (2#fs = 2# " 60Hz).!1 is constant for the above assumption. The parametersfs, Hi, Pmi, Di, Ei, Gii, Gij , and Bij are in per unitsystem except for Hi and Di in second, and for fs in Helz.The mechanical input power Pmi to generator i and themagnitude Ei of internal voltage in generator i are assumedto be constant for transient stability studies [1], [2]. Hi isthe inertia constant of generator i, Di its damping coefficient,and they are constant. Gii is the internal conductance, andGij + jBij the transfer impedance between generators iand j; They are the parameters which change with networktopology changes. Note that electrical loads in the test systemare modeled as passive impedance [11].

B. Numerical Experiment

Coupled swing dynamics of 10 generators in thetest system are simulated. Ei and the initial condition(!i(0),"i(0) = 0) for generator i are fixed through powerflow calculation. Hi is fixed at the original values in [11].Pmi and constant power loads are assumed to be 50% at theirratings [22]. The damping Di is 0.005 s for all generators.Gii, Gij , and Bij are also based on the original line datain [11] and the power flow calculation. It is assumed thatthe test system is in a steady operating condition at t = 0 s,that a line-to-ground fault occurs at point F near bus 16 att = 1 s!20/(60Hz), and that line 16–17 trips at t = 1 s. Thefault duration is 20 cycles of a 60-Hz sine wave. The faultis simulated by adding a small impedance (10"7j) betweenbus 16 and ground. Fig. 10 shows coupled swings of rotorangle !i in the test system. The figure indicates that all rotorangles start to grow coherently at about 8 s. The coherentgrowing is global instability.

C. Remarks

It was confirmed that the system (11) in the New Eng-land test system shows global instability. A few comments

0 2 4 6 8 10-5

0

5

10

15

!i /

ra

d

10

02

03

04

05

0 2 4 6 8 10-5

0

5

10

15

!i /

ra

d

TIME / s

06

07

08

09

Fig. 10. Coupled swing of phase angle !i in New England test system.The fault duration is 20 cycles of a 60-Hz sine wave. The result is obtainedby numerical integration of eqs. (11).

are provided to discuss whether the instability in Fig. 10occurs in the corresponding real power system. First, theclassical model with constant voltage behind impedance isused for first swing criterion of transient stability [1]. This isbecause second and multi swings may be affected by voltagefluctuations, damping effects, controllers such as AVR, PSS,and governor. Second, the fault durations, which we fixed at20 cycles, are normally less than 10 cycles. Last, the loadcondition used above is different from the original one in[11]. We cannot hence argue that global instability occurs inthe real system. Analysis, however, does show a possibilityof global instability in real power systems.

IV. TOWARDS A CONTROL FOR GLOBAL SWING

INSTABILITY

Global instability is related to the undesirable phenomenonthat should be avoided by control. We introduce a keymechanism for the control problem and discuss controlstrategies for preventing or avoiding the instability.

A. Internal Resonance as Another Mechanism

Inspired by [12], we here describe the global instabilitywith dynamical systems theory close to internal resonance[23], [24]. Consider collective dynamics in the system (5).For the system (5) with small parameters pm and b, the set{(!,") # S1 " R | " = 0} of states in the phase plane iscalled resonant surface [23], and its neighborhood resonantband. The phase plane is decomposed into the two parts:resonant band and high-energy zone outside of it. Here theinitial conditions of local and mode disturbances in Sec. IIindeed exist inside the resonant band. The collective motionbefore the onset of coherent growing is trapped near theresonant band. On the other hand, after the coherent growing,it escapes from the resonant band as shown in Figs. 3(b),4(b), 5, and 8(b) and (c). The trapped motion is almostintegrable and is regarded as a captured state in resonance[23]. At a moment, the integrable motion may be interruptedby small kicks that happen during the resonant band. That is,the so-called release from resonance [23] happens, and thecollective motion crosses the homoclinic orbit in Figs. 3(b),4(b), 5, and 8(b) and (c), and hence it goes away fromthe resonant band. It is therefore said that global instability

!"#$%&'''%()(*%(+,-.,*%/012-3*%)0-4%5677*%899: !"#$%&'

(')$

Authorized licensed use limited to: Univ of Calif Santa Barbara. Downloaded on June 10, 2009 at 14:48 from IEEE Xplore. Restrictions apply.

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single wide-area comm link

single long range interaction ⇒ nearly centralizedperformance

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Optimal wide-area control

linearized dynamics: x = Ax + B1 d + B2 u

objective function: J = limt→∞

E(xT (t)Qx(t) + uT (t)Ru(t)

)

memoryless controller: u = −K x

? no structural constraints

globally optimal controller:

ATP + P A − P B2R−1BT2 P + Q = 0

Kc = R−1BT2 P

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Sparsity-promoting optimal control

minimize J(K) + γ∑

i, j

Wij |Kij|

←−

←−

varianceamplification

sparsity-promotingpenalty function

? γ > 0 − performance vs sparsity tradeoff

? Wij ≥ 0 − weights (for additional flexibility)

Lin, Fardad, Jovanovic, IEEE TAC ’13 (in press; arXiv:1111.6188)

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Parameterized family of feedback gains

K(γ) := argminK

(J(K) + γ g(K))

ALGORITHM: alternating direction method of multipliers

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Performance index

• Energy of power network without inter-area modes

? inspired by slow coherency theory

J := limt→∞

E(θT (t)Qθ θ(t) + θT (t) θ(t) + uT (t)u(t)

)

Qθ := ε I +

(I − 1

N11T

)

? other choices possible

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Open-loop dynamics• Dominant inter-area modes with local PSSs

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Mode 1

10all others

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Mode 2

4, 5, 6, 7

1, 2, 3, 8, 9

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Mode 3

4, 5, 6, 7, 9

2,3

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Mode 4

6,74,5

others 0.5

1

1.5

2

30

210

60

240

90

270

120

300

150

330

180 0

Mode 5

others1,8

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Performance vs sparsity

(J − Jc) /Jc card (K) /card (Kc)

10−4 10−3 10−2 10−1 1000

0.4

0.8

1.2

1.6

γ

perc

ent

10−4 10−3 10−2 10−1 1000

20

40

60

80

γ

γ = 1relative to Kc−−−−−−−−−−→

{1.6% performance loss

5.5% non-zero elements in K

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• Signal exchange network

γ = 0.0289, card (K) = 90

γ = 1, card (K) = 37

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0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Mode 1

10all others

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Mode 1

10

others

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

t [s]

t [s]

t [s]

t [s]

✓(t)

[Hz]

✓(t)

[Hz]

✓ 10(t

)�

✓ i(t

)[r

ad]

✓ 10(t

)�

✓ i(t

)[r

ad]

generator 10 generator 10

local PSS control local PSS control & wide-area control

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• Robustness?

powernetwork

dynamics

local control loops

...

dynamics with local control

wide-area controluwac(t)

⌘(t)

x(t)

�g

K⇤�

gain uncertainty

system noise

�m

+

+

-

multiplicative uncertainty

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19multivariable phase margin

10−4 10−3 10−2 10−1 10067

68

69

70

γ

multivariable gain reduction margin multivariable gain amplification margin

10−4 10−3 10−2 10−1 100

0.18

0.19

0.2

γ10−4 10−3 10−2 10−1 100

5

5.2

5.4

5.6

5.8

γ

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Summary and outlook

• SPARSITY-PROMOTING OPTIMAL CONTROL

? Performance vs sparsity tradeoffLin, Fardad, Jovanovic, IEEE TAC ’13 (in press; arXiv:1111.6188)

? Softwarewww.umn.edu/∼mihailo/software/lqrsp/

• WIDE-AREA CONTROL OF POWER NETWORKS

? Remedy against inter-area oscillations

? IEEE New England power grid model

• OPEN QUESTIONS

? Extension to structure-preserving descriptor models

? Theoretic analysis of robustness degradation

? Exploit the rotational symmetry of the models