decentrlised load freq

8/8/2019 decentrlised load freq

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This work was sponsored in part by National Science Foundation under grant ECS-9870041 and in part by US DOE/EPSCoR WV State Implementa-

tion Award.

The authors are with the Lane Department of Computer Science & Elec-

trical Engineering, West Virginia University, Morgantown, WV 26506.

Decentralized Load Frequency Control for

Load Following Services

Dulpichet Rerkpreedapong, Student Member, IEEE , and Ali Feliachi, Senior Member, IEEE

Abstract --This paper proposes a decentralized controller for

the load frequency control operated as a load following service.

Decentralization is achieved by developing a model for the inter-

face variables, which consist of frequencies of other subsystems.

To account for the modeling uncertainties, a local Kalman filter

is designed to estimate each subsystem’s own and interface vari-

ables. The controller uses these estimates, optimizes a given per-

formance index, and allocates generating units’s outputs accord-

ing to a deregulation scenario. Two test systems are given to

illustrate the proposed methodologies.

Index Terms--Load frequency control, Automatic generation

control, Decentralized control, Kalman filter, Estimation, Ancil-

lary services, Deregulation.

I. INTRODUCTION

HE power system consists of several interconnected con-trol areas where each one is traditionally responsible for

its native load and scheduled interchanges with neighboring

areas. Load frequency control (LFC) or automatic generation

control (AGC) is the mechanism by which the energy balanceis maintained. Under deregulation, such a mechanism can be

used as a load following service operated by the regulating

units according to a given contract. The regulating unit in acontrol area changes may change its output to match the

power demands in other areas, as the contract desires.

Conventionally, the area control error (ACE), which is a

combination of a frequency error (∆f) and a tie-line power

error (∆Ptie), reflects the control area’s performance, and is

used as the input to the load frequency controllers. Such con-trollers are PI (Proportional-Integral) controllers whose pa-

rameters are tuned using lengthy simulations and trial-and-

error approaches. Several optimization techniques have been proposed to solve this problem, but they require information

about the entire system rather than local information [1-2].

This paper proposes a completely decentralized LFC

scheme for load following services. A model for the interfacevariables, which consist of frequencies of other control areas

or subsystems, is developed. To account for the modeling

uncertainties, a local Kalman filter is designed to estimateeach subsystem’s own and interface variables from only local

measurements, namely area frequency and tie-line power. The

deregulation scenario considered here assumes that generat-ing units in each area supply a portion of the regulated power

according to their load following contracts [3-4]. Therefore,

the parameters of the proposed controllers are simultaneouslydesigned to control the generated power of each generating

unit to meet contractual requirement. The effectiveness of the

proposed controllers is demonstrated using two test systems,each consisting of a three-area power system. The first system

has one generator in each area, and the second has one area

with two units operating under a deregulation scenario.

II. DYNAMIC MODEL

A large interconnected power system consists of a number

of subsystems or control areas. Each area can be modeled ingreat details depending on the generators models and their

prime movers. But, to illustrate the proposed idea, a simpledynamic model, shown in Fig. 1, is presented in this section.

The test system has a more elaborate model.

HisT+1

1

TisT+1

1+ +

DiP∆

TiP∆

∑∆ j

ijtieP

Pii sTD +

1

iR

1

if ∆

iB

+

iACE

iCi uP =∆

ViP∆

Governor T urbine

Fig. 1. Block diagram of the ith-generating unit.

P Ti: turbine power P Vi: governor valve P Ci: governor set point P Di: power demand

f i: frequency ΑCE i: Area control error P tie,ij: tie-line power between area i and j

∆: deviation from nominal values

The state space model for this system is given by:

Diiiiiiiii P F z Gu B x A x ∆+++=& (1)

where

∑

−−

−

−−

=

≠=

0100

00002

001

01

0011

0

01

01

1

i

N

i j j

ij

Hi Hii

TiTi

Pi Pi Pi

i

i

B

T

T T R

T T

T T T

D

A

π

,

=

0

0

1

0

0

Hii

T B

T

1252

0-7803-7322-7/02/$17.00 © 2002 IEEE



[ ]T iG 02000 π −= ,

T

Pii

T F

−= 0000

1

∫ ∑ ∆∆∆∆=

≠=

i

N

i j j

ijtieViTiiT i ACE P P P f x

1,

, ∑ ∆⋅=

≠=

N

i j j

jiji f T z 1

xi: State variables of the ith area

ui: Input of the ith area, z i: Interface variablesT ij: Synchronizing power coefficient of tie-line i-jT Pi: Generator’s time constant

T Ti: Turbine’s time constant, T Hi: Governor’s time constant Ri: Droop characteristic, Di: Damping coefficient N: Number of interconnected areas

III. CONTROL DESIGN

The proposed controllers are designed for a given state

space model using an LQR (linear quadratic regulator) approach. It is known that the LQR has good gain and phase

stability margins, but an accurate model is needed and all of

its state variables are essential for its implementation. This isnot suitable for a decentralized control structure because of

the interface variables. In this paper, local state feedback

gains ( K i) are designed using LQR, but an additional feed-forward gain ( K Di) is separately designed to cancel the effects

of the interfaces [5].

A. Control Design with Available State and Interface

Variables

In this section, it is assumed that both state and interface

variables are available for feedback. Then, using the state

space model (1), a controller (ui) is designed as:

i Di xii Diiii z K u z K x K u +=+−= (2)

where K Di: Interface cancellation gain

K i : Stabilizing feedback gain, xiu : Stabilizing input

then the closed loop system is expressed by:

( ) Diiii Dii xiiiii P F z G K Bu B x A x ∆++++=& (3)

The gain ( K Di) interfaces are designed later to cancel the in-terface variables. Hence, the closed-loop system now has the

form:

Dii xiiiii P F u B x A x ∆++=& (4)

For a load frequency control problem, a non-zero set point,

i.e., steady state condition, is present when there is a change

in power demand ( ∆ P Di). The set point is given by:

Diioii

oii

oi P F u B x A x ∆++== 0& (5)

yielding:oii x x = (6)

oCi

oi

o xi P uu ∆== (7)

Define oiii x x x −=′ (8)

oi xii uuu −=′ (9)

Substitute eq. (8) and (9) into (4), a new system is obtained:

iiiii u B x A x ′+′=′& (10)

Subsequently, the controller parameters ( K i) in (2) aredesigned using LQR with the following performance index.

( )∫ ′′+′′=∞

0

dt ur u xQ x J iiT

iiiT

ii (11)

where J i: Performance index of the ith subsystemQi : System weighting matrix

r i: Input weighting matrix

The optimal controller is:iii x K u ′−=′ (12)

or oii

oiii xi x K u x K u ++−= (13)

In a deregulated power system, the generated power fromgenerating units will be allocated according to given load

following contracts shown as

∆

∆

∆

=

∆

∆

∆

dcm

dc

dc

nmnn

m

m

Gn

G

G

P

P

P

P

P

P

M

4 4 4 4 34 4 4 4 21

OM

2

1

21

22221

11211

2

1

α

α α α

α α α

α α α

(14)

where

∆ P Gi: Required change in pu MW of the ith generating unit

∆ P dcj : Change in demand of the jth distribution company

α ij: Contract factor that indicates ratio of required change

in pu MW of the ith generating unit to change in de-

mand (∆ P dcj) of the jth distribution company

∆

∆

∆

=

∆

∆

∆

Gn

G

G

N DN

D

D

P

P

P

P

P

P

M

L

MOM

L

M

2

1

2

1

2

1

00

00

00

β

β

β

(15)

[ ]ik i ×= 1111 L β (16)

∆ P Di : Total change in demand for which the ith area are

responsiblek i: Number of generation units of the ith area

Consequently, designing the controller parameters ( K i)must also satisfy contract-based constraints. At a desired set

point,

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oii

oGi

oCi

oi x K P P u −=∆=∆= (17)

( ) Diiiiioi P F K B A x ∆−−= −1

(18)

whereoi x : Set point of state variables

oiu : Set point of inputs

The input weighting matrix (r i) used for designing K i will

be selected by “lsqnonlin”, a search routine embedded in the

optimization toolbox in MATLAB, to satisfy eq. (17) and(18).

Load following

Contracts

(αij)

∆Pdi β i (List of generating

units of each area)

∆PGi, ∆PDi

Lsqnonlin

System

parameters

Ai, Bi, Fi

ri, K i

Fig. 2. Flowchart of determination of control parameters ( K i).

Next, the interface cancellation gain ( K Di) is designed tocancel the effects of the interface on the integral of the area

control error, which is one of the state variables.

∫ == iiii xC ACE y~~ (19)

The effect of the interface on the considered output ( i y~ ) is

( ) ( ) ( ) ii Diiiiii zi z G K B K B AC y +−−=∞ −1~~ (20)

then the interface cancellation gain ( K Di) is selected to make

eq. (20) equals to zero:

( ) ( ) iiiii Diiiiii G K B AC K B K B AC 11 ~~ −− −−=− (21)

( )[ ] ( ) iiiiiiiiii Di G K B AC B K B AC K 111 ~~ −−− −−−= (22)

From eq. (21), interface cancellation gain ( K Di), however, can be obtained as far as the number of inputs are not less than

that of the outputs.

B. Control Design with Estimation of State and Interface

Variables

In an interconnected power system, not all the state vari-

ables are measurable, and the interface variables cannot be

obtained from local measurements. In this paper, a localKalman filter is used to overcome this limitation by estimat-

ing the required state and interface variables using only avail-

able measurements at the expense of some performance deg-

radation. The structure of the proposed controller using esti-mates obtained from a Kalman filter is illustrated in Fig. 3.

Subsystem

(area #i)

Kalman

Filter

-K i+

K Di

ui(t) = ∆PCi

zi(t)

yi(t)

wi(t) vi(t)

zi(t)^

x̂i(t)

xi(t)

PDi

Fig. 3. Decentralized control structure for the ith subsystem.

wi : Plant noise, vi: Measurement noiseui: Control input, yi: Output

To design a Kalman filter that will estimate both local andinterface variables a dynamical model for these variables is

desired. This model is obtained by introducing the dynamics

of the interface variable ( z i), a combination of the deviationsof frequencies from the other areas, in the following form:

fii w z =& (23)

where w fi is a fictitious white noise. The reason for this as-sumption comes from the nature of the area frequencies

whose deviations keep oscillating around zero, and are

bounded when NERC’s performance standards are met.However, the variance of the fictitious noise must be properly

chosen to increase the accuracy of the above model. In this paper, the value of this variance is chosen at 0.001 by trial-and-error.

Augmenting the model given by (1) by adding the interface

dynamics given by (23) gives the following model:

Diiiiiiiii P F W Gu B x A x ∆+++=& (24)

iiii v xC y += (25)

where

=

i

ii

z

x x ,

=

fi

ii w

wW ,

=

00

iii

G A A ,

=

0

ii

B B ,

=

0

ii

F F

T

iG

=

100000

000100,

=

10000

0100000001

iC , [ ]0ii C C =

i x : Augmented state vector

A Kalman filter, based on (24) and (25), is designed to

estimate the augmented state vector. It is given by:

Diiiiiiiii

ii P F y Lu B x A

z

x x ∆+++=

= ˆˆ

ˆ

ˆˆ&

&&(26)

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iiii C L A A −=ˆ (27)

where Li is the Kalman gain that can be determined by aMATLAB function called “KALMAN.”

The full dynamical model of the ith subsystem including

the dynamics of the actual system in (1) and the Kalman es-timator in (26) is expressed in (28).

Dii

ii

ii

i

i

i

i

iii

i

i

i P

F

F z

Gu

B

B

x

x

AC L

A

x

x∆

+

+

+

=

0ˆˆ

0&̂&

(28)

In Fig. 3, the input of the subsystem using the estimates of state variables and interface is defined as

i Diiii z K x K u ˆˆ +−= (29)

IV. PERFORMANCE ANALYSIS

The stability of the entire interconnected system, called

here the composite or actual system, when the proposed con-trollers are implemented, is of a major concern rather thanthat of individual subsystems. The composite system has the

following state-space model and control input:

D P F Bu Ax x ∆++=& (30)

Kxu −= (31)

whereT

T N

T T T N

T T x x x x x x x

=

&L

&&&L&& ˆˆˆ

2121

[ ]T N uuuu L21= , [ ]T

DN D D D P P P P ∆∆∆=∆ L21

=

N N N

N N N

N

N

AC L

AC L

AC L

AGG

G AG

GG A

A

ˆ0000

0ˆ000

00ˆ00

00

00

000

222

111

21

2221

1121

LL

MOMMOM

LL

LL

OMMOM

M

LL

T

T N

T N

T T

T T

B B

B B

B B

B

=

LLOMOM

MM

LL

00

00

0000

22

11

jijiij S T GG = , [ ]4 4 34 4 21

L

j xof rowsof number

jS 0001=

=

N

NxN

K

K

K

K

L

OM

M

L

0

0

00

0 2

1

, [ ] Diii K K K −=

The system matrix ( A) can be written as

G A A~~

+= (32)

where

A~

: Ideal system matrix without interface consideration, i.e.,

A A =~

where any 0=ijG

G

~

: Interface matrix

The closed loop system is:

Dcl D P F x A P F x BK A x ∆+=∆+−= )(& (33)

( ) G AG BK A A cl cl ~~~~

+=+−= (34)

The performance of the composite closed-loop system ( Acl )

is analyzed based on its eigenvalues. Let µ be any eigenvalue

of the composite closed-loop system, i.e. an eigenvalue of Acl ,

and λ i be the ith eigenvalue of cl A~

. The objective of this sec-

tion is to estimate the maximum departure of µ from the ei-

genvalues of cl A~ . For this purpose let:

{ }ncl diag DT AT ~211

,,,~

λ λ λ L==− (35)

then

Γ+=+= −−− DT GT T AT T AT cl cl ~~ 111

(36)

Gershgorin Circle Theorem [6] states:

∑ Γ≤−∈=ℑ=

n

jijii d C d

~

1

: λ (37)

iℑ : Gershgorin disk’s region

C : Complex domain

ijΓ : (i-j) entity of the matrixΓ From Gershgorin’s theorem, a Gershgorin disk’s region

gives an estimate of the maximum departure of the composite

closed-loop system eigenvalues (µ) from the eigenvalues (λi)

of the closed-loop control area model. Since the strength of the interconnection is preset, one can guarantee closed-loop

stability of the composite system by properly designing the

area controllers.

V. CASE STUDY

A power system, which consists of three control areas in-

terconnected through a number of tie-lines as shown in Fig. 4

is used to illustrate the proposed idea.

Area 1 Area 2

Area 3

tie-line

Fig. 4. A three-area power system.

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The generating units and their prime mover models within

subsystems are shown as

Σ

Σ Σ Σ

K1 K2 K3 K4

Σ

Σ

∆Pc

Governor

Generator

Turbine

∆f ∆PT

+

_

+

∑∆ tiePDP∆

21

1

sT+

3

2

T

T

R

1

3

2

1 T

T

−

+

+

41

1

sT+ 51

1

sT+

+

+

+

+

+

+

61

1

sT+ 71

1

sT+

_ _

11

1

sT+

PsTD+1

Fig. 5. Generating unit and prime mover models.

Table 1. Data for a three-area power system

Data Area 1 Area 2 Area 3

Rating (MW) 1000 750 2000

Droop characteristic: R (%) 5 4 5

Damping: D (pu MW/Hz) 20 15 18

Constant of inertia: H (sec) 5 5 5

T1 2.8 3 2.5

T2 1 0 0

T3 0.15 1 1

T4 0.2 0.4 0.5

T5 6 0 5

T6 7 0 0

T7 0.5 0 0

K1 0.2 1 0.4K2 0.2 0 0.6

K3 0.4 0 0

K4 0.2 0 0

Synchronizing power coefficient of tie line i-j:

T12 = 60 MW/rad, T13 = 200 MW/rad, T23 = 100 MW/rad

Fig. 6. Eigenvalues of the closed-loop ideal system and actual system.

The eigenvalues of the closed-loop, ideal )clA~

and com-

posite (Acl), systems are obtained but only eigenvalues closeto the imaginary axis are plotted in Fig 6. These two sets of

eigenvalues are very close, and thus justify the application of

the proposed controllers as a feasible and effective decentral-ized control structure.

VI. SIMULATION

The performance of the proposed controllers is assessed

through simulation of two test systems: 1) a three-area power system with a single generator in each area, and 2) a test sys-

tem similar to the previous one except area 1 now has two

generating units operating under a deregulation scenario.

A. Test system #1

In this test system, the contract matrix (α) is identity, and

each area has a single generating unit whose data are given inTable 1. The following unit-step changes in power demands

are applied: ∆Pdc1 = 200 MW, ∆Pdc2 = -100 MW and ∆Pdc3 =

150 MW. The MVA base is 2000. The area control error

(ACE) of the closed-loop ideal system ( )clA~

and the closed-

loop composite system are shown in Fig. 7.

Fig. 7. Area control error for ideal and actual systems.

Later, the frequency deviation (∆f) of each area regulated

by the proposed controllers is shown in Fig. 8.

Fig. 8. Frequency deviation of actual system.

O : Ideal system

X : Actual system

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B. Test system #2

This test system is more realistic than the previous one. Now area 1 has an additional generating unit identical to the

one in area 2. It is used to demonstrate the ability of the pro-

posed controllers to allocate generating units’s outputs ac-cording to a given load following contract described as

pu

P

P

P

P

P

P

P

dc

dc

dc

G

G

G

G

−=

∆

∆

∆

=

∆

∆

∆∆

6.0

0325.0

0475.0

05.0

8.000

1.08.00

1.004.0

02.06.0

3

2

1

4

3

2

1

4 4 34 4 21α

Later the identical changes in demands in the previous test

system are applied. Then the total change in demand for

which each area has to be responsible can be obtained as

pu

P

P

P

P

P

P

P

G

G

G

G

D

D

D

=

∆

∆

∆

∆

=

∆

∆

∆

0.06

0.0325-

0975.0

1000

0100

0011

4

3

2

1

3

2

1

After the simulation is completed, the changes in turbine

power (∆PT) of each unit are shown in Fig. 9.

Fig. 9. Changes in turbine power for test system #2.

The area control error and frequency deviation of each area

are shown as Fig. 10 and 11 respectively.

Fig. 10. Area control error for test system #2

Fig. 11. Frequency deviation for test system #2.

VII. CONCLUSION

This paper proposes a decentralized controller for the load

frequency control operated as a load following service. Decen-

tralization is achieved by developing a model for the interfacevariables, which is a combination of frequencies of other sub-

systems. To account for the modeling uncertainties, a local

Kalman filter is designed to estimate each subsystem’s ownand interface variables. The controller uses these estimates,

optimizes a given performance index, and allocates generat-ing units’s outputs according to a deregulation scenario. The

performance of the proposed controllers is assessed through

eigenanalysis and simulation of two test systems, each con-

sisting of a three-area power system. The first system has onegenerator in each area, and the second has one area with two

units operating under a deregulation scenario. It is shown that

the proposed technique gives good results.

VIII. REFERENCES

[1] C. E. Fosha, Jr. and O. I. Elgerd, “The Megawatt-Frequency ControlProblem: A New Approach Via Optimal Control Theory,” IEEE Trans-actions on Power Systems, vol. 89, no. 4, pp. 563-577, April 1970.

[2] M. L. Kothari, N. Sinha and M. Rafi, “Automatic Generation Control of an Interconnected Power System Under Deregulated Environment,”

Power Quality’ 98, pp. 95-102, 1998.

[3] R. D. Christie and A. Bose, “Load Frequency Control Issues in Power System Operations after Deregulation”, IEEE Transaction on Power Systems, Vol. 11, No. 3, pp. 1191-1200, August 1996.

[4] R. D. Christie and A. Bose, “Load Frequency Control In Hybrid Elec-

tric Power Markets,” Proceedings of the 1996 IEEE International Con- ference on Control Applications, pp. 432-436, September, 1996.

[5] J. B. Burl, Linear Optimal Control , Addison Wesley Longman, Inc.,1999.

[6] G. H. Golub and A.F. Van Loan, “Matrix Computations,” 2nd edition,

Baltimore: The John Hopkins university Press, 1989.

IV. BIOGRAPHIES

Dulpichet Rerkpreedapong received his MSEE from the CSEE Depart-

ment, West Virginia University in 1999. He is currently working towards a

Ph.D. in Electrical Engineering at WVU. His research interests are in power systems control and operation, and power systems restructuring.

Ali Feliachi received the MS and PhD degree in electrical engineering

from Georgia Tech in 1979 and 1983 respectively. He joined the faculty of Electrical and Computer Engineering at West Virginia University in January

1984 where he is now a Full Professor and the holder of the endowed Electric

Power Systems Chair position. His research interests are in modeling,

simulation, control and estimation of large-scale systems with emphasis onelectric power systems.

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