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30

CCHHAAPPTTEERR 33

MMOODDEELLLLIINNGG OOFF PPOOWWEERR SSYYSSTTEEMMSS FFOORR AAGGCC WWIITTHH

IINNTTEEGGRRAALL CCOONNTTRROOLL AANNDD OOPPTTIIMMAALL CCOONNTTRROOLL

33..11 IINNTTRROODDUUCCTTIIOONN

For AGC studies it is necessary to obtain appropriate models of the

interconnected power systems. In present research work, models of the following

types of power systems have been considered for AGC studies [3-4, 6-8, 11, 35-42].

1) Two area thermal-thermal (non reheat)

2) Two area thermal-thermal (reheat)

3) Two area thermal-hydro

4) Three area thermal-thermal-hydro

The models of above mentioned interconnected power systems with integral

control scheme, state space modeling of these power systems to design optimal

controllers and stability studies of these power system models have been dealt with

in this chapter. The discrete versions of these power system models have also been

obtained.

The models of above mentioned interconnected power systems have been

used subsequently in Chapter 5 for illustrating the application of artificial neural

networks as controllers for AGC.



31

33..22..11 MMOODDEELL OOFF AA TTWWOO AARREEAA TTHHEERRMMAALL--TTHHEERRMMAALL ((NNOONN RREEHHEEAATT))

PPOOWWEERR SSYYSSTTEEMM WWIITTHH IINNTTEEGGRRAALL CCOONNTTRROOLLLLEERR

Perturbed model of a two area thermal-thermal (non reheat) power system

with conventional integral controller scheme is shown in Fig. 3.1.

1------------1 + sTg1

1------------1 + sTt1

Kp1------------1 + sTp1

1-----

S

1------------1 + sTg2

1------------1 + sTt2

Kp2------------1 + sTp2

-1

-----

S

-1

-KT

∆Pg1 ∆Pt1

∆Pg2 ∆Pt2

∆PD1

∆PD2

∆Ptie(1,2)

−

−

+

−

+

−

+

−

u1ACE1

1-----R1

−∆ f 1

∆ f 2

B1

1-----

S +

1-----R2

2πT0

B2

+

+

+

+

GovernorSteam Turbine

Non Reheat Power System



AREA 1(THERMAL NON REHEAT)

TIE LINE

12

3


45

6

7

8

9

u2

x1x2x3

x4x5x6

x7

LoadDisturbance

(d1)

LoadDisturbance

(d2)

Integral Controller

Integral Controller

-KT

ACE2

Fig. 3.1: Two area thermal-thermal (non reheat) system with integral controller

The system state equations with reference to transfer function blocks from 1

to 7 (equations for 71xto x ) are same as in state space model of the same power

system (Section 3.3.1, Fig. 3.5). With integral control, the equations for control inputs

21&uu are as given below:

Area 1 (Block 8):

)()( 71111 x x BK ACE K u T T +−=−=

Area 2 (Block 9):

)()( 74222 x x BK ACE K u T T −−=−=



32

33..22..22 MMOODDEELL OOFF AA TTWWOO AARREEAA TTHHEERRMMAALL––TTHHEERRMMAALL ((RREEHHEEAATT)) PPOOWWEERR

SSYYSSTTEEMM WWIITTHH IINNTTEEGGRRAALL CCOONNTTRROOLLLLEERR

Perturbed model of a two area thermal-thermal (reheat) power system with

conventional integral controller scheme is shown in Fig. 3.2.

Kp1----------1+sTp1

1----S

Kp2----------1+sTp2

-1-1

∆PD1

∆PD2

∆Ptie(1,2)

−

−

+

−

+

−

+

−

ACE1

1-----R1

−

∆ f 1

∆ f 2

B1

+

ACE2

1-----R2

B2

+

+

+

+

+

−

AREA 1(THERMAL)

TIE LINE

∆Pt21----S

Σ Σ Σ

Σ

ΣΣΣ

1-----------1+sTg1

1----------1+sTt1

∆Pg1 ∆Pt11+sKr1Tr1---------------

1 + sTr1

∆Pr1

AREA 2(THERMAL)

1-----------1+sTg2

1----------1+sTt2

∆Pg2 1+sKr2Tr2---------------

1 + sTr2

∆Pr2

2πΤ0−−−−

S

Power SystemSteam

TurbineReheater StageGovernorIntegral Controller

Power SystemSteam

TurbineReheater StageGovernorIntegral Controller

-KT

x1x2x3x4

x5x6x7x8

x9

12

34

5678

9

10

11

u1

u2-KT

Fig. 3.2: Two-area thermal-thermal (reheat) power system with integral controller


to 9 (equations for 91 xto x ) are same as in the state space model of the same power



Area 1 (Block 10):

)()(91111x x BK ACE K u T T +−=−=

Area 2 (Block 11):

)()(95222 x x BK ACE K u T T −−=−=



33

33..22..33 MMOODDEELL OOFF AA TTWWOO AARREEAA TTHHEERRMMAALL––HHYYDDRROO PPOOWWEERR SSYYSSTTEEMM WWIITTHH

IINNTTEEGGRRAALL CCOONNTTRROOLLLLEERR

Perturbed model of a two area thermal-hydro power system with


1------------1 + sTg1

1------------1 + sTt1

Kp1------------1 + sTp1

1-----

S

1----------1 + sT1

1 - sTw-------------1+0.5sTw

Kp2------------1 + sTp2

-1-1

-KT

∆Pg1 ∆Pt1

∆Ptw

∆PD1

∆PD2

∆Ptie(1,2)

−

−

+

−

+

−

+

−

ACE1

1-----R1

−∆ f 1

∆ f 2

B1

1-----

S +

ACE2

1-----R2

2πTo

------

s

B2

+

+

+

+

+

−



Governorstage 2 Water Turbine Power System

AREA 2(HYDRO)

AREA 1(THERMAL NONREHEAT)

TIE LINE

Σ Σ Σ

Σ

Σ

ΣΣ1+sT2

-----------1 + sT3

Governorstage 1

∆PG2∆PG1

Integral Controller

Integral Controller

x1x2x3

x4x5x6x7

x8

9

u1

u2

10

12

3

4567

-KH

8

Fig. 3.3: Two area thermal-hydro power system with integral controller





Area 1 (Block 9):

)()(81111 x x BK ACE K u T T +−=−=

Area 2 (Block 10):

)()( 84222 x x BK ACE K u H H −−=−=



34

33..22..44 MMOODDEELL OOFF AA TTHHRREEEE AARREEAA TTHHEERRMMAALL––TTHHEERRMMAALL--HHYYDDRROO PPOOWWEERR

SSYYSSTTEEMM WWIITTHH IINNTTEEGGRRAALL CCOONNTTRROOLLLLEERR

Perturbed model of a three area thermal-thermal-hydro power system with


1------------1 + sTg1

1------------1 + sTt1

Kp1------------1 + sTp1

1-----S

1------------1 + sTg2

1------------1 + sTt2

Kp2------------1 + sTp2

-KT

∆Pg1 ∆Pt1

∆Pg2 ∆Pt2

∆PD1

∆PD2

∆Ptie(1)

−

−

+

−

+

−

+

ACE1

1-----R1

−

∆ f 1

∆ f 2

B1

1-----S

-KT+

ACE2

+

+

+

+






1 + sT2------------1 + sT3

1 - sTw---------------1 + 0.5 sTw

Kp3------------1 + sTp3

∆PG2 ∆Ptw−

+

−

−

∆ f31-----S

-KH+

u3ACE3+

+

Water Turbine Power System

------------s

2πT12 ------------s

2πT13

------------s

2πT23

1-----R2

B2

∆PD31

-----R3

B3

+

−

+ −

+

+

+ −

+

+

a12 = -1

a13 = -1

a23 = -1

+

+

−

∆Ptie(2)

∆Ptie(3)

AREA 3(HYDRO)

1------------1 + sT1

Governorstage 1

∆PG1


4

5

6

78910

12

3

13

1112

14

15

16

u2

u1

x1x2x3

x4x5x6

x7x8x9x10

x11x12

x13

TIE LINE

TIE LINE

TIE LINE

Governorstage 2

Fig. 3.4: Three area thermal-thermal-hydro power system with integral controller



system (Section 3.3.4, Fig. 3.8). With integral control, the control inputs 321&, uuu are:

Area 1 (Block 14):

)()(12111111x x x BK ACE K u T T ++−=−=

Area 2 (Block 15):

)()( 13114222 x x x BK ACE K u T T +−−=−=

Area 3 (Block 16):

)()(13127333x x x BK ACE K u H H −−−=−=



35

33..33 SSTTAATTEE SSPPAACCEE RREEPPRREESSEENNTTAATTIIOONN OOFF PPOOWWEERR SSYYSSTTEEMM MMOODDEELLSS FFOORR

OOPPTTIIMMAALL CCOONNTTRROOLL

33..33..11 SSTTAATTEE SSPPAACCEE MMOODDEELL OOFF TTWWOO AARREEAA TTHHEERRMMAALL--TTHHEERRMMAALL ((NNOONN RREEHHEEAATT)) PPOOWWEERR SSYYSSTTEEMM

For the two area thermal–thermal (non reheat) power system, the state space model

with full state feedback (9 state feedback) has been developed as shown in Fig. 3.5.

1------------1 + sTg1

1------------1 + sTt1

Kp1------------1 + sTp1

1----S

Kp2------------1 + sTp2

-1

1-----S

-1

∆Pg1 ∆Pt1

∆PD1

∆PD2

∆Ptie(1,2)

−

−

+

−

+

−

+

−

ACE1

1-----R1

−

∆ f 1

∆ f 2

B1

+

ACE2

1-----R2

2πT0

B2

+

+

+

+

+

−

AREA 2(THERMAL)

AREA 1(THERMAL)

TIE LINE

1------------1 + sTg2

1------------1 + sTt2

∆Pg2 ∆Pt2

x1 x2 x3

x7

x5 x6

x8

x9 x4

u1

u2

d1

d2

1----S

123

6 5 4

7

8

9

Σ Σ Σ

Σ

ΣΣΣ

Fig. 3.5: State space model of two area thermal-thermal (non reheat) power system

Different variables have been defined as:

State Variables:

11f x ∆= 12

Pt x ∆= 13 Pg x ∆= 24f x ∆= 25 Pt x ∆= 26 Pg x ∆=

)2,1(7 tieP x ∆= = dt ACE x 18 = dt ACE x 29

Control inputs: 1u and 2

u

Disturbance inputs: 11 DPd ∆= and 22 DPd ∆=



36

State equations:

From the transfer function blocks labeled from 1 to 9 (Fig. 3.5);

For block 1:

)(1721111

d x xK xT xPP

−−=+

i.e., 1

1

1

7

1

1

2

1

1

1

1

1

1d

T

K x

T

K x

T

K x

T x

P

P

P

P

P

P

P

−−+−=

For block 2:

3212 x xTt x =+

i.e., 3

1

2

1

2

11 x

Tt x

Tt x +−=

For block 3:

11

1

313

1u x

R xTg x +

−=+

i.e., 1

1

3

1

1

11

3

111u

Tg x

Tg x

Tg R x +−

−=

For block 4:

)(2752424 d x xK xT x PP −+=+

i.e., 2

2

2

7

2

2

5

2

2

4

2

41 d

T K x

T K x

T K x

T x

P

P

P

P

P

P

P

−++−=

For block 5:

6525 x xTt x =+

i.e., 6

2

5

2

5

11 x

Tt x

Tt x +−=

For block 6:

24

2

6261 u x

R xTg x +−=+

i.e., 2

2

6

2

4

22

6

111u

Tg x

Tg x

Tg R x +−−=



37

For block 7:

4

0

1

0

722 xT xT x π π −=

For block 8:

7118 x x B x +=

For block 9:

7429 x x B x −=

Above equations are arranged in vector matrix form called as the ‘State Equation’:

d Bu Ax x Γ ++= ,

where, A(9×9) is State Matrix, B(9×2) is Control Matrix & (9×2) is Disturbance

Matrix

The matrices A, B and are:

−

−

=Γ

=

−

−

−−

−

−

−−

−

−−

=

00

00

00

00

00

0

00

00

0

;

00

00

00

10

00

00

01

00

00

;

00100000

00100000

000002002

0001

01

000

00011

0000

0001

000

0000001

01

00000011

0

0000001

2

2

1

1

2

1

2

1

00222

22

2

2

2

2

2

111

11

1

1

1

1

1

P

P

P

P

P

P

P

P

P

P

P

P

P

P

T K

T

K

Tg

Tg

B

B

B

T T

TgTg R

Tt Tt

T

K

T

K

T

TgTg R

Tt Tt

T

K

T

K

T

A

π π

The State Vector ‘x’ (9×1), Control Vector ‘u’ (2×1) and the Disturbance Vector ‘d’ (2×1) are:

[ ]T x x x x x x x x x x987654321= ; u=

2

1

u

u; d =

2

1

d

d



38

DESIGN OF OPTIMAL CONTROLLER

In optimal control, the control inputs are chosen as a linear combination of

feedback from all the nine system states (921

,........,, x x x ) as given below:

9192121111.......... xk xk xk u +++=

9292221212.......... xk xk xk u +++=

where, ‘K’ (2×9) is the feedback gain matrix given by;

=

292827262524232221

191817161514131211

k k k k k k k k k

k k k k k k k k k K

The system State Equation is:

Bu Ax x += ……. (For a step load change of a constant magnitude, ‘ d Γ . ’ = 0)

The output equation is:

DuCx y +=

However, for a feedback control system, the matrix D is assumed zero.

Hence; Cx y = where C (2 × 9) is the Output Matrix.

Finally, the state space model of the system under consideration takes a form as;

Bu Ax x += and Cx y =

The control inputs are linear combinations of system states given by; Kxu −=

Determination of the Feedback Gain Matrix (K):

The design of an optimal controller is to determine the feedback matrix ‘K’ in

such a way that a certain Performance Index (PI) is minimized while transferring the

system from an initial arbitrary state0)0( ≠ x

to origin in infinite time i.e.,0)( =∞ x

.Generally the PI is chosen in quadratic form as:

( )∞

+=

02

1dt u Ru xQ xPI

T T



39

where, ‘Q’ is a real, symmetric and positive semi-definite matrix called as ‘state

weighting matrix’ and ‘R’ is a real, symmetric and positive definite matrix called as

‘control weighting matrix’.

The matrices Q and R are determined on the basis of following system requirements.

1) The excursions (deviations) of ACEs about steady values are minimized. In this

model, these excursions are;

711)2,1(111 x x BP f B ACE tie +=+∆= and742)2,1(222 x x BP f B ACE tie −=−∆=

2) The excursions of dt ACE about steady values are minimized. In this model,

these excursions are 8 x & 9

x .

3) The excursions of control inputs 21uand u about steady values are minimized.

Under these considerations, the PI takes a form;

( ) ( ) ( ) ( ) ( ) ( )[ ]∞

++++−++=

0

2

2

2

1

2

9

2

8

2

742

2

7112

1dt uu x x x x B x x BPI

i.e., [ ]∞

++++−+++=

0

2

2

2

1

2

9

2

8742

2

4

2

2

2

7711

2

1

2

1 2222

1dt uu x x x x B x B x x x B x BPI

This gives the matrices Q (9×9) and R (2×2) as:

−

−

=

100000000

010000000

0020000

000000000

000000000

0000000

000000000

000000000

0000000

21

2

2

2

1

2

1

B B

B B

B B

Q

=

10

01 R

The matrices A, B, Q & R are known.

The optimal control is given by Kxu −=

‘K’ is the feedback gain matrix given by;

S B RK T 1−

=



40

where, ‘S’ is a real, symmetric and positive definite matrix which is the unique

solution of matrix Riccati Equation:

01

=+−+− QS BSBRSAS A

T T

The closed loop system equation is; x A x BK AKx B Ax x C =−=−+= )()(

The matrix )( BK A AC

−= is the closed loop system matrix. The stability of closed

loop system can be tested by finding eigenvalues of C A .

SYSTEM ANALYSIS USING MATLAB

After substituting values of parameters as given in Appendix - I, the state equationsand the matrices A, B, Q & R are:

[ ]17211605.0 d x x x x −−+−=

3225.25.2 x x x +−=

13135.125.122083.5 u x x x +−−=

[ ]27544605.0 d x x x x −++−=

655 5.25.2 x x x +−=

26465.125.122083.5 u x x x +−−=

41744422.044422.0 x x x −=

718 425.0 x x x +=

749425.0 x x x −=



41

=

−

−

−−

−

−

−−

−

−−

=

00

00

00

5.12000

00

05.12

00

00

;

00100425.0000

00100000425.0

000004442.0004442.0

0005.1202083.50000005.25.20000

0060605.0000

0000005.1202083.5

0000005.25.20

0060000605.0

B A

=

−

−

=1001;

100000000

010000000

00200425.000425.0

000000000

000000000

00425.000180625.0000

000000000

000000000

00425.000000180625.0

RQ

MATLAB program to obtain S, K & Ac:

(‘MATLAB 6’ software has been used to design the optimal controllers).

A = [-0.05 6 0 0 0 0 -6 0 0; 0 -2.5 2.5 0 0 0 0 0 0; -5.20833 0 -12.5 0 0 0 0 0 0; 0 0 0 -0.05 6 0

6 0 0; 0 0 0 0 -2.5 2.5 0 0 0; 0 0 0 -5.20833 0 -12.5 0 0 0; 0.44422 0 0 -0.44422 0 0 0 0 0;

0.425 0 0 0 0 0 1 0 0; 0 0 0 0.425 0 0 -1 0 0]

B = [0 0; 0 0; 12.5 0; 0 0; 0 0; 0 12.5; 0 0; 0 0; 0 0]

Q = [0.180625 0 0 0 0 0 0.425 0 0; 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0; 0 0 0 0.180625 0 0 -

0.425 0 0; 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0; 0.425 0 0 -0.425 0 0 2 0 0; 0 0 0 0 0 0 0 1 0; 0 0

0 0 0 0 0 0 1]

R = [1 0; 0 1]

S = care(A,B,Q,R)

K = inv(R)*B'*S

Ac = A-B*K



42

eig(S)

eig(A)

eig(Ac)

Output of the MATLAB program:

−−

−−

−−−−

−−−

−−−−

−−−−−

−−−−

−−−−−

−−−

=

854.10088.01527.008.04615.03305.00008.03305.0

0088.0854.11527.00008.00226.008.04615.01025.0

1527.01527.06086.00219.00673.01025.00219.00673.0425.0

08.000219.00123.00664.00338.00016.00092.0005.0

4615.0008.00673.00664.03689.02122.00092.00571.00398.0

3305.00226.01025.00338.02122.01797.0005.00398.00462.0

008.00219.00016.00092.0005.00123.00664.00338.0

008.04615.00673.00092.00571.00398.00664.03869.02122.0

0226.03305.01025.0005.00398.00.0462-0.03380.21220.1797

S

Matrix ‘S’ is found to be real, positive definite & symmetric. Its all eigenvalues are

real and positive: 0 ; 0 ; 0.0045 ; 0.0287 ; 0.2385 ; 0.3482 ; 0.656 ; 2.0516 ; 2.111

−−−

−−−−=

102737.01538.08294.04226.002.01156.0063.0

012737.002.01156.0063.01538.08294.04226.0K

Hence the control inputs:

872737.0

602.0

51156.0

4063.0

31538.0

28294.0

14226.0

1x x x x x x x xu −++++−−−=

972737.0

61538.0

58294.0

44226.0

302.0

21156.0

1063.0

2x x x x x x x xu −−−−−++=

The closed loop system matrix ‘AC’ is:

−

−

−−−−−

−

−

−−−−

−

−

=

00100425.0000

00100000425.0

000004442.0004442.0

5.1204208.3423.143673.104908.102504.04444.17871.0

0005.25.20000

0060605.0000

05.124208.32504.04444.17871.0423.143673.104908.10

0000005.25.20

006000060.05-

C A



43

The eigenvalues of open loop system matrix (state matrix) ‘A’ are:

0; 0; -13.068; -13.052; -0.38+3.189i; -0.38-3.189i; -0.991+2.262i; -0.991-2.262i; -1.2376

Two eigenvalues are zero and remaining have negative real parts indicating that, the

system is marginally stable before applying the optimal control strategy.The eigenvalues of closed loop system matrix ‘AC’ are:

-13.0594; -13.0758; -1.034+3.4078i; -1.034-3.4078i; -1.4791+2.5810i; -1.4791-2.581i;

-1.3521; -0.7439; -0.6887

All eigenvalues of ‘AC’ have negative real parts indicating that the system is

asymptotically stable after applying optimal control strategy.

33..33..22 SSTTAATTEE SSPPAACCEE MMOODDEELL OOFF TTWWOO AARREEAA TTHHEERRMMAALL--TTHHEERRMMAALL

((RREEHHEEAATT)) PPOOWWEERR SSYYSSTTEEMM

The state space model of two area thermal–thermal (reheat) power system,

with full state feedback (11 state feedback) has been developed as shown in Fig. 3.6.

Kp1----------1+sTp1

1----S

Kp2----------

1+sTp2

-1-1

∆PD1

∆PD2

∆Ptie(1,2)

−

−

+

−

+

−

+

−

ACE1

1-----R1

−

∆ f 1

∆ f 2

B1

+

ACE2

1-----R2

2πTo

-------

s

B2

+

+

+

+

+

−

AREA 1(THERMAL REHEAT)

TIE LINE

∆Pt2

x1

x9

x10

x11 x5

u1

u2

d1

d2

1----

S

12

5

9

10

11

Σ Σ Σ

Σ

ΣΣΣ

1-----------1+sTg1

1----------1+sTt1

∆Pg1∆Pt1

x3 x4

4 31+sKr1Tr1---------------1 + sTr1 x2

∆Pr1

AREA 2(THERMAL REHEAT)

61

-----------

1+sTg2

1----------

1+sTt2

∆Pg2

x7 x8

8 71+sKr2Tr2---------------

1 + sTr2 x6

∆Pr2

Fig. 3.6: State space model of two area thermal-thermal (reheat) power system



44

State Variables:

11f x ∆= 12

Pt x ∆= 13 Pr∆= x 14 Pg x ∆= 25 f x ∆= 26 Pt x ∆=

27 Pr∆= x 28 Pg x ∆= )2,1(9 tieP x ∆= = dt ACE x 110 = dt ACE x 211

Control inputs: 21uand u

Disturbance inputs:11 DPd ∆= and

22 DPd ∆=

State equations:

For block 1:

)( 1921111 d x xK xT x PP −−=+

i.e., 1

1

19

1

12

1

11

1

1

1d

T

K x

T

K x

T

K x

T x

P

P

P

P

P

P

P

−−+−=

For block 2:

3212 x xTt x =+

i.e., 3

1

2

1

2

11 x

Tt x

Tt x +−=

For block 3:

4114313 xTr Kr x xTr x +=+

i.e.,

+−

−++−= 1

1

4

1

1

11

14

1

3

1

3

11111u

Tg x

Tg x

Tg RKr x

Tr x

Tr x

i.e., 1

1

14

1

1

1

3

1

1

11

13

11u

Tg

Kr x

Tg

Kr

Tr x

Tr x

Tg R

Kr x +

−+−−=

For block 4:

11

1

414

1u x

R xTg x +

−=+

i.e., 1

1

4

1

1

11

4

111u

Tg x

Tg x

Tg R x +−

−=

For block 5:

)( 2962525 d x xK xT x PP −+=+

i.e., 22

2

92

2

62

2

52

5

1

d T

K

xT

K

xT

K

xT xP

P

P

P

P

P

P−++−=

For block 6:

7626 x xTt x =+

i.e., 7

2

6

2

6

11 x

Tt x

Tt x +−=



45

For block 7:

8228727 xTr Kr x xTr x +=+

i.e.,

+−−++−= 2

2

8

2

5

22

28

2

7

2

7

11111u

Tg x

Tg x

Tg RKr x

Tr x

Tr x

i.e., 211

2

28

2

2

2

7

2

5

22

27 u

Tg

Kr x

Tg

Kr

Tr x

Tr x

Tg R

Kr x +

−+−−=

For block 8:

25

2

828

1u x

R xTg x +−=+

i.e., 2

2

8

2

5

22

8

111u

Tg x

Tg x

Tg R x +−−=

For block 9:

5

0

1

0

9

22 xT xT x π π −=

For block 10:

91110 x x B x +=

For block 11:

95211 x x B x −=

The matrices A (11×11) and B (11×2) are:

=

−

−

−−

−−−

−

−

−−

−

−−

−

−−

=

00

00

002

10

2

20

00

00

0

1

1

0

1

1

00

00

;

0010002

0000

00100000001

0000000

20000

2

000

2

100

22

10000

000

2

2

2

1

2

10

22

20000

0000

2

1

2

100000

00

2

200

2

2

2

10000

0000000

1

100

11

1

0000000

1

1

1

1

1

10

11

1

00000000

1

1

1

10

00

1

1000000

1

1

1

1

Tg

Tg

Kr

Tg

Tg

r K

B

B

B

T T

TgTg R

Tg

Kr

Tr Tr Tg R

Kr

Tt Tt

PT

PK

PT

PK

PT

TgTg R

Tg

Kr

Tr Tr Tg R

Kr

Tt Tt

P

T

PK

P

T

PK

P

T

A

π π



46

State Vector (x) = [ ]T x x x x x x x x x x x 1110987654321

Control Vector (u) =

2

1

u

u


The control inputs are:

11111101012121111 .......... xk xk xk xk u−−

++++=

11112101022221212 .......... xk xk xk xk u−−

++++=

where,

=

−−

−−

112102292827262524232221

111101191817161514131211

k k k k k k k k k k k

k k k k k k k k k k k K

Hence, Kxu −=

The system state equation is: Bu Ax x +=

The output equation is: Cx y =


( )∞

+=

02

1dt RuuQx xPI

T T

( ) ( ) ( ) ( ) ( ) ( )[ ]∞

++++−++=

0

2

2

2

1

2

11

2

10

2

952

2

9112


[ ]∞

++++−+++=

0

2

2

2

1

2

11

2

10952

2

5

2

2

2

9911

2

1

2

1 2222


This gives the symmetric matrices Q (11×11) and R (2×2) as:



47

−

−

=

10000000000

01000000000

002000000

00000000000

00000000000

00000000000

000000000

00000000000

00000000000

00000000000

000000000

21

2

2

2

1

2

1

B B

B B

B B

Q

= 10

01 R


After substituting values of parameters as given in Appendix - I, the state equations

and the matrices A, B, Q & R are:

[ ]19211 605.0 d x x x x −−+−=

3225.25.2 x x x +−=

143131625.40625.41.073437.1 u x x x x +−−−=

14145.125.122083.5 u x x x +−−=

[ ]29655

605.0 d x x x x −++−=

7665.25.2 x x x +−=

28757 1625.40625.41.073437.1 u x x x x +−−−=

2858 5.125.122083.5 u x x x +−−=

51944422.044422.0 x x x −=

9110 425.0 x x x +=

9511425.0 x x x −=



48

=

−

−

−−

−−−

−

−

−−

−−−

−

−−

=

00

00

00

5.120

1625.40

00

00

05.12

01625.4

00

00

;

001000425.00000

0010000000425.0

00000044422.000044422.0

0005.12002083.50000

0000625.41.0073437.10000

00005.25.200000

00600605.00000

00000005.12002083.5

00000000625.41.0073437.1

000000005.25.20

006000000605.0

B A

=

−

−

=10

01

10000000000

01000000000

002000425.0000425.0

00000000000

00000000000

00000000000

00425.0000180625.0000000000000000

00000000000

00000000000

00425.00000000180625.0

RQ


A = [-0.05 6 0 0 0 0 0 0 -6 0 0; 0 -2.5 2.5 0 0 0 0 0 0 0 0; -1.734375 0 -0.1 -4.0625 0 0 0 0 0 0

0; -5.20833 0 0 -12.5 0 0 0 0 0 0 0; 0 0 0 0 -0.05 6 0 0 6 0 0; 0 0 0 0 0 -2.5 2.5 0 0 0 0; 0 0 0 0

-1.734375 0 -0.1 -4.0625 0 0 0; 0 0 0 0 -5.20833 0 0 -12.5 0 0 0; 0.44422 0 0 0 -0.44422 0 0 0

0 0 0; 0.425 0 0 0 0 0 0 0 1 0 0; 0 0 0 0 0.425 0 0 0 -1 0 0]

B = [0 0; 0 0; 4.1625 0; 12.5 0; 0 0; 0 0; 0 4.1625; 0 12.5; 0 0; 0 0; 0 0]

Q = [0.180625 0 0 0 0 0 0 0 0.425 0 0; 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0

0 0 0 0 0; 0 0 0 0 0.180625 0 0 0 -0.425 0 0; 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0; 0 0 0

0 0 0 0 0 0 0 0; 0.425 0 0 0 -0.425 0 0 0 2 0 0; 0 0 0 0 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 0 0 0 1]

R = [1 0; 0 1]

S = care(A,B,Q,R)



49

K = inv(R)*B'*S

Ac = A-B*K

eig(A)

eig(S)eig(Ac)


−−−

−−−−

−−−−−

−−−−−−−−

−−−−−−−

−−−−

−−−−−−−

−−−−

−−−−−

−−−−−

=

5523.22827.03311.08619.08284.20729.16283.01766.05304.00436.00186.0

2827.05523.23311.01766.05304.00436.00186.08619.08284.20729.16283.0

3311.03311.07592.22909.00713.18116.00283.02909.00713.18116.00283.0

8619.01766.02909.04697.15859.44893.01815.00457.015.00581.00366.0

8284.25304.00713.15859.4348.147307.16626.015.04894.01752.01163.00729.10436.08116.04893.07307.12523.16208.00581.01752.00238.00095.0

6283.00186.00283.01815.06626.06208.04122.00366.01163.00095.00621.0

1766.08619.02909.00457.015.00581.00366.04697.15859.44893.01815.0

5304.08284.20713.115.04894.01752.01163.05859.4348.147307.16626.0

0436.00729.18116.00581.01752.00238.00095.04893.07307.12533.16208.0

0186.06283.00283.00366.01163.00095.00621.01815.06626.06208.04122.0

S

The matrix ‘S’ has been real, positive definite & symmetric. Its eigenvalues are:

0; 0; 0; 0.0293; 0.5654; 0.6722; 2.1545; 2.5134; 3.117; 16.6845; 17.094

All the eigenvalues of matrix ‘S’ are real and positive.The feedback gain matrix ‘K’ has been obtained as:

−−

−−−=

10823.07172.03995.20873.14896.0053.01621.00026.0026.0

01823.0053.01621.00026.0026.07172.03995.20873.14896.0K

Hence the control inputs are:

109876543211 823.0053.01621.00026.0026.07172.03995.20873.14896.0 x x x x x x x x x xu−++−−−+−−−=

119876543212823.07172.03995.20873.14896.0053.01621.00026.0026.0 x x x x x x x x x xu −−+−−−+−−−=



50

The closed loop control matrix ‘AC’ has been obtained as:

−

−

−−−−−−−−−

−−−−−−−−−

−

−

−−−−−−−−

−−−−−−−−

−

−−

=

001000425.00000

0010000000425.0

0000004442.00004442.0

5.1202877.105355.39935.295918.133278.11663.00269.20323.03253.0

1625.404258.30773.10878.105261.47721.32208.06749.00108.01083.0

00005.25.200000

00600605.00000

05.122877.1663.00269.20323.03253.05355.39935.295918.133278.11

01625.44258.32208.06749.00108.01083.00773.10878.105261.47721.3

000000005.25.20

006000000605.0

C A

The eigenvalues of open loop system matrix ‘A’ are:

0; 0; -12.6985; -12.6923; -0.1716+2.5868i; -0.1716-2.5868i; -2.0178; -1.0223+0.7052i;-1.0223-0.7052i; -0.0968; -0.4068

Two eigenvalues are zero and remaining have negative real parts indicating that, the

system is marginally stable before applying the optimal control strategy.

The eigenvalues of closed loop system matrix ‘AC’ are:

-12.6995; -12.6933; -0.4538+2.6334i; -0.4538-2.6334i; -2.0199; -1.1766+1.0781i;

-1.1766-1.0781i; -0.839; -0.2703+0.1338i; -0.2703-0.1338i; -0.2937

The real parts of all the eigenvalues of ‘AC’ are negative indicating that, the system is

asymptotically stable after applying the optimal control strategy.

33..33..33 SSTTAATTEE SSPPAACCEE MMOODDEELL OOFF TTWWOO AARREEAA TTHHEERRMMAALL--HHYYDDRROO PPOOWWEERR

SSYYSSTTEEMM

State space model of two area thermal–hydro power system with full state feedback

(10 state feedback) has been developed as shown in Fig. 3.7.



51

1

------------1 + sTg1

1

------------1 + sTt1

Kp1

------------1 + sTp1

1

-----S

1----------1 + sT1

1 - sTw-------------1+0.5sTw

Kp2------------1 + sTp2

-1

1-----S

-1

∆Pg1 ∆Pt1

∆Ptw

∆PD1

∆PD2

∆Ptie(1,2)

−

−

+

−

+

−

+

−

ACE1

1-----R1

−∆ f 1

∆ f 2

B1

1-----S

+

ACE2

1-----R2

2πT0

B2

+

+

+

+

+

−

1

AREA 2(HYDRO)

AREA 1(THERMAL)

TIE LINE

Σ Σ Σ

Σ

Σ

ΣΣ1+sT2-----------1 + sT3

∆PG2∆PG1

x1

23

7 6 5 4

8

9

10

x2 x3 x9

x10 x7 x6 x5

x4

x8

u1

u2

d1

d2

Fig. 3.7: State space model of two area thermal-hydro power system

State Variables:

11f x ∆= 12

Pt x ∆= 13 Pg x ∆= 24 f x ∆= Ptw x ∆=5 26 GP x ∆=

17 GP x ∆= )2,1(8 tieP x ∆=

= dt ACE x 19

= dt ACE x 210

Control inputs:21

uand u

Disturbance inputs: 11 DPd ∆= and 22 DPd ∆=

State equations:

For block 1:

)( 1821111 d x xK xT x PP −−=+

i.e., 1

1

18

1

12

1

11

1

1

1d

T

K x

T

K x

T

K x

T

xP

P

P

P

P

P

P

−−+−=

For block 2:

3212 x xTt x =+

i.e., 3

1

2

1

2

11 x

Tt x

Tt x +−=



52

For block 3:

11

1

313

1u x

R xTg x +

−=+

i.e., 11

31

111

3

111

uTg xTg xTg R x+−

−=

For block 4:

)( 2852424 d x xK xT x PP −+=+

i.e., 2

2

28

2

25

2

24

2

4

1d

T

K x

T

K x

T

K x

T x

P

P

P

P

P

P

P

−++−=

For block 5:

6655 5.0 xTw x xTw x −=+

∴ 6655 25.0

1

5.0

1 x x

Tw x

Tw x −+−=

∴

+

−+−

−−+

−= 2

31

27

31

2

3

6

3

4

312

2655

112

22u

T T

T x

T T

T

T x

T x

T T R

T x

Tw x

Tw x

i.e., 2

31

27

331

26

3

54

312

25

2222222u

T T

T x

T T T

T x

T Tw x

Tw x

T T R

T x −

−+

++−=

For block 6:

727636 xT x xT x +=+

∴ 7

3

27

3

6

3

6

11 x

T

T x

T x

T x ++−=

+−−++−= 2

1

7

1

4

123

27

3

6

3

6

11111u

T x

T x

T RT

T x

T x

T x

i.e.,2

31

27

31

2

3

16

3

14

312

26

uT T

T x

T T

T

T x

T x

T T R

T x +

−+−−=

For block 7:

24

2

717

1

u x R xT x +−=+

i.e., 2

1

7

1

4

12

7

111u

T x

T x

T R x +−−=

For block 8:

40

10

8 22 xT xT x π π −=



53

For block 9:

8119 x x B x +=

For block 10:

84210 x x B x −=

The matrices A(10×10) and B(10×2) are:

−

=

−

−

−−

−

−−

−

+

−

−

−−

−

−−

=

00

00

001

10

31

20

31

22

0

00

0

1

100

00

0010002

000

0010000001

0000000

2000

2

000

1

100

12

1000

000

31

2

3

1

3

10

312

2000

000

3

2

31

22

3

222

312

22

000

00

2

200

2

2

2

1000

0000000

1

10

11

1

0000000

1

1

1

10

00

1

100000

1

1

1

1

T

T T

T

T T

T

Tg

B

B

B

T T

T T R

T T

T

T T T T R

T T T T

T

T TwTwT T R

T

PT

PK

PT

PK

PT

TgTg R

Tt Tt

PT

PK

PT

PK

PT

A

π π

State Vector (x) = [ ]T x x x x x x x x x x 10987654321


2

1

u

u



101012121111 .......... xk xk xk u−

+++=

101022221212 .......... xk xk xk u−

+++=

where,

=

−

−

102292827262524232221

101191817161514131211

k k k k k k k k k k k k k k k k k k k k K

hence, Kxu −=

The system state equation is: Bu Ax x +=

The output equation is: Cx y =



54

To determine the Feedback Gain Matrix (K):

( )∞

+=

02

1dt RuuQx xPI T T

( ) ( ) ( ) ( ) ( ) ( )

[ ]

∞

++++−++=

0

2

2

2

1

2

10

2

9

2

842

2

8112


[ ]∞

++++−+++=

0

2

2

2

1210

29842

24

2

228811

2

1

2

1 2222



−

−

=

1000000000

0100000000

00200000

0000000000

0000000000

0000000000

00000000

0000000000

0000000000

00000000

21

2

2

2

1

2

1

B B

B B

B B

Q

=

10

01 R




[ ]18211605.0 d x x x x −−+−=

322 5.25.2 x x x +−=

1313 5.125.122083.5 u x x x +−−=

[ ]28544605.0 d x x x x −++−=

276545 002106.01979.02.220008778.0 u x x x x x −−+−=

2001053.0

709894.0

61.0

4000439.0

6u x x x x ++−−=

274702053.002053.0008555.0 u x x x +−−=



55

41844422.044422.0 x x x −=

819425.0 x x x +=

8410425.0 x x x −=

−=

−

−

−−

−−

−−

−

−−

−

−−

=

00

00

00

02053.00

001053.00

002106.00

00

05.12

00

00

001000425.0000

001000000425.0

00000044422.00044422.0

00002053.000008555.0000

00009894.01.00000439.0000

0001979.02.220008778.0000

00600605.0000

00000005.1202083.5

00000005.25.20

00600000605.0

B A

=

−

−

=10

01

10000000000100000000

002000425.000425.0

0000000000

0000000000

0000000000

00425.0000180625.0000

0000000000

0000000000

00425.0000000180625.0

RQ


A = [-0.05 6 0 0 0 0 0 -6 0 0; 0 -2.5 2.5 0 0 0 0 0 0 0; -5.2083 0 -12.5 0 0 0 0 0 0 0; 0 0 0 -0.05 6 0 0 6

0 0; 0 0 0 0.0008778 -2 2.2 -0.1979 0 0 0; 0 0 0 -0.000439 0 -0.1 0.09894 0 0 0; 0 0 0 -0.008555 0 0 -

0.02053 0 0 0; 0.44422 0 0 -0.44422 0 0 0 0 0 0; 0.425 0 0 0 0 0 0 1 0 0; 0 0 0 0.425 0 0 0 -1 0 0]

B = [0 0; 0 0; 12.5 0; 0 0; 0 -0.002106; 0 0.001053; 0 0.02053; 0 0; 0 0; 0 0]

Q = [0.180625 0 0 0 0 0 0 0.425 0 0; 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0; 0 0 0 0.180625 0 0 0 -

0.425 0 0; 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0; 0.425 0 0 -0.425 0 0 0 2 0 0; 0 0

0 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 0 0 1]

R = [1 0; 0 1]



56

S = care(A,B,Q,R)

K = inv(R)*B'*S

Ac = A-B*K

eig(A)

eig(S)

eig(Ac)


−

−−−−−−

−

−−−

−

−

−

−

−

=

8.94.67.77.34908.81.303.02.0

4.67.74.57.2676621.04.03.0

7.74.59.81.243.757.87.21.06.06.0

7.347.261.249.3963.5207.298.903.02.0

90763.753.5205.11524.847.272.09.06.0

8.867.87.294.845.93.31.05.04.0

1.327.28.97.273.32.101.01.0

01.01.002.01.0001.01.0

3.04.06.03.09.05.01.01.05.03.0

2.03.06.02.06.04.01.01.03.03.0

S


0; 0; 0; 0.3; 0.5; 1.5; 2.5; 7.3; 135.9; 1439.2

All the eigenvalues of matrix ‘S’ are real and positive.

The feedback gain matrix ‘K’ has been obtained as:

−−−−

−=

7877.0616.05555.06338.87167.116794.0224.00009.00059.00052.0

616.07877.09674.05813.09204.19205.01842.02095.01572.16597.0K


109876543211616.07877.09674.05813.09204.19205.01842.02095.01572.16597.0 x x x x x x x x x xu −−−+−−−−−−=

19876543212 7877.0616.05555.06338.87167.116794.0224.00009.00059.00052.0 x x x x x x x x x xu −+−−−−−++=



57

The closed loop system matrix ‘AC’ has been obtained as:

=

−

−

−−−−−−

−−−−−

−−−

−

−−−−−−−−−

−

−−

101000425.0000

001000000425.0

0000004442.0004442.0

0162.00126.00114.01978.02405.00139.00132.000001.00001.0

0008.00006.00006.00898.01123.00007.00007.0000

0017.00013.00012.01797.02247.29986.10013.0000

00600065.0000

7003.78466.90926.122665.70053.245056.113025.21187.154648.144543.13

00000005.25.20

00600000605.0

C A


0; 0; -13.06; -0.4002+2.8873i; -0.4002-2.8873i; -0.6199+1.2494i; -0.6199-1.2494i;

-0.0601+0.0207i; -0.0601-0.0207i; -2.0001

Two eigenvalues are zero and the remaining have negative real parts indicating that,

the system is marginally stable before applying the optimal control strategy.


-13.0676; -1.0386+3.1050i; -1.0386-3.1050i; -0.9071+1.4148i; -0.9071-1.4148i;

-2.0001; -0.7583; -0.0778+0.1130i; -0.0778-0.1130i; -0.1545The real parts of all the eigenvalues of ‘AC’ are negative indicating that, after

applying the optimal control strategy, the system is asymptotically stable.

33..33..44 SSTTAATTEE SSPPAACCEE MMOODDEELL OOFF TTHHRREEEE AARREEAA TTHHEERRMMAALL--TTHHEERRMMAALL--

HHYYDDRROO PPOOWWEERR SSYYSSTTEEMM

The state space model of three area thermal–thermal-hydro power system with full

state feedback (16 state feedback) has been developed as shown in Fig. 3.8.



58

1------------1 + sTg1

1------------1 + sTt1

Kp1------------1 + sTp1

1-----S

1------------1 + sTg2

1------------1 + sTt2

Kp2------------1 + sTp2

∆Pg1 ∆Pt1

∆Pg2 ∆Pt2

∆PD1

∆PD2

∆Ptie(1)

−

−

+

−

+

−

+

ACE1

1-----R1

−

∆ f 1

∆ f 2

B1

+

ACE2

+

+

+

+






1 + sT2------------1 + sT3

1 - sTw---------------1 + 0.5 sTw

Kp3------------1 + sTp3

∆PG2 ∆Ptw−

+

−

∆ f3

+

ACE3+

+

Governor Water Turbine Power System

------------s

2πT12 ------------s

2πT13

------------s

2πT23

1-----R2

B2

∆PD31

-----

R3

B3

+

−

+ −

++

+ −

+

+

a12 = -1

a13 = -1

a23 = -1

+

+

−

∆Ptie(2)

∆Ptie(3)

x1 x2 x3

x4 x5 x6

x7 x8 x9

1------------1 + sT1

Governor

∆PG1

x10

x11

x12

x13

u1

x14

1-----S x15

u2

1-----S x16

u3

d2

d3

d1

AREA 3(HYDRO)

−

1214 3

456

78910

1112

13

15

16


Fig. 3.8: State space model of three area thermal-thermal-hydro power system

State Variables:

11 f x∆=

12 Pt x∆=

13 Pg x∆=

24 f x∆=

25 Pt x∆=

26 Pg x∆=

37 f x ∆= Ptw x ∆=8 29 GP x ∆= 110 GP x ∆=

)2,1(11 tieP x ∆= )3,1(12 tieP x ∆=

)3,2(13 tieP x ∆= = dt ACE x 114 = dt ACE x 215 = dt ACE x 316

Control inputs:321

, uand uu

Disturbance inputs:11 DPd ∆= , 22 DPd ∆= and 33 DPd ∆=

State equations:

For block 1:

)( 1121121111 d x x xK xT x PP −−−=+

i.e., 1

1

112

1

111

1

12

1

11

1

1

1d

T

K x

T

K x

T

K x

T

K x

T x

P

P

P

P

P

P

P

P

P

−−−+−=



59

For block 2:

3212 x xTt x =+

i.e., 3

1

2

1

2

11 x

Tt x

Tt x +−=

For block 3:

11

1

313

1u x

R xTg x +

−=+

i.e., 1

1

3

1

1

11

3

111u

Tg x

Tg x

Tg R x +−−=

For block 4:

[ ]2131152424d x x xKp xTp x −−+=+

i.e., 2

2

213

2

211

2

25

2

24

2

4

1d

Tp

Kp x

Tp

Kp x

Tp

Kp x

Tp

Kp x

Tp x −−++

−=

For block 5:

6525x xTt x =+

i.e.,6

2

5

2

5

11 x

Tt x

Tt x +−=

For block 6:

24

2

626

1u x

R xTg x +−=+

i.e., 2

2

6

2

4

22

6

111u

Tg x

Tg x

Tg R x +−−=

For block 7:

[ ]3131283737

d x x xKp xTp x −++=+

i.e., 3

3

313

3

312

3

38

3

37

3

7

1d

Tp

Kp x

Tp

Kp x

Tp

Kp x

Tp

Kp x

Tp x −+++−=

For block 8:

99885.0 xTw x xTw x −=+

i.e., 3

31

210

331

29

3

87

313

28

2222222u

T T

T x

T T T

T x

T Tw x

Tw x

T T R

T x −

−+

++−=



60

For block 9:

10210939 xT x xT x +=+

i.e., 3

31

210

31

2

3

9

3

7

313

29

11u

T T

T x

T T

T

T

x

T

x

T T R

T x +

−+−−=

For block 10:

37

3

10110

1u x

R xT x +−=+

i.e., 3

1

10

1

7

13

10

111u

T x

T x

T R x +−−=

For block 11:

41211211 22 xT xT xπ π −=

For block 12:

7131131222 xT xT x π π −=

For block 13:

7234231322 xT xT x π π −=

For block 14:

12111114 x x x B x ++=

For block 15:

13114215 x x x B x +−=

For block 16:

13127316x x x B x −−=



61

The matrices A (16×16) and B (16×3) are:

−−

−

−

−

−

−−

−−−

−

+−

−

−−

−

−−

−−

−

−−

=

0001100003

000000

0001010000002

000

0000110000000001

00000000023

20023

2000

00000000013

20000013

2

00000000000012

20012

2

000000

1

100

13

1000000

000000

31

2

3

1

3

10

313

2000000

000000

3

2

31

22

3

222

313

22

000000

000

3

3

3

3000

3

3

3

1000000

0000000000

2

10

22

1000

0000000000

2

1

2

10000

000

2

20

2

200000

2

2

2

1000

0000000000000

1

10

11

1

0000000000000

1

1

1

10

0000

1

1

1

100000000

1

1

1

1

B

B

B

T T

T T

T T

T T R

T T

T

T T T T R

T

T T T

T

T TwTwT T R

T

Tp

Kp

Tp

Kp

Tp

Kp

Tp

TgTg R

Tt Tt

Tp

Kp

Tp

Kp

Tp

Kp

Tp

TgTg R

Tt Tt

Tp

Kp

Tp

Kp

Tp

Kp

Tp

A

π π

π π

π π

−

=

000

000

000

000

000

0001

100

31

200

31

22

00

000

0

2

10

000

000

00

1

1000

000

T

T T

T

T T

T

Tg

Tg

B



62

State Vector (x) =

[ ]T x x x x x x x x x x x x x x x x 16151413121110987654321


3

2

1

uu

u



16161151512121111.......... xk xk xk xk u

−−++++=

16162151522221212.......... xk xk xk xk u

−−++++=

16163151532321313 .......... xk xk xk xk u−−

++++=

where,

=

−−−−−−−

−−−−−−−

−−−−−−−

13153143133123113103393837363534333231

12152142132122112102292827262524232221

11151141131121111101191817161514131211

K K K K K K K K K K K K K K K K



K

Hence, Kxu −=

The system state equation:

Bu Ax x +=

The output equation:

Cx y =


( )

∞

+=

021 dt RuuQx xPI T T

( ) ( ) ( )[ ]∞

++++++−−++−+++=

0

2

3

2

2

2

1

2

16

2

15

2

14

2

131273

2

131142

2

1211112

1dt uuu x x x x x x B x x x B x x x BPI

[ ]∞

++++−+++=

0

2

2

2

1

2

11

2

10952

2

5

2

2

2

9911

2

1

2

1 2222




63


=

−−

−

−−

−−

−

=

100

010

001

;

1000000000000000

0100000000000000

001000000000000000021100000000

00012100000000

00011200000000

0000000000000000

0000000000000000

0000000000000000

0000000000000

0000000000000000

0000000000000000

0000000000000

0000000000000000

0000000000000000

0000000000000

32

31

21

33

2

3

222

2

11

2

1

R

B B

B B

B B

B B B

B B B

B B B

Q




[ ]11211211

605.0 d x x x x x −−−+−=

322 5.25.2 x x x +−=

13135.125.122083.5 u x x x +−−=

[ ]21311544

605.0 d x x x x x −−++−=

6555.25.2 x x x +−=

26465.125.122083.5 u x x x +−−=

[ ]31312877 605.0 d x x x x x −+++−=

3109878 002106.019789.02.22000877.0 u x x x x x −−+−=

310979 001053.009894.01.0000438.0 u x x x x ++−−=

310710 02053.002053.0008555.0 u x x x +−−=

411144422.044422.0 x x x −=

7112 44422.044422.0 x x x −=



64

7413 44422.044422.0 x x x −=

1211114425.0 x x x x ++=

1311415425.0 x x x x +−=

1312716425.0 x x x x −−=

=

−−

−

−

−

−

−−

−−

−−

−

−−

−

−−

−−

−

−−−

00110000425.0000000

00101000000425.0000

00011000000000425.0

000000004442.0004442.0000

000000004442.0000004442.0

000000000004442.0004442.0

0000002053.000008555.0000000

0000009894.01.00000438.0000000

0000019789.02.22000877.0000000

0066000605.0000000

0000000005.1202083.5000

0000000005.25.20000

0060600000605.0000

0000000000005.1202083.5

0000000000005.25.20

0006600000000605.0

A

−=

000

000

000

000

000

000

020534.000

001053.000

002106.000

000

05.120

000

000

005.12

000

000

B



65

=

−−

−

−−

−−

−

=

100

010

001

;

10000000000000000100000000000000

0010000000000000

000211000425.000425.0000

000121000425.000000425.0

000112000000425.000425.0

0000000000000000

0000000000000000

0000000000000000

000425.0425.00000180625.0000000

0000000000000000

0000000000000000

000425.00425.0000000180625.0000

0000000000000000

0000000000000000

0000425.0425.0000000000180625.0

RQ


A = [-0.05 6 0 0 0 0 0 0 0 0 -6 -6 0 0 0 0; 0 -2.5 2.5 0 0 0 0 0 0 0 0 0 0 0 0 0; -5.2083 0 -12.5 0 0 0 0 0

0 0 0 0 0 0 0 0; 0 0 0 -0.05 6 0 0 0 0 0 6 0 -6 0 0 0; 0 0 0 0 -2.5 2.5 0 0 0 0 0 0 0 0 0 0; 0 0 0 -5.2083 0

-12.5 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 -0.05 6 0 0 0 6 6 0 0 0; 0 0 0 0 0 0 0.0008778 -2 2.2 -0.19789 0 0

0 0 0 0; 0 0 0 0 0 0 -0.00044 0 -0.1 0.0989 0 0 0 0 0 0; 0 0 0 0 0 0 -0.0085 0 0 -0.02053 0 0 0 0 0 0;

0.4442 0 0 -0.4442 0 0 0 0 0 0 0 0 0 0 0 0; 0.4442 0 0 0 0 0 -0.4442 0 0 0 0 0 0 0 0 0; 0 0 0 0.4442 0 0

-0.4442 0 0 0 0 0 0 0 0 0; 0.425 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0; 0 0 0 0.425 0 0 0 0 0 0 -1 0 1 0 0 0; 0 0 0

0 0 0 0.425 0 0 0 0 -1 -1 0 0 0]

B = [0 0 0; 0 0 0; 12.5 0 0; 0 0 0; 0 0 0; 0 12.5 0; 0 0 0; 0 0 -0.0021; 0 0 0.001; 0 0 0.0205; 0 0 0; 0 0 0;

0 0 0; 0 0 0; 0 0 0; 0 0 0]

Q = [0.1806 0 0 0 0 0 0 0 0 0 0.425 0.425 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0; 0 0 0 0.1806 0 0 0 0 0 0 -0.425 0 0.425 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0.1806 0 0 0 0 -0.425 -0.425 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0.425 0 0 -0.425 0 0 0 0 0 0 2 1 -1 0 0 0; 0.425 0

0 0 0 0 -0.425 0 0 0 1 2 1 0 0 0; 0 0 0 0.425 0 0 -0.425 0 0 0 -1 1 2 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]

R = [1 0 0; 0 1 0; 0 0 1]

S = care(A,B,Q,R)



66

K = inv(R)*B'*S

Ac = A-B*K

eig(S)

eig(A)eig(Ac)


=

−−

−−−−−−−−−−−

−−−−−−−−−−

−−−

−−

−−−−−−

−−−−−−−−

−−

−−

−−−−−−−−

−−−

−−−

−−−−

−−−−

−−−−

−−−

2554.124593.44593.41382.51382.503068.381133.946409.80003.30281.01707.01404.00281.01707.01404.0

4593.4908.30688.27076.18434.11358.03565.159299.401732.30514.10747.04486.03591.00053.0027.00112.0

4593.40688.2908.38434.17076.11358.03565.159299.401732.30514.10053.0027.00112.00747.04486.03591.0

1382.57076.18434.15.13095.6903.29247.132933.407813.43623.10039.01326.03639.00345.02453.02702.0

1382.58434.17076.13095.65.1903.29247.132933.407813.43623.10345.02453.02702.00039.01326.03639.0

01358.01358.0903.2903.29065.100000306.01127.00938.00306.01127.00938.03068.383565.153565.159247.139247.1303328.3734112.4623456.267647.80238.01472.01259.00238.01472.01259.0

1133.949299.409299.402933.402933.4004112.4624577.9562068.683254.220697.04038.02807.00697.04038.02807.0

6409.81732.31732.37813.47813.403456.262068.682592.74631.20229.01682.02081.00229.01682.02081.0

0003.30514.10514.13623.13623.107647.83254.224631.29933.00045.00256.00065.00045.00256.00065.0

0281.00747.00053.00039.00345.00306.00238.00697.00229.00045.00161.00889.00479.0001.00062.00057.0

1707.04486.0027.01326.02453.01127.01472.04038.01682.00256.00889.05081.0315.00062.00335.00292.0

1404.03591.00112.03639.02702.00938.01259.02807.02081.00065.00479.0315.02896.00057.00292.00142.0

0281.00053.00747.00345.00039.00306.00238.00697.00229.00045.0001.00062.00057.00161.00889.00479.0

1707.0027.04486.02453.01326.01472.01472.04038.01682.00256.00062.00335.00292.00889.05081.0315.0

1404.00112.03591.02702.03639.00938.01259.02807.02081.00065.00057.00292.00142.00.0479 0.3150.2896

S


0; 0; 0; 0; 0; 0; 0.4; 0.4; 0.6; 1.1; 1.7; 2.1; 2.7; 6.6; 122; 1230.8

All the eigenvalues of matrix ‘S’ are real and positive.

The feedback gain matrix ‘K’ has been obtained as:

=

−−−−−−−−

−−−

−−−−

8675.03517.03517.03183.03183.000975.83587.105975.01983.00005.0003.00027.00005.0003.00027.0

3517.09337.00663.00483.04309.03826.02976.08709.02863.00564.02017.01107.15992.00129.00777.00717.0

3517.00663.09337.04309.00483.03826.02976.08709.02863.00564.00129.00777.00717.02017.01107.15992.0

K


161514131211109

876543211

3517.00663.09337.04309.00483.03826.02976.08709.0

2863.00564.00129.00777.00717.02017.01107.15992.0

x x x x x x x x

x x x x x x x xu

−+−−−++−

−+−−−−−−=

161514131211109

876543212

3517.09337.00663.00483.04309.03826.02976.08709.0

2863.00564.02017.01107.15992.00129.00777.00717.0

x x x x x x x x

x x x x x x x xu

−−+−−−+−

−+−−−−−−=

1615141312109

876543213

8675.03517.03517.03183.03183.00975.83587.10

5975.01983.00005.0003.00027.00005.0003.00027.0

x x x x x x x

x x x x x x x xu

−++−−−−

−−+++++=



67

The closed loop system matrix ‘AC’ is:

=

−−

−

−

−

−

−−−−−−−

−−−−−−

−−−−

−

−−−−−−−−−−−−−

−

−−

−−−−−−−−−−−−

−

−−−

000110000425.0000000

000101000000425.0000

000011000000000425.0

0000000004442.0004442.0000

0000000004442.0000004442.0

0000000000004442.0004442.0

0178.00072.00072.00065.00065.001868.02127.00123.00126.000001.00001.000001.00001.0

0009.00004.00004.00003.00003.000904.01109.00006.00006.0000000

0018.00007.00007.00007.00007.001808.02218.29987.10013.0000000

00066000605.00000003968.46718.118282.06035.03861.57825.47194.38858.10579.37052.00214.158837.136981.121616.0971.08966.0

00000000005.25.20000

00060600000605.0000

3968.48282.06718.113861.56035.07825.47194.38858.10579.37052.01616.0971.08966.00214.158837.136981.12

00000000000005.25.20

00006600000000605.0

A


0; 0; 0; -13.0601; -13.0445; -0.2686+3.602i; -0.2686-3.602i; -0.2372+3.1718i;

-0.2372-3.1718i; -0.7827+1.6214i; -0.7827-1.6214i; -1.4683; -2.0; -0.0843; -0.0362; 0

Four eigenvalues are zero and the remaining have negative real parts indicating

that, the system is marginally stable before applying the optimal control strategy.


-13.0677; -13.0514; -0.9698+3.8136i; -0.9698-3.8136i; -0.8646+3.3335i;

-0.8646-3.3335i; -1.1245+1.8141i; -1.1245-1.8141i; -1.685; -2; -0.7362; -0.7338;

-0.0741+0.1067i; -0.0741-0.1067i; -0.1492; 0

It can be seen that after applying optimal control, the system is asymptotically

stable.

33..44 EEQQUUAATTIIOONNSS OOFF PPOOWWEERR SSYYSSTTEEMM MMOODDEELLSS IINN DDIISSCCRREETTEE FFOORRMM FFOORR

IINNTTEEGGRRAALL CCOONNTTRROOLL AANNDD OOPPTTIIMMAALL CCOONNTTRROOLL

State equations and control equations for power system models under consideration

have been obtained in sections 3.2.1 to 3.2.4 for integral control and in sections 3.3.1to 3.3.4 for optimal control. For development of ANN controllers with MATLAB

programming, the power system equations are required in discrete time form, which

have been obtained as given below. The sampling time is 0.01 second and ‘k’ denotes

the sample number (iteration number).



68

33..44..11 EEQQUUAATTIIOONNSS FFOORR TTWWOO AARREEAA TTHHEERRMMAALL--TTHHEERRMMAALL ((NNOONN RREEHHEEAATT))

PPOOWWEERR SSYYSSTTEEMM

Equations for Integral Control:

[ ])()()(06.0)(9995.0)1( 17211 k d k xk xk xk x −−+=+

)(025.0)(975.0)1(322k xk xk x +=+

)(125.0)(875.0)(05208.0)1( 1313 k uk xk xk x ++−=+

[ ])()()(06.0)(9995.0)1( 27544 k d k xk xk xk x −++=+

)(025.0)(975.0)1(655 k xk xk x +=+

)(125.0)(875.0)(05208.0)1(2646 k uk xk xk x ++−=+

)()(0044422.0)(0044422.0)1( 7417 k xk xk xk x +−=+

)()(002.0)(00085.0)1( 1711 k uk xk xk u +−−=+

)()(002.0)(00085.0)1( 2742 k uk xk xk u ++−=+

Equations for Optimal Control:

[ ])()()(06.0)(9995.0)1(17211 k d k xk xk xk x −−+=+

)(025.0)(975.0)1(322k xk xk x +=+

)(125.0)(875.0)(05208.0)1(1313

k uk xk xk x ++−=+

[ ])()()(06.0)(9995.0)1( 27544 k d k xk xk xk x −++=+

)(025.0)(975.0)1(655 k xk xk x +=+

)(125.0)(875.0)(05208.0)1(2646 k uk xk xk x ++−=+

)()(0044422.0)(0044422.0)1( 7417 k xk xk xk x +−=+

)()(01.0)(00425.0)1( 8718 k xk xk xk x ++=+

)()(01.0)(00425.0)1( 9749 k xk xk xk x +−=+

)(8

)(7

2737.0)(6

02.0

)(5

1156.0)(4

063.0)(3

1538.0)(2

8294.0)(1

4226.0)1(1

k xk xk x

k xk xk xk xk xk u

−++

++−−−=+

)(9

)(7

2737.0)(6

1538.0

)(5

8294.0)(4

4226.0)(3

02.0)(2

1156.0)(1

063.0)1(2

k xk xk x

k xk xk xk xk xk u

−−−

−−++=+



69

33..44..22 EEQQUUAATTIIOONNSS FFOORR TTWWOO AARREEAA TTHHEERRMMAALL--TTHHEERRMMAALL ((RREEHHEEAATT))



[ ])()()(06.0)(9995.0)1( 19211 k d k xk xk xk x −−+=+

)(025.0)(975.0)1( 322 k xk xk x +=+

)(041625.0)(040625.0)(999.0)(0173437.0)1(14313 k uk xk xk xk x +−+−=+

)(125.0)(875.0)(052083.0)1(1414k uk xk xk x ++−=+

[ ])()()(06.0)(9995.0)1( 29655 k d k xk xk xk x −++=+

)(025.0)(975.0)1(766k xk xk x +=+

)(041625.0)(040625.0)(999.0)(0173437.0)1( 28757 k uk xk xk xk x +−+−=+

)(125.0)(875.0)(052083.0)1(2858k uk xk xk x ++−=+

)()(0044422.0)(0044422.0)1( 9519 k xk xk xk x +−=+

)()(002.0)(00085.0)1( 1911 k uk xk xk u +−−=+

)()(002.0)(00085.0)1( 2952 k uk xk xk u ++−=+


[ ])()()(06.0)(9995.0)1( 19211 k d k xk xk xk x −−+=+

)(025.0)(975.0)1(322k xk xk x +=+

)(041625.0)(040625.0)(999.0)(0173437.0)1(14313k uk xk xk xk x +−+−=+

)(125.0)(875.0)(052083.0)1(1414k uk xk xk x ++−=+

[ ])()()(06.0)(9995.0)1(29655 k d k xk xk xk x −++=+

)(025.0)(975.0)1(766 k xk xk x +=+

)(041625.0)(040625.0)(999.0)(0173437.0)1( 28757 k uk xk xk xk x +−+−=+

)(125.0)(875.0)(052083.0)1( 2858 k uk xk xk x ++−=+

)()(0044422.0)(0044422.0)1(9519 k xk xk xk x +−=+

)()(01.0)(00425.0)1( 109110 k xk xk xk x ++=+



70

)()(01.0)(00425.0)1( 119511 k xk xk xk x +−=+

)()(823.0)(053.0)(1621.0)(0026.0

)(026.0)(7172.0)(3995.2)(0873.1)(4896.0)1(

109876

543211

k xk xk xk xk x

k xk xk xk xk xk u

−++−−

−+−−−=+

)()(823.0)(7172.0)(3995.2)(0873.1)(4896.0)(053.0)(1621.0)(0026.0)(026.0)1(

119876

543212

k xk xk xk xk xk xk xk xk xk xk u

−−+−−−+−−−=+

33..44..33 EEQQUUAATTIIOONNSS FFOORR TTWWOO AARREEAA TTHHEERRMMAALL--HHYYDDRROO PPOOWWEERR SSYYSSTTEEMM


[ ])()()(06.0)(9995.0)1(18211k d k xk xk xk x −−+=+

)(025.0)(975.0)1( 322 k xk xk x +=+

)(125.0)(875.0)(052083.0)1(1313 k uk xk xk x ++−=+

[ ])()()(06.0)(9995.0)1(28544 k d k xk xk xk x −++=+

)(00002106.0)(001979.0)(022.0)(98.0)(000008778.0)1( 276545 k uk xk xk xk xk x −−++=+

)(00001053.0)(0009894.0)(999.0)(00000439.0)1(27646 k uk xk xk xk x +++−=+

)(0002053.0)(9997947.0)(00008555.0)1( 2747 k uk xk xk x ++−=+

)()(0044422.0)(0044422.0)1( 8418 k xk xk xk x +−=+

)()(002.0)(00085.0)1(1811 k uk xk xk u +−−=+

)()(0002.0)(000085.0)1( 2842 k uk xk xk u ++−=+


[ ])()()(06.0)(9995.0)1( 18211 k d k xk xk xk x −−+=+

)(025.0)(975.0)1(322 k xk xk x +=+

)(125.0)(875.0)(052083.0)1( 1313 k uk xk xk x ++−=+

[ ])()()(06.0)(9995.0)1( 28544 k d k xk xk xk x −++=+

)(00002106.0)(001979.0)(022.0)(98.0)(000008778.0)1( 276545 k uk xk xk xk xk x −−++=+

)(00001053.0)(0009894.0)(999.0)(00000439.0)1(27646k uk xk xk xk x +++−=+

)(0002053.0)(9997947.0)(00008555.0)1( 2747 k uk xk xk x ++−=+



71

)()(0044422.0)(0044422.0)1( 8418 k xk xk xk x +−=+

)()(01.0)(00425.0)1(9819k xk xk xk x ++=+

)()(01.0)(00425.0)1( 108410 k xk xk xk x +−=+

)(616.0)(7877.0)(9674.0)(5813.0)(9204.1

)(9205.0)(1842.0)(2095.0)(1572.1)(6597.0)1(

109876

543211

k xk xk xk xk x

k xk xk xk xk xk u

−−−+−

−−−−−=+

)(7877.0)(616.0)(5555.0)(6338.8)(7167.11

)(6794.0)(224.0)(0009.0)(0059.0)(0052.0)1(

109876

543212

k xk xk xk xk x

k xk xk xk xk xk u

−+−−−

−−++=+

33..44..44 EEQQUUAATTIIOONNSS FFOORR TTHHRREEEE AARREEAA TTHHEERRMMAALL-- TTHHEERRMMAALL -- HHYYDDRROO



[ ])()()()(06.0)(9995.0)1(11211211k d k xk xk xk xk x −−−+=+

)(025.0)(975.0)1(322k xk xk x +=+

)(125.0)(875.0)(052083.0)1(1313k uk xk xk x ++−=+

[ ])()()()(06.0)(9995.0)1(21311544k d k xk xk xk xk x −−++=+

)(025.0)(975.0)1( 655 k xk xk x +=+

)(125.0)(875.0)(052083.0)1(2646k uk xk xk x ++−=+

[ ])()()()(06.0)(9995.0)1(31312877k d k xk xk xk xk x −+++=+

)(00002106.0)(0019789.0)(022.0)(98.0)(000008778.0)1( 3109878 k uk xk xk xk xk x −−++=+

)(00001053.0)(0009894.0)(999.0)(00000438.0)1(310979 k uk xk xk xk x +++−=+

)(0002053.0)(999794.0)(00008555.0)1( 310710 k uk xk xk x ++−=+

)()(0044422.0)(0044422.0)1(114111

k xk xk xk x +−=+

)()(0044422.0)(0044422.0)1( 127112 k xk xk xk x +−=+

)()(0044422.0)(0044422.0)1(137413

k xk xk xk x +−=+

[ ] )()()()(425.0002.0)1(1121111k uk xk xk xk u +++−=+

[ ] )()()()(425.0002.0)1( 2131142 k uk xk xk xk u ++−−=+



72

[ ] )()()()(425.00002.0)1( 3131273 k uk xk xk xk u +−−−=+


[ ])()()()(06.0)(9995.0)1(11211211k d k xk xk xk xk x −−−+=+

)(025.0)(975.0)1( 322 k xk xk x +=+

)(125.0)(875.0)(052083.0)1( 1313 k uk xk xk x ++−=+

[ ])()()()(06.0)(9995.0)1(21311544 k d k xk xk xk xk x −−++=+

)(025.0)(975.0)1( 655 k xk xk x +=+

)(125.0)(875.0)(052083.0)1(2646k uk xk xk x ++−=+

[ ])()()()(06.0)(9995.0)1( 31312877 k d k xk xk xk xk x −+++=+

)(00002106.0)(0019789.0)(022.0)(98.0)(000008778.0)1( 3109878 k uk xk xk xk xk x −−++=+

)(00001053.0)(0009894.0)(999.0)(00000438.0)1(310979k uk xk xk xk x +++−=+

)(0002053.0)(999794.0)(00008555.0)1( 310710 k uk xk xk x ++−=+

)()(0044422.0)(0044422.0)1(114111

k xk xk xk x +−=+

)()(0044422.0)(0044422.0)1(127112 k xk xk xk x +−=+

)()(0044422.0)(0044422.0)1(137413 k xk xk xk x +−=+

)()(01.0)(01.0)(00425.0)1(141211114

k xk xk xk xk x +++=+

)()(01.0)(01.0)(00425.0)1( 151311415 k xk xk xk xk x ++−=+

)()(01.0)(01.0)(00425.0)1(161312716 k xk xk xk xk x +−−=+

)(3517.0)(0663.0)(9337.0)(4309.0

)(0483.0)(3826.0)(2976.0)(8709.0)(2863.0)(0564.0

)(0129.0)(0777.0)(0717.0)(2017.0)(1107.1)(5992.0)1(

16151413

121110987

6543211

k xk xk xk x

k xk xk xk xk xk x

k xk xk xk xk xk xk u

−+−−

−++−−+

−−−−−−=+

)(3517.0)(9337.0)(0663.0)(0483.0

)(4309.0)(3826.0)(2976.0)(8709.0)(2863.0)(0564.0

)(2017.0)(1107.1)(5992.0)(0129.0)(0777.0)(0717.0)1(

16151413

121110987

6543212

k xk xk xk x

k xk xk xk xk xk x


−−+−

−−+−−+

−−−−−−=+

)(8675.0)(3517.0)(3517.0

)(3183.0)(3183.0)(0975.8)(3587.10)(5975.0)(1983.0

)(0005.0)(003.0)(0027.0)(0005.0)(003.0)(0027.0)1(

161514

131210987

6543213

k xk xk x

k xk xk xk xk xk x


−++

−−−−−−

+++++=+



33..55 CCOONNCCLLUUDDIINNGG RREEMMAARRKKSS

The models of interconnected power systems comprising of areas of different

characteristics have been developed with integral as well as optimal control

strategies. The optimal controllers have been designed and the control equations in

continuous time have been obtained for all the power system models under

consideration. All these models have been studied for system stability and it has

been ensured that they are asymptotically stable with parameters values as given in

Appendix – I, after applying optimal control strategy. Also, for all the power system

models under consideration, the discrete time equations for system states as well as

control inputs have been obtained for both integral control and optimal control

strategies. These equations have been used for development of ANN controllers,

which has been discussed in Chapter 5.

chaper 3_c b bangal

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