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EDC4 1.0 Introduction There are several methods for penalty factor calculation. We will review some of them in these notes. These notes correspond to Section 4.2 in your text. 2.0 Power flow Jacobian method This method is described in [1, pg. 424-425]. The method presented here is similar to the method presented in Section 4.2.4.2 of W&W. Consider a power system with total of n buses of which bus 1 is the swing bus, buses 1…m are the PV buses, and buses m+1…n are the PQ buses. Consider that losses must be equal to the difference between the total system generation and the total system demand:

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DGL PPP −= (8) Recall the definition for bus injections, which is

DiGii PPP −= (9) Now sum the injections over all buses to get:

DG

n

iDi

n

iGi

n

iDiGi

n

ii

PPPP

PPP

−=−=

−=

∑∑

∑∑

==

==

11

11)(

(10)

Therefore,

∑=

=n

iiL PP

1 (11) Now differentiate with respect to a particular bus angle θk (where k is any bus number except 1) to obtain:

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nkPPPPPP

k

n

k

m

k

m

kkk

L ,...,2,121 =∂∂

++∂∂

+∂∂

+∂∂

+∂∂

=∂∂ +

θθθθθθKK

(12) Assumption to the above: All voltages are fixed at 1.0 (this relieves us from accounting for the variation in power with angle through the voltage magnitude term). Now let’s assume that we have an expression for losses PL as a function of generation PG2, PG3,…,PGm, i.e.,

PL=PL(PG2, PG3,…,PGm) (13) Then we can use the chain rule of differentiation to express that

nkPP

PPPP

PPP

k

m

Gm

GL

kG

GL

k

L ,...,2,)()( 2

2

=∂∂

∂∂

++∂∂

∂∂

=∂∂

θθθK

(14) Subtracting eq. (12) from eq. (14), we obtain, for k=2,…,n:

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k

n

k

m

Gm

GL

k

m

G

GL

kk

k

m

Gm

GL

kG

GL

k

L

k

n

k

m

k

m

kkk

L

PPP

PPPP

PPPP

PP

PPPP

PPP

PPPPPP

θθ

θθθ

θθθ

θθθθθθ

∂∂

++∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂∂

++⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂∂

+∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

++∂∂

∂∂

−=∂∂

−

∂∂

++∂∂

+∂∂

+∂∂

+∂∂

=∂∂

+

+

K

K

K

KK

1

2

21

2

2

121

)(1)(10

_________________________________________

)()(

Now bring the first term to the left-hand-side, for k=2,…,n Writing the above

k

n

k

m

Gm

GL

k

m

G

GL

kk

PPP

PPPP

PPPP

θθ

θθθ

∂∂

++∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂∂

++⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂∂

=∂∂

−

+ K

K

1

2

21 )(1)(1

The above equation, when written for k=2,…,n, can be expressed in matrix form as

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⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

−=

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

−

∂∂

−

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

n

Gm

GL

G

GL

n

n

n

m

n

nm

P

P

PPP

PPP

PPP

PPP

θ

θ

θθθ

θθθ

1

2

12

2

222

2

1

1

)(1

)(1

M

M

M

KK

MKMKM

KK

(15)

The matrix on the left is just the transpose of the power flow Jacobian JPθ submatrix, and it is square. The right-hand-side of (15) may be found by simply differentiating the real power flow equation for bus 1 with respect to each angle. The result will be

( ) ( )[ ]iiiiii

BGVVP θθθθθ

−−−=∂∂

111111 cossin|||| (16)

The solution vector contains the penalty factors in the first m-1 terms. 3.0 Loss formula The method of loss formula results in an approximate expression given by

000 BPBPBPP GT

GTGL ++= (17)

where PG is the vector of generation

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⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

Gm

GTG

P

PP M

1

(18)

Develop of the coefficient matrices in (17) has been done in several ways. Your W&W text utilizes a method developed by Meyer [2] (see Appendix B of chapter 4). I have appended in the Appendices of these notes another method based on the work of Kron, which is partially articulated in the book by El-Harawry and Christenson. Note some important similarities in the methods: 1. Both are dependent on the following

assumptions: • Each bus can be clearly

distinguished as either a load bus or a generation bus.

• Reactive generation varies linearly with generation, i.e., Qgk=Qgo+fkPgk.

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2. Both end up with expressions for PL of the same form.

3. Both expressions for PL are dependent on the elements of the Zbus matrix.

But there is one major difference between the formulations in that Kron’s approach makes no assumption regarding conforming loads. However, the method of W&W (Meyers) does, i.e., in Meyer’s approach, all loads must increase or decrease uniformly. We assume that we have the so-called B-coefficients in what follows.

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9

10

11

12

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Appendix [1] A. Bergen and V. Vittal, “Power System Analysis,” [2] W. Meyer, “Efficient computer solution for Kron and Kron-Early Loss Formulas,” Proc of the 1973 PICA conference, IEEE 73 CHO 740-1, PWR, pp. 428-432.