Comparing two power flow optimization algorithms by estimating technico-economic equivalent states

COMPARING TWO POWER FLOW OPTIMIZATION ALGORITHMS

BY ESTIMATING TECHNICO-ECONOMIC EQUIVALENT STATES

Luu Huu Vinh Quang

Power System Department

University of Technology – VietNam National University – HCM City.

ABSTRACT

This paper mentions the modified bus impedence matrix model applying to power flow

optimization, and compares this model with the augmented rectangular system model with an eigen-

image matrix. The modified expressions to compute the bus PV voltages are shown and a new

p_power loss coefficient expression is proposed for the power flow optimization computation. To

compare the different models of power flow optimization we propose a new modified estimative

matrix for a method estimating the technico-economic equivalent states, and appreciating the optimum

conditions of p_power generation between electric generators in a power system using the largest

eigenvalue of an estimative matrix relating to the p_power loss coefficients and the system economic

incremental fuel cost.

model, the second is related to eigen-image

vector power flow model [1].

1. INTRODUCTION

Power flow is the most frequently

performed study for power system. The power

flow problem involves determining voltage and

line flow, in a large electrical network, for a

given load and generation schedule. Many

important contributions have been made on this

along the years. Any improvement to its method

on storage, convergence, and reliability is of

value.

The method introduced in [2] is used to

determine the size of the technico-economic

equivalent area, therefore, we can compare the

two methods by estimating the accuracy of the

realization of optimum condition in power

system operation.

2. MODIFIED ZBUS MODEL AND

P_POWER FLOW OPTIMIZATION.

The bus impedence matrix (zbus) power

flow model is one of the known models for

power flow calculation. In this paper we present

a modified zbus power flow algorithm and its

development applying in power flow

optimization. The combination of many agents

will cause not only one optimum operation point

but also a technico-economic equivalent area

where exist many technico-economic equivalent

operation points. These agents, such as

generator fuel cost functions, the bus load

powers, the network resistances, inductances

and capacitances, etc. are approximately

determined, and the optimization algorithm with

some simplified assumtion can result in small

errors. We will compare new loss coefficient

expressions that are developed in the two

p_power optimization algorithms, in which the

first is related to modified zbus power flow

Let’s examine an electrical power system

comprising m generators. In all practical cases,

the fuel cost of generator i can be represented as

quadratic function C_iof p_power generation P_ig

(1)

C_i= a_iP²+ b_iP_ig+ c_i; i=1,2,…,m.

ig

The economic p_power dispatching problem

is to minimize the overall generating cost C_i. In

case of applying the zbus model, it can develop

into the following equations [3]

(2)

P

P_2g

b₁

b₂

1g

a₁

1

B_o1

B_o2

B₁₁

B₂₁

B₁₂

...

B_1m

B_2m

1

a₂

B₂₂...

:

1

2

:

;

:

a_m

b_m

B_m1

B_m2

...

B_mm

1

B_om

P_mg

(5)

where P_igis generator p_power;

is system

2

N

incremental fuel cost;

coefficients [3]

B_ijare the loss

1)

(P_jT^(k

Q⁽_j^{k 1)}T_X^(k

)

R

j

1

V_o

V_j^(k

1)

(3)

V_i^(k)

;

Nb

R_k

2

B_ij

(Re[C_ki]Re[C_kj

]

Im[C_ki]Im[C_kj]) ;

N

_{k 1}(0.5(V_iV_j))²

1)

V_o

(P_jT^(k

Q⁽_j^{k 1)}T_R^(k

)

X

j

1

1)

V_j^(k

N

(4)

B_oi

2

B_ijP_Lj;

j 1

(6)

N

(P_jT^(k

Q⁽_j^{k 1)}T_R^(k

)

1)

X

j

1

where Nb is the branch quantity of network; N

is the bus number quantity of network; R_kis

the resistance of branch k linking two buses i

and j; V_i, V_jare the voltages at buses i, j ;

Re[C_ki], Re[C_kj], Im[C_ki], Im[C_kj] are the real

and imaginery parts of elements of current

distribution factor matrix C; P_Ljis the p_power

load at bus j.

(k)

i

arctan

N

1)

V_o

(P_jT^(k

Q⁽_j^{k 1)}T_X^(k

)

R

j

1

(k 1)

T_R^{(k 1)}r_ijcos

T_X^{(k 1)}r_ijsin

x_ijsin

x_ijcos

;

(7)

(8)

j

(k 1)

j

where k is the number of iteration; V_i, are the

i

We can carry out the power system steady-

state optimization as shown in figure 1

bus voltage modul and argument; V_ois the

modul of slack bus voltage; N is the independent

bus quantity of electric network; r_ij, x_ijare the

real and imaginary elements of zbus matrix; P_j,

Q^(k-1)_jare the active and reactive bus powers;

During iterative calculation, the bus

q_power values are specified at the PQ bus,

(Q^(k-1)_j=Q^(o)), and the new modified expressions

j

to adjust the reactive power value at the PV bus

j are proposed as follow

Q⁽_j^k)V sin ^{(k 1)}I_pV cos ^{(k 1)}I_q

(9)

;

ie

j

(k 1)

P_jcos

Q⁽_j^{k 1)}sin

V_j^{(k 1)}

(10)

j

I_p

;

(11)

(k 1)

P_jsin

Q⁽_j^{k 1)}cos

j

I_q

V_j^(k

1)

(k 1)

r_jjV^{(k 1)}sin

x_jjV^{(k 1)}cos

z²_jj

V_je

V_j^(k

j

1

;

1)

where V_jeis the expected voltage modul at the

bus j;

The bus q_power at the PV bus j must be

satisfied

The load flow calculation is to determine

voltage and line flow, in a large electrical

network, for a given load and generation

schedule. Using the zbus matrix model

[4],[5],[6],[8] we can obtain the expressions to

compute the bus voltage as follow

Q_jQ⁽_j^{k 1)}

Q⁽_j^{k 1)}Q_j;

(12)

if the inequality (12) is not satisfied for the bus

j, then the bus reactive power must be fixed at

the limit, and this bus can be traited as a PQ bus

in the next iteration.

In the case, to control the voltage at PV bus

j, the expected bus voltage at this bus can be

adjusted so that an inequality is satisfied

models for the power flow solution. Applying

this power flow model to p_power optimization

we can develop the following equations

(18)

P_1g

P_2g

b₁

b₂

a₁

1

2

B₁₁

2B₁₂

a₂

...

2B_1m

2B_2m

(k 1)

r_jjcos

(13)

j

V_jV_j^{(k 1)}

Q⁽_j^{k 1)}V_j;

2B₂₁

2

B₂₂...

:

(V_j^{(k 1) 2}

)

:

;

:

a_m

b_m

2B_m1

2B_m2

...

2

B_mm

In case of branch m (linking two buses i, j)

1

P_mg

which characterized branch’s transfo-ratio

different from one unit, the bus voltages

depends upon the bus currents so the injected

transfo-model currents can be added to the bus

current, and we can compute the injected

transfo-model currents as follow

where B_ijdenotes the elements of a loss

coefficient matrix which can be found from [7]

(19)

cos(

V V_jcos _icos

)

j

i

B_ij

i

j

(14)

^NC_ksJ_sC_kiJ

s 1

^NC_ksJ_sC_kjJ

Re[J_iT] (t_m1)(V_jcos G_mV_jsin B_m)

j

Nb

s 1

(t²_m1)(V cos G_mV sin B_m) ;

2

;

i

k 1

J

(15)

Im[J_iT] (t_m1)(V_jcos B_mV_jsin G_m)

j

where m is the generator bus quantity; N is the

load bus quantity; Nb is the branch quantity;

C_ks, C_ki, C_kjare the elements of current

distribution factor matrix; J_sis the load current

at bus s; J is the sum of generator currents;

(t²_m1)(V cos B_mV sin G_m) ;

i

(16)

Re[J_jT

]

(1 t_m)(V cos G_mV sin _iB_m);

i i i

(17)

V_i,V_jare the generator voltages;

,

i

are the

Im[J_jT

(1 t_m)(V cos B_mV sin G_m);

j

i

angles between current and voltage of

generators i, j; cos , cos _jare the power factors

of generators i, j.

i

where Re[J_iT], Im[J_iT], Re[J_jT], Im[J_jT] are the

real and imaginary parts of injected transfo-

model currents at the buses i and j; t_mis the per-

unit transfo-ratio ; G_m, B_mare the real and

imaginary parts of admittance of transformer

branch m;

The process of power system steady-state

optimization implemented by search-method,

as the modified zbus model, can be also refered

to flow-chart shown on figure 1. Using these

algorithms to solve load flow problem we obtain

the same result of bus voltages in overall power

system. The main difference is to compare the

influence on the optimization results of two

different forms of equations (2) and (18) with

the different expressions of loss coefficients

(3),(4),(19), involved in determining the

technico-economic equivalent states [2] by

calculating the R magnitude which determines

the proximity state level to the optimum point

3.

AUGMENTED

RECTANGULAR

SYSTEM MODEL WITH THE EIGEN-

IMAGE ADMITTANCE MATRIX.

A new mathematical model [1] is proposed

for solving power flow problem using

augmented rectangular system with the system

eigen-image admittance matrix, a method to

create a mathematical model with a rectangular

jacobian matrix consisting of its almost un-

changed elements and also to receive a kind of

diagonal jacobian matrix for the simplicity of

the process of calculation. The results of its

application prove the advantages of this new

proposed model by comparison with the another

m

(20)

R

P_i²;

i 1

where P_iis the p_power deviations from

optimum p_power magnitude.

Assume that the existing P_i(i=1,2,…,m)

satisfy to an inequality

B₁₁a₁

B₂₁

B₁₂

B₂₂a₂

...

B_1m

B_2m

...

1

2

E₁

;

...

B_m1

m

P_i²R²;

(21)

B_m2

... B_mma_m

m

i 1

where

(28)

m

then the system fuel cost increment is

2B_ijP_LjB_oi

j 1

C_maxHR²;

;

( P_i0);

(22)

i

P_i

where H is a number which must be found to

appreciate the sytem fuel cost increment. If a

Once the largest eingenvalue

matrix E₁is found, we can determine the

system fuel cost increment

of the

1max

maximum eigenvalue

of the estimative

max

matrix is computed, then it will be accepted [5]

as the number H in (22)

m

(29)

C

P²;

i

1max

i 1

(23)

H=

max

4. EXAMPLE OF NUMERICAL RESULTS

in this case we have

4.1 Example 1.

m

(24)

C

P²;

i

max

i 1

Let’s investigate a 220kV power system

consisting of 4 generators and of 7 loads. The

power system’s schema is shown in figure 2

the system fuel cost increment C is determined

as a quadratic function of the

p_power output deviations

estimative matrix E

generator

P_iwith an

(25)

B₁₁a₁

B₂₁

...

B₁₂

B₂₂a₂...

...

B_1m

B_2m

...

E

;

...

B_m1

B_m2

... B_mma_m

in general, the generator bus quantity n is less

than the bus total quantity in the whole power

system. The loss coefficient matrix B must be a

square matrix of (m×m) size as the form of (19).

Applying the (3) and (4), we can obtain the

modified expression of the system fuel cost

increment as follow

The linedata is given in table 1

m m

m

i

^ma_iP_i²;

(26)

C

B

P_jP_i

P_i²

i

ij

i 1j 1

i 1

in case of (26), to estimate the equivalent state,

we propose a new modified expression of

estimative matrix E₁to stand for the matrix E

The numerical results of initial state

parameters calculation are presented in table 2

(27)

The modal matrix of the estimative matrix is

The eigenvalues are

The generator fuel cost functions are given

as follow

And we have

.0084;

C₁=0.007P²_1g+3.9P_1g+155, $/h; (50≤P_1g≤275)MW;

C₂=0.006P²_2g+4.5P_2g+149, $/h; (50≤P_2g≤250)MW;

C₃=0.008P²_3g+3.2P_3g+145, $/h; (50≤P_3g≤300)MW;

C₇=0.0065P²_7g+3.5P_7g+147, $/h; (50≤P_7g≤300)MW;

Using modified zbus model (MZbM) to

optimize the power system steady-state,

analogically, we have the main numerical results

of the problem of economic p_power generation

as presented in table 4

Using the augmented regtangular system

model (ARSM) with eigen-image matrix to

carry out the power system steady-state

optimization, we obtain the main numerical

results of the problem of economic p_power

generation as presented in table 3

The loss coefficient sub-matrix is

The loss coefficient matrix is

the system incremental fuel cost is

=7.15218;

b

Using the modified zbus model, we obtain the

estimative matrix E₁as follow

the system incremental fuel cost is

_ARSM=6.8957;

In this case the estimative matrix E is

The modal matrix of the estimative matrix E₁is

4.2. Example 2.

Let’s investigate a standard IEEE 30bus

power system consisting of 4 generators. The

power system’s schema is shown in figure 3

In this case the eigenvalue matrix is

And we obtain H₁= 0013;

Applying (24) to estimate fuel cost

increment, we obtain C_ARSM=0.13$/h. Using

(29) to estimate fuel cost increment, we

obtain approximative

C_MZbM=0.2027$/h. It

Investigating the economic equivalent state,

we have the numrical results as follow

means that both optimum results corresponding

to their mathematical models are technico-

economic equivalent.

Comparing the economic state parameters of

a power system loading 825MW, such as

incremental fuel costs _xxx, fuel costs C_xxxand

transmission p_power losses P_xxx, from the

results of the two models above, we notice very

little differences between their magnitudes. The

numerical results are follow

The technico-economic equivalent parameters

are shown in table 5

C = C_ARSM– C_MZbM= 0.052 $/MWh;

=

–

= -0.256 $/MWh;

ARSM

MZbM

The ARM model estimative parameters are

d P = P_ARSM– P_MZbM= 0.046 MW;

If the maximum system fuel cost increment

is accepted in range of 1$/h, then the technico-

economic equivalent states must be found in a

globoil space (precision of economic equivalent

state) with radius R=10.91;

Comparing the optimum generator p_power

outputs of two cases presented above, we can

obtain the differences as follow

the number H is 0.2461, and the system fuel

cost increment is C_ARSM=10.6468$/h;

│ P₁│=│196.09 –193.57│= │ 2.52│< R;

│ P₂│=│187.12 –188.08│= │-1.96│< R;

│ P₃│=│216.76 –218.90│= │-2.24│< R;

│ P₇│=│239.17– 238.55│= │ 0.62│< R;

The MZb model estimative parameters are

The numerical results presented in the table

3 and table 4 show the voltage state parameters

set together are almost unchanged.

International Advanced Symposium on

Electrical Engineering.

3. Luu Huu Vinh Quang. Economic p_power

dispaching and load q_power compensation

in power system. Authorship testimonial

4. Glenn W. Stagg, Ahmed H.El-Abiad.

Computer method in power system

analysis. McGraw-Hill 1983.

5. V.A. Venikov. Electrical Power System

Computing Programming and Optimization.

Moscow High School Edition 1973.

the number H₁is 0.2394, and the system fuel

cost increment is C_MZbM=10.3570$/h;

Comparing

the

system

fuel

cost

increment C, we can conclude the both

optimum results corresponding to their

mathematical models are technico-economic

equivalent.

6. John J.Grainger & William D.Stevenson,JR.

Power system analysis. McGraw-Hill 1994.

7. Luu Huu Vinh Quang. A new P_Power loss-

coefficient expression and economic active

5. CONCLUSION

The modified zbus power flow model is

developed as (5)-(6)-…-(15)-(16)-(17).

power generation of thermal plans.

th

A new augmented regtangular system power

flow model [1] is proposed to power flow

optimization with new expression of loss

coefficient [7].

Proceedings of the 8 conference on science

and technology. p.1-6. Session Electrical

Engineering and Power systems. University

of Technology VNU HCMC.

This paper develops a method estimating the

technico-economic equivalent state [2] with an

estimative matrix which is determined by a

square and symmetric loss coefficient matrix of

(n×n) size in which n is the generator quantity

of a power system, including the slack bus

generator [7]. A new loss coefficient expression

[3] and a new modified expression (27) of

estimative number H are proposed. Using the H

and R magnitudes we can analyse the influence

of power system parameters on the technico-

economic equivalent state in electrical power

system operation with an estimative matrix E

obtained from the results of a power system

state optimization solving.

8. Hadi Saadat. Power system analysis.

McGraw-Hill 1999.

REFERENCES

1. Luu Huu Vinh Quang. Electrical Power

System steady-state simulation using

Augmented Rectangular System Model with

the Eigen-Image Admittance Matrix.

Authorship testimonial N^o1175/2004 /QTG -

Copyright Office of VN.

2. Luu Huu Vinh Quang. A method for

estimating the technico-economic equivalent

states in power system operation.

th

Proceedings of the 10 conference on

science and technology - University of

Technology VNU HCMC

-

2007