Voltage Stability Evaluation of The Khouzestan Power System in Iran Using CPF Method and Modal

Analysis

Farbod Larki Dept. of Technical Office of Network

Khouzestan Regional Electrical Company Ahvaz, Iran

Mahmood Joorabian Dept. of Engineering

Shahid Chamran University Ahvaz, Iran

Homayoun Meshgin Kelk, Mojtaba Pishvaei Dept. of Electrical Engineering

Tafresh University Tafresh, Iran

AbstractVoltage stability is an important factor that needs to be taken into consideration during the planning and operation of power systems in order to avoid voltage collapse and subsequently partial or full system blackout. The study of the voltage collapse phenomenon can provide a way to prevent this event from happening. There have been many methods developed to study the criteria of voltage collapse phenomenon but static analysis probably provide the best way to study this phenomenon. This paper presents Continuation Power Flow (CPF) method to assess voltage stability of the Khouzestan power system in Iran for both normal operation and for contingencies. In rest of this work Modal analysis is used to complete the voltage stability evaluation. The voltage stability of the system has been analyzed during the peak load condition of the summer 2007. The calculation identifies the weakest bus where remedial action may be needed for voltage support.

Keywords-voltage statbility; voltage collaps; continuation power flow; modal analysis; contingency

I. INTRODUCTION Recently network blackouts related to voltage collapse tend

to occur from lack of the reactive power support in heavily stressed conditions, which are usually triggered by system faults. Therefore, the voltage collapse problem is related to a reactive power planning problem including contingency analysis. Voltage stability analysis is essential for a secure power system operation. There are two types of the voltage stability based on the time frame of simulation: static voltage stability and dynamic voltage stability. Dynamic analysis uses for transient stability studies that dynamic of load and generator is important. Static analysis involves only the solution of algebraic equations and therefore is computationally less extensive than dynamic analysis. Static voltage stability is ideal for the bulk of studies in which voltage stability limit for many pre-contingency and post-contingency cases must be determined. Voltage stability static analysis has been based on the curves calculation, or on the singularity of the PF Jacobian

matrix. For systems with constant loads, the gradual load increment leads to a Saddle-Node Bifurcation (SNB), which corresponds to the Maximum Loading Point (MLP). Under these conditions, the Power Flow (PF) Jacobian matrix becomes singular and, consequently, the conventional Newtons Refson methods show numerical difficulties. Although these methods allow the computation of points very close to MLP. Several methods have been proposed with the purpose of avoiding the singularity of the Jacobian matrix, through the modification of the conventional Newtons method. The reformulation of the PF equations aims to remove the matrix singularity at the MLP and, consequently, the numerical problems that arise in its vicinity. Continuation Power Flow (CPF) methods, which are based on predictor-corrector algorithms, have been proposed using the full Jacobian matrices. This can obtain losses of the reactive powers in the buses accurately, simply and speedy. The computation of MLP is important for the knowledge of voltage stability margins, as well as for modal analysis. The latter is used to identify system areas with voltage stability problems and to determine the most adequate reactive power support strategy. Modal analysis is most effectively used for system reinforcement studies or improved control measures when made at the MLP or near it [1]. Modal analysis can be used to select the best sites for installing new reactive compensation equipment, and to determine the most effective actions such as generation redispatch and load shedding to alleviate voltage conditions. CPF method and modal analysis will be discussed briefly in this paper and will be applied to study voltage stability of the Khouzestan power system in Iran.

Rest of this paper is organized as follows: Section II illustrates the static voltage stability, Continuation Power Flow (CPF) method and modal analysis. Section III presents the Case study. Section IV presents the simulation results. Section V concludes the paper.

This work is financially supported bye Khouzestan Regional Electrical Company, Iran, Ahvaz

978-1-4244-4813-5/10/$25.00 2010 IEEE

II. STATIC VOLTAGE STABILITY In static voltage stability, slowly developing changes in the

power system occur that eventually lead to a shortage of reactive power and declining voltage. This phenomenon can be seen from the plot of the power transferred versus the voltage at receiving end. The plots are popularly referred to as P-V curve or Nose curve. As the power transfer increases, the voltage at the receiving end decreases. Eventually, the critical (nose) point, the point at which the system reactive power is short in supply, is reached where any further increase in active power transfer will lead to very rapid decrease in voltage magnitude. Before reaching the critical point, the large voltage drop due to heavy reactive power losses can be observed. The only way to save the system from voltage collapse is to reduce the reactive power load or add additional reactive power prior to reaching the point of voltage collapse [2].

Voltage Stability studies are of growing importance for the design and operation of power systems. The purposes of analysis techniques are to identify system conditions causing voltage instability, to find loading margin of the system and to specify parameters affecting the voltage stability of the system. In static voltage stability studies, two analysis techniques, namely Continuation Power Flow and modal analysis are well known.

A. Continuation Power Flow Method The power flow Jacobian becomes singular at the point of

collapse. Therefore, the conventional power flow fails to give solution at the collapse point. To determine the solution of power flow, the continuation method has been used extensively. The continuation method systematically increases the loading level or bifurcation parameter, until bifurcation or point of collapse is determined (Ajjarapu and Christy, 1992). In power systems, continuation methods typically trace the voltage profile of the system up to the maximum loading point of the system [3].

Continuation power flow methods, which are based on predictor-corrector algorithms, have been proposed using the full Jacobian matrices. This method can obtain losses of reactive powers in the buses accurately, simply and speedy. This methods may be used to trace the path of a power system from a stable equilibrium point up to a bifurcation point is reached [4]. The CPF method uses the successive power flow solution to fully compute the voltage profiles up to collapse point to determine the loading margin. The power flow model is represented by [5]:

0),(),(

),( =

=

xQxP

xf (1)

where f(x, ) is power flow equation and x represents the state variables and is a system parameter, used to drive a system from one equilibrium point to another. This type of model has been employed for numerous voltage collapse studies, with being considered as the system load/generation increase factor or power transfer level. The system load change drives the system to collapse in the following way:

)1()1(

,,0,

,,0,

iQiDiD

iPiDiD

KQQKPP

+=

+= (2)

where iDP ,0 and iDQ ,0 represent the initial active and reactive

loads at bus i and constants iPK , and iQK , represent the active

and reactive load increase direction of bus i, respectively.

CPF method consists of three steps. The first part is a predictor step, the second is a corrector step and the third is a parameterization step. The predictor and corrector steps have been illustrated in the bifurcation diagram of Fig. 1. From an initial solution ),( 00 x , a step x , and change of the parameter are first determined. From the new point,

xxx += 01 the new equilibrium point 2x , is calculated. The continuation power flows trace the solution of the power flow equations 0),( =xf .

Figure 1. Illustration of predictor and corrector scheme for CPF method

B. Modal Analysis Modal or eigenvalue analysis of the power flow Jacobian

matrix, which is obtained at the MLP, can be used to identify buses vulnerable to voltage collapse. Modal analysis involves calculation of eigenvalues and eigenvectors of the power flow (reduced) Jacobian matrix [5]. Power flow equations and Jacobian matrix are presented by:

=

=

=

VQQVPP

JJJJ

J

VJ

QP

QVQ

PVP

(3)

It is practically known that when the system voltage stability is affected by both P and Q. However, at each operating point we may keep P constant and evaluate voltage stability the incremental relationship between Q and V. Although incremental changes in P are neglected in the formulation, the effects of changes in system load or power transfer level are taken into account by studying the incremental relationship between Q and V at different

operating conditions. Based on the above considerations, in (3), let 0=P . Then [6]:

VJQVJJ

JJQ

R

QVQ

PVP

=

=

0 (4)

where RJ called reduced jacobian matrix:

][ 1 PVPQQVR JJJJJ

= (5)

from (4) we can write

QJV R =1

(6)

voltage stability characteristics of the system can be identified by computing the eigenvalues and eigenvectors of the reduced Jacobian matrix defined by (5) :

=RJ (7) Where Assume and are, respectively the right and left eigenvectors of the Jacobian corresponding to the eigenvalue then the ith modal reactive power variation is:

iimi KQ = (8)

where =

=

n

jjiiK

1

22 1 the corresponding ith modal voltage variation is:

mi

i

mi QV = 1

(9)

Bus participations: Participation of bus k to mode i is:

kikikiP = (10)

Where ki is the kth element of the ith column right eigenvector and

ki is the kth element of the ith row left

eigenvector. The suffix i indicates a particular mode, i , corresponding to the eigenvalue

i . A component with higher participation indicates that this component's contribution to this mode is large [7].

III. CASE STUDY Simulated case study is Khouzestan power system in Iran

and drawn in Fig. 2. Khouzetan zone is greatest commercial

and industrial zone in Iran that it is located in south-west of country (Khouzestan province). In this region, there are greatest iron and steel factories, oil and gas industries. Hence, power network of this state has been very important. Khouzestan system has been 4 power stations in 400KV voltage level and 14 transmission lines and 7 PQ bus substations. In this system generation unit are modeled as standard PV buses considering reactive power limits and load are represented as constant PQ loads and simulated condition of network is in peak load conditions (30 July, 2007).data of this system is presented in Table I.

Figure 2. Kouzestan power system in Iran

TABLE I. DATA OF KOUZESTAN POWER SYSTEM IN IRAN

Q load P load Q gen P gen phase V Bus

[p.u.] [p.u.] [p.u.] [p.u.] [rad] [p.u.]

-0.53 -3.74 3.441 -7.31 0 1.02 Bus 1

-2.59 -6.33 0 0 -0.003 0.982 Bus 2

0.67 1.61 0 0 -0.016 0.968 Bus 3

1.27 2.15 0 0 -0.166 0.946 Bus 4

1.93 4.39 0 0 0.043 0.964 Bus 5

-0.14 -1.34 0.235 2.09 0.11 0.975 Bus 6

-0.65 -1.15 0.345 8.05 0.105 1.01 Bus 7

-0.21 -0.1 0.002 8.3 0.101 0.988 Bus 8

1.37 3.4 0 0 -0.144 0.932 Bus 9

1.45 5.07 0 0 -0.162 0.967 Bus 10

-3.6 6.79 0 0 -0.153 0.971 Bus 11

IV. SIMULATION RESULTS In this research PSAT tool box [8] and MATLAB software

are used to analyze static voltage stability and calculate MLP (

max ). Fig. 3 shows -V curves (P-V curves) for Kouzestan power system and Table II, illustrates Voltage magnitude of buses at the collapse point of base case. It is noticed from Fig. 3 that,

max is 1.744 p.u.. Usually, placing adequate reactive power support at the

weakest bus enhances static voltage stability margins. The weakest bus is defined as the bus, which is nearest to experiencing a voltage collapse. In this paper, the weakest bus is one that has a smallest voltage in maximum loading point (

max ). Voltage magnitude at Bus 9 is 0.6435p.u. in collapse point. So bus 9 (Abadan city) is the weakest bus and Bus 4 (Mahshar city) is near to experiencing a voltage collapse too (V4=0.6728 p.u.).

Figure 3. -V curves (P-V curves) for Kouzestan power system

TABLE II. VOLTAGE MAGNITUDE OF BUSES AT THE COLLAPSE POINT OF BASE CASE

Bus no. Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 V [p.u.] 0.8064 0.7767 0.6728 0.820773 0.8806 Bus no. Bus 7 Bus 8 Bus 9 Bus 10 Bus 11

V [p.u.] 0.9784 0.8955 0.6435 0.7239 0.7333

Now voltage stability of system is surveyed in contingency condition by CPF method. Table III shows the results of single generation unit's outages applying CPF method. When generation unit in bus 7 is out, the system has lowest MLP ( 3687.1= ). Outage of bus 7 is identified as the most critical contingency between contingencies of other generation unit outages and in this condition weakest bus is bus 9.

TABLE III. THE RESULTS OF SINGLE GENERATION UNIT OUTAGES

Generation unit outage

Bus No. with lowest voltage

magnitude

Lowest voltage magnitude in MLP

(p.u.) max

(p.u) Bus 7 Bus 9 0.86993 1.3687 Bus 8 Bus 9 0.83056 1.441 Bus 6 Bus 9 0.63416 1.6762

In other situation, we study effects of line outage on the voltage stability. Table IV shows the line outages results by applying CPF method.

TABLE IV. THE RESULTS OF LINE OUTAGES

It is observed that the position of the weakest bus in line contingencies is fixed in 85% of outages (bus 9). The outage of the line ER 905 (connected to bus 1 to 11) has lowest

max . Outage of the line ER 905 is identified as the most critical contingency between contingencies of other line outages and the weakest bus is bus 9. When the line EH 906 is out, voltage stability is increased, because the load shedding is occurred on the bus 10, so this line outage is not used for identification of the weakest bus.

We can apply modal or eigenvalue analysis to complete results of this paper. Table V shows the eigenvalue of the Jacobian (J) matrix of the test system. The bus that has Minimum eigenvalue is the weakest bus [6], [7]. Minimum eigenvalue of the system is 15.78 for Bus 9.

TABLE V. EIGENVALUES OF THE JACOBIAN MATRIX

Eigevalue Most Associated Bus Real part Imaginary

Part

Eig Jlfr # 1 bus 11 1168.244 0

Eig Jlfr # 2 bus 2 512.1721 0

Eig Jlfr # 3 bus 9 15.78831 0

Eig Jlfr # 4 bus 5 207.4358 0

Eig Jlfr # 5 bus 4 184.8551 0

Eig Jlfr # 6 bus 10 78.23398 0

Eig Jlfr # 7 bus 3 99.00437 0

Eig Jlfr # 8 bus 1 PV bus PV bus

Eig Jlfr # 9 bus 6 PV bus PV bus

Eig Jlfr #10 bus 7 PV bus PV bus

Eig Jlfr #11 bus 8 PV bus PV bus

Line outage

Bus_No. with lowest

voltage magnitude

Lowest voltage magnitude in MLP (p.u.)

max (p.u)

ER 905 Bus 9 0.74588 1.1754 KR 908 Bus 9 0.79024 1.2656 ES 904 Bus 4 0.54149 1.2814 EW 935 Bus 4 0.68373 1.2943 DT 915 Bus 9 0.55401 1.3553 WR 901 Bus 9 0.67289 1.4993 DU 927 Bus 9 0.64879 1.5815 DK 912 Bus 9 0.66485 1.6091 WT 900 Bus 9 0.60969 1.6094 DW 921 Bus 9 0.6224 1.6618 TU 933 Bus 9 0.64148 1.6726 HU 934 Bus 9 0.63915 1.6635 DS 914 Bus 9 0.60112 1.7315 EH 906 Bus 9 0.61291 2.2565

Table VI shows the bus participations of the modal analysis for Khouzestan power system. A component with higher participation indicates that this component's contribution to this mode is large [6], [7]. For example: according to table VI, buses 4 and 5 have high bus participations and these buses affect more than other buses on the voltage stability of bus 4 or mode 5 (Eig Jlfr# 5). In other word, voltage stability of bus 4 dependents on reactive power support of the buses 4 and 5.

Bus 9 is the weakest bus and voltage stability of this bus (Eig Jlfr# 3) is dependent to reactive power support of the buses 9, 4, 10 and 11. Bus 9 has maximum Bus participation. This bus affects more than other buses on its voltage stability that this result is obvious and natural.

TABLE VI. RESULTS OF MODAL ANALYSIS AND BUS PARTICIPATIONS OF TEST SYSTEM

V. CONCLUSION In this paper, we specified voltage stability of the

Khouzestan power system by applying continuation power flow (CPF) and calculation of Maximum Loading Point (MLP) in two cases, base case and contingency. The weakest bus of the Khouzestan power system is determined in these conditions (bus 9 in Abadan city). Corresponding to the analysis, position of the weakest bus in contingencies is fixed in 85% of outages at Khouzestan power system. Modal analysis or eigenvalue analysis is used to complete results of this paper. Weakest bus (bus 9) has Minimum eigenvalue of the system. Bus participations of the system are determined by modal analysis. We surveyed effects of other buses on the voltage stability of the weakest bus using modal analysis.

REFERENCES

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[2] A.Sode-Yome,N.Mithulananthan,Kwang Y.Lee. Static Voltage Stability Margin Enhancement Using STATCOM,TCSC,and SSSC, IEEE/PES Transmission and Distribution Conference & Exhibition: Asia and Pacific Dalian, China, 2005

[3] Ajjarapu, V. and Christy, Colin, "The Continuation Power Flow: A Tool to Study Steady State Voltage Stability," IEEE Transactions on Power Systems, Vol. 7, No. 1, pp. 416-423, Feb. 1992

[4] A. Kazemi, B. Badrzadeh, Modeling and simulation of SVC and TCSC to study their limits on maximum loadability point, Electrical Power and Energy Systems Jurnal , Elzevier, vol 26, pp. 619-626, 2004.

[5] V. Ajjarapu, Computational Techniques for Voltage Stability Assessment and Control, Springer, 2006.

[6] P. Kundur, Power System Stability and Control, New York, NY: McGraw-Hill, 1994

[7] B. Gao, G. K. Morison, and P. Kundur, Voltage stability evaluation using modal analysis, IEEE Trans. Power Syst.7: 1529-1536, 1992

[8] M. Young, The Technical Writer's Handbook. Mill Valley, CA: University Science, 1989.

[9] F. Milano, Power System Analysis Toolbox, Version 2.1.3, Software and Documentation, Feb. 14, 2009.

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