Determination of maximum load margin using fuzzy logic

Electrical Power and Energy Systems 52 (2013) 231–246

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Determination of maximum load margin using fuzzy logic Sourav Mallick ⇑, P. Acharjee, S.P. Ghoshal, S.S. Thakur Department of Electrical Engineering, National Institute of Technology, Durgapur 713 209, India

a r t i c l e

i n f o

Article history: Received 14 November 2011 Received in revised form 4 July 2012 Accepted 29 March 2013 Available online 28 April 2013 Keywords: Fuzzy logic Power flow Maximum load margin Sparse constant array Decoupling properties

a b s t r a c t In the modern power systems, maximum load margin (MLM) plays an important role from the point of view of system stability. The knowledge of MLM helps the system operator to take proper decisions regarding load margin. In this paper, three new schemes using Fuzzy Logic (FL) is developed along with a new formation of sparse constant array. The proposed schemes are tested on IEEE 5-bus, 14-bus, 30bus, 57-bus and 118 bus test systems under different practical security constraints. The iterative process can be started with random initialization using proposed FL schemes which is not possible using N–R technique. The results are compared with the same of Newton–Raphson (N–R) method technique and the standard fuzzy logic controllers. The comparison indicates the superiority of proposed FL schemes over traditional N–R technique. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In the electrical power systems, the network complexity increases day by day and the problem of stable operation of the system increases continuously in power systems which is one of the most challenging problems that have been studied over the last decades due to its importance in stable, secure and reliable power system operation [1–6]. Although the nature of the system voltage is essentially dynamic, change of voltage profile can be treated as a static problem if the parameters of the systems change slowly. Therefore, power flow studies deal with the calculations of bus voltages, their phase angles, the active and the reactive power flows through various branches, generators and loads under steady state conditions. In the NR method [1–6], the number of iterations is very less because of the quadratic convergence characteristics. Another advantage of NR method is that it is independent of the system size. But the problems of NR method are the power differences and the elements of the Jacobian are to be computed per iteration [1,2]. Triangularization has also been done in each iteration, so that the time taken per iteration is considerably longer. Stott et al. proposed one decoupled form of NR method to overcome the lacunae of the NR method [7]. Thereafter, several variants/improvements of Newton’s method as well as the decoupled form of NR method have been suggested in [8–20]. The Fuzzy Logic (FL) is a methodology to solve problems which are too complex to be understood quantitatively. It is based on fuz⇑ Corresponding author. Tel.: +91 9434205918. E-mail addresses: [email protected] (S. Mallick), parimal.acha [email protected] (P. Acharjee), [email protected] (S.P. Ghoshal), [email protected] (S.S. Thakur). 0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.03.033

zy set theory, introduced by Prof. L. Zadeh [21,22]. Use of fuzzy sets in logical expression is known as Fuzzy Logic (FL) that has been the subject of important investigations. At the beginning of the nineties, FL was firmly grounded in terms of its theoretical foundations and used in various fields. The central assertion underlying this approach is that entities in the real world simply do not fit into neat categories. Vlachogiannis used fuzzy logic [23,24] in solving the power flow problem. Lo et al. [25] used fuzzy logic to the adjustment of variable parameters to satisfy different constraints in load flow studies. A few notable works have been contributed by several authors in this field [26–36]. In order to maintain the reliability of an electric power system at an appropriate level, it is essential that voltage stability is accurately assessed. Voltage stability is, therefore, one of the most important and burning issues of bulk power transfer systems. It has become synonymous with modern day Energy Management Systems (EMSs). Hence, the determination of maximum load margin (MLM) to attain the bifurcation point in the graph of voltage magnitude versus system load is necessary for the system operator. Numerous papers in this field show the importance of MLM in EMS [37–45]. In this paper, three new schemes of FL and the formation of sparse constant array are introduced. The aim is to attain better MLM point as well as to form an efficient algorithm to solve the power flow problem with flat initialization as well as random initialization. The organization of the paper is as follows: In Section 2, the MLM is described briefly followed by the security constraints of power system. Fuzzy load flow controller (FLC) is discussed in Section 3. The standard fuzzy power flow scheme and the proposed fuzzy schemes are discussed in Section 4. Section 5 depicts the computational procedure with the flow chart. The simulation and

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Nomenclature n NS Npq Npv NSL Nline fbi tbi xi ri Ysh tt Y

S P Q V d G B Vi di PGi Q Gi Sik

number of buses number of buses excluding slack bus number of load buses number of generator buses slack bus number number of lines starting bus number of line ‘i’ end bus number of line ‘i’ reactance of line ‘i’ per km resistance of line ‘i’ per km half line charging transformer tap settings bus admittance matrix

apparent power active power injected reactive power injected voltage magnitude in p.u. phase angle in radian conductance in p.u. susceptance in p.u. voltage magnitude at node i phase angle at node i active power generation at node i reactive power generation at node i power flow in the line between node i and node j in MVA

results and discussion are presented in Sections 6 and 7, respectively, with the concluding remarks in Section 8.

several algorithms to solve SLFE. These two algorithms are discussed in the Appendix section.

2. Problem formulation

2.1. Maximum load margin

A balanced three-phase system is assumed, represented by its positive sequence network of lumped series and shunt parameters. For the purpose of analysis, it is convenient to regard loads as negative generators and to lump the generator and loads connected together at the buses. Thus, at any given bus ‘i’, the net complex power injected is given by,

In order to prevent the occurrence of voltage instability, it is essential to analyze voltage stability of power systems. The maximum load margin (MLM) is an efficient way to evaluate the steady state voltage stability. Load margin (LM) is a rational index from engineering point of view, to provide the system operator a more practical sense of security margin in terms of system loading. To solve the MLM problem, a balanced steady state system is considered. Hence, voltage, current and other parameters are considered as phasors. Voltage limits as well as reactive power capabilities of PV buses are considered as in-equality constraints but no thermal limit is taken into consideration. At each PV bus, during the iterations, value of reactive power Q is calculated and is checked against the upper and lower limits. If none of the limit is violated, it remains as PV bus, otherwise it is treated as load bus (PQ) for that iteration and the reactive power at that bus is fixed at the limit that is violated.

Si ¼ Pi þ jQ i ¼ V i Ii

for i ¼ 1; 2; . . . n

ð1Þ

Substituting,

Ii ¼

n X Y ik :V k

ð2Þ

k¼1

Si ¼ P i  jQ i ¼ V i

n X Y ik V k

ð3Þ

k¼1

Defining, V i ¼ jV i jejdi and Yik = Gik + jBik and separating real and imaginary parts,

Pi ¼

n X jV i jjV k jfGik cosðdi  dk Þ þ Bik sinðdi  dk Þg

ð4Þ

k¼1

Qi ¼

n X

jV i jjV k jfGik sinðdi  dk Þ  Bik cosðdi  dk Þg

ð5Þ

k¼1

Eqs. (4) and (5) are referred to as Static Load Flow Equations (SLFE). The four important power flow variables in a complex nonlinear power system problem are phase angle (d), voltage magnitude (|V|), reactive power (Q) and active power (P). Each bus can be characterized by four variables Pi, Qi, d and |Vi| resulting in a total of 4n variables of a n-bus system. Eqs. (4) and (5) can be solved for 2n variables if the remaining 2n variables are specified. Practical considerations allow a power system analyst to fix a priori two variables at each bus. The solution for the remaining 2n bus variables is difficult as (4) and (5) are nonlinear algebraic equations as bus voltages are involved in product form and sine and cosine terms are present. Explicit solution is not possible. Solution can only be obtained by iterative numerical techniques. Newton–Raphson Load Flow (NRLF) technique and Fast Decoupled Load Flow (FDLF) technique are very popular iterative techniques among

2.1.1. Security Constraints (SCs) and handling of security constraints In power flow problem, the unknown power flow variables, called state variables (w), are to be obtained taking into account both equality (w(w,u)) and in-equality (u(w,u)) constraints of power systems. In order to solve the MLM problem, the security constraints (SCs) are considered as: (i) limits of reactive power generation for generator buses, (ii) phase angle limits excluding slack bus, (iii) voltage magnitude limits for load buses, (iv) limits of real power generation for generator buses including slack bus and (v) line power flow limits. The control variables (u) such as transformer tap settings, real power generations and generator terminal voltages help to improve power flow according to requirement. The SCs are given below.

dmin 6 di 6 dmax i i

i 2 NS

V min 6 V i 6 V max i i min PGi 6 PGi 6 P max Gi max Q min Gi 6 Q Gi 6 Q Gi min max Sik 6 Sik 6 Sik

i 2 Npq i 2 Npv þ N SL i 2 N pv i 2 Nline

The SCs can be included using the following steps. The phase angle can be varied within dmax and dmin. If the phase angle of the load bus crosses any of its limits, then its phase angle is kept fixed at that limiting value which is violated i.e. if upper limit

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S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246 Table 1 The different security limits for fuzzy method. Security parameter

Voltage magnitude (p.u.) Phase angle (rad)

Case 1

Case 2

Case 3

Case 4

Maximum limit

Minimum limit

Maximum limit

Minimum limit

Maximum limit

Minimum limit

Maximum limit

Minimum limit

1.1

0.9

1.1

0.8

1.1

0.7

No limit

No limit

0

0.45

0

0.60

0

0.75

No limit

No limit

(dmax) is violated, the phase angle of the PQ bus is to be kept fixed at (dmax). Similarly, if phase angle of the load bus becomes less than its lower limit (dmin), then its phase angle is to be fixed to (dmin). Similar steps can be applied for other constraints such as voltage magnitudes, real power generations, and line power flows. But, for constraint of reactive power generation, it is handled differently. The specified voltage for the PV bus is kept constant if the reactive power generation of the generator does not cross its limit; otherwise the PV bus is treated as PQ bus keeping the reactive power output fixed to its violated limit. Table 1 portrays the limits of different SC parameters defined as different Cases 1, 2, 3 and 4, used to find respective the MLM points using AC power flow model.

D|V| of Q–|V| loop are, respectively, the input DU and the output DW to the FLC. According to (B.2), the change in |V| is dependent on reactive power mismatch (DQ). The most important aspect of the FLC is the maximum power parameters (DUmax = DPmax or, DQmax) to determine the range of scale mapping for reassigning the input signals into corresponding fuzzy signals at each iteration. 4. Techniques of FLC In this paper, the standard fuzzy power flow (SFPF) scheme and three newly proposed schemes (Scheme I, Scheme II and Scheme III) have been used in FLC to solve SLFE. In all these schemes of FLC, triangular membership functions have been used because of the simplicity in construction and implementation. The constructions of the triangular membership functions are explained in the following sub-sections.

3. Fuzzy load flow Using the decoupling properties of the power flow equations, the nonlinear SLFE can be linearized as

4.1. SFPF scheme

½DU ¼ ½B½DW

ð6Þ In SFPF scheme, seven membership functions have been used. The input signals DU are fuzzified into consequent fuzzy signals using the following seven linguistic variables; large negative (LN), medium negative (MN), small negative (SN), zero (ZR), small positive (SP), medium positive (MP) and large positive (LP). The standard seven membership functions are shown in Fig 2. Each of three points of the fuzzified inputs (left bottom limit, top, right bottom limit) is shown in Table 2. The left bottom limit of LN is 1; the central bottom point is DUmax and the right bottom point is DU3max . The MN ranges from DUmax to 0 with central bottom point at DU2max . The SN ranges from DU3max to 0 with central U max bottom point at DU6max . The left bottom limit of ZR is D12 . The central bottom point of ZR is 0 and right bottom limit of ZR is DU12max . The SP ranges from 0 to DU3max with central bottom point at DU6max . The MP ranges from 0 to DUmax with central bottom point at DU2max . The left bottom limit of LP is DU3max ; the central bottom point is DUmax and the right bottom point is 1. Table 3 presents the rule base of the FLC, having seven rules in accordance with seven linguistic variables, mentioned in Table 2. At every iteration, the fuzzified outputs DWfuz are directly proportional to fuzzified inputs DUfuz i.e. the crisp outputs Dd and D|V| are directly proportional to the crisp inputs DP and DQ, respec-

Eq. (6) denotes that the correction of state vector (DW) at each node of the system is directly proportional to DU. The scheme of fuzzy load flow (FLF) algorithm is based on (6). It is considered to be a linear function. The diversion of the proposed scheme (FLF) from FDLF is that the repeated updates of the state vectors (d and |V|) have been performed using individual fuzzy logic control (FLC) (Fig. 1) schemes. The schemes are explained in Section 4. The FLC can be expressed as

DW ¼ fuzðDUÞ

ð7Þ

The basic idea of the FLC is shown in Fig. 1. It consists of four principal components: a fuzzification interface (Fuzzifier), Process logic and a defuzzification interface (Defuzzifier). Fuzzifier uses ranges of input membership functions to fuzzify the crisp input DU which becomes the fuzzy input signal DUfuz. Process logic transforms DUfuz; to the corresponding fuzzy output signal DWfuz with the help of rule base. This fuzzy output signal is defuzzified with the help of Defuzzifier to get the crisp output value, DW. DP and Dd of P–d loop are, respectively, the input DU and the output DW to the FLC. According to (B.1), the change in d is mainly dependent on active power mismatch (DP). Similarly, DQ and

fuz

ΔU Crisp input values

ΔU fuz Fuzzifier Fuzzy input signal

Process logic Rule Base

ΔW fuz Fuzzy output signal

Fig. 1. Schematic block diagram of FLC.

Defuzzifier

ΔW Crisp output values

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S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246

Fig. 2. Membership functions for input signals DU for seven memberships.

Table 2 Range of linguistic variables for input DU(DP or, DQ) in terms of DUmax for SFPF scheme and R for Scheme I and Scheme II for fuzzifying to get DUfuz. Linguistic variable name

Left bottom limit

Central bottom point

Right bottom point

LN

1

DUmax(R)

DU max R ð3Þ 3

MN

DUmax(R) DU max R ð3Þ 3 DU max R ð 12 Þ 12

DU max R ð2Þ 2 DU max R ð6Þ 6

0

SN

0

DU max R 12 ð12Þ DU max R 3 ð3Þ

ZR SP

0

DU max R 6 ð6Þ DU max R ð2Þ 2

0

MP

0

LP

DU max R 3 ð3Þ

DUmax(R)

DUmax(R)

1

Table 3 Rule base using seven linguistic variables corresponding to SFPF scheme, Scheme I and Scheme II. Rule number

Input

Input linguistic variable

Output

Output linguistic variable

1 2 3 4 5 6 7

DUfuz DUfuz DUfuz DUfuz DUfuz DUfuz DUfuz

LN MN SN ZR SP MP LP

DWfuz DWfuz DWfuz DWfuz DWfuz DWfuz DWfuz

LN MN SN ZR SP MP LP

tively. The maximum corrective action of fuzzified output DWmax is proportional to the corresponding fuzzified input DUmax is given as

DW max



dU I ¼ dW I

1

Using this DWmax, the ranges of output variable are shown in Table 4 and the corresponding membership functions are shown in Fig. 3. 4.2. Scheme I

 DU max;I

ð8Þ

where UI expresses the real or reactive power balance equation at node-I with maximum real or reactive power mismatch of the system, XI, represents the voltage angle or magnitude at node-I.

The proposed Scheme I has seven linguistic variables, similar to SFPF. But the main difference of the proposed scheme is in the ranges of the input membership functions as given by

Table 4 Range of linguistic variables of the SFPF scheme, Scheme I and Scheme II for output defuzzifying output DWfuz to get DW. Linguistic variable name

Left bottom limit

Central bottom point

Right bottom point

LN MN SN ZR SP MP LP

1 DWmax DWmax/3 DWmax/12 0 0 DWmax/3

DWmax DWmax/2 DWmax/6 0 DWmax/6 DWmax/2 DWmax

DWmax/3 0 0 DWmax/12 DWmax/3 DWmax 1

S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246

235

Fig. 3. Membership functions for output signals DW.

R ¼ aDU max ¼



aDPMAX for P—dloop aDQ MAX for Q —jVjloop

ð9Þ

scheme is in the calculation of DWmax which is shown in detail in Section 4.5. 4.3. Scheme II

where a = 1.25. The ranges in terms of R of the input memberships are shown in the bracketed terms in Table 2. The rule base and the output memberships are shown in Tables 3 and 4, respectively. In Fig. 4, the memberships for Scheme I are presented. The ranges of the output memberships are same as the SFPF scheme. The modification of the proposed Scheme I from the SFPF

The proposed Scheme II has also seven linguistic variables, similar to SFPF. But the proposed Scheme II differs from the Scheme I in range multiplier a. In this scheme, a value is chosen as 1.5. The ranges in terms of R of the input memberships are shown in the bracketed terms in Table 2. The rule base and the output memberships are shown in Tables 3 and 4, respectively. In Fig. 4, the mem-

Fig. 4. Membership functions for input signals R.

Fig. 5. Membership functions for input signals DU for five memberships.

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S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246 Table 5 Range of linguistic variables for input DU(DP or, DQ) in terms of DUmax for Scheme III for fuzzifying to get DUfuz. Linguistic variable name

Left bottom limit

Central bottom point

Right bottom point

LN MN ZR MP LP

1 DUmax DUmax/3 0 DUmax/3

DUmax DUmax/2 0 DUmax/2 DUmax

DUmax/3 0 DUmax/3 DUmax 1

bership functions for Scheme II are presented in terms of R. The ranges of the output memberships are same as the SFPF scheme. The modification of the proposed Scheme II from the SFPF scheme is in the calculation of DWmax which is shown in detail in Section 4.5. 4.4. Scheme III In Scheme III, the input signals DU, same input signals as those of the SFPF, are fuzzified into consequent fuzzy signals using the following five linguistic variables; large negative (LN), medium negative (MN), zero (ZR), medium positive (MP) and large positive (LP). The membership functions of the input signal for the Scheme III are shown in Fig. 5 and respective left bottom limit, top and right bottom limits are shown in Table 5. The left bottom limit of LN is 1; the central bottom point is DUmax and the right bottom point is DU3max . The MN ranges from DUmax to 0 with central bottom point at DU2max . The left bottom limit of ZR is DU3max . The central bottom point of ZR is 0 and right bottom limit of ZR is DU3max . The MP ranges from 0 to DUmax with

central bottom point at DU2max . The left bottom limit of LP is DU3max ; the central bottom point is DUmax and the right bottom point is 1. So, the difference from the seven linguistic variables and seven memberships is that the SP and SN variables are omitted and the left and right bottom limit points are changed to DU3max and DU3max , respectively in case of Scheme III. Table 6 shows the rule base for Scheme III with five rules in accordance with five linguistic variables, presented in Table 5. The output memberships are five in numbers and same as the input memberships. These are shown in Fig. 6. The ranges of output memberships are shown in Table 7. The proposed Scheme III also uses the newly developed technique of calculation of DWmax which is shown in detail in Section 4.5. 4.5. Formation of constant arrays [B1], [B2] and calculation of DWmax In SFPF scheme, the maximum corrective action of fuzzified output DWmax is proportional to the corresponding fuzzified input DUmax and this correction uses the elements from the Jacobian

Table 6 Rule base using five linguistic variables corresponding to Scheme III. Rule number

Input

Input linguistic variable

Output

Output linguistic variable

1 2 3 4 5

DUfuz DUfuz DUfuz DUfuz DUfuz

LN MN ZR MP LP

DWfuz DWfuz DWfuz DWfuz DWfuz

LN MN ZR MP LP

Fig. 6. Membership functions for input signals DW. Table 7 Range of linguistic variables of Scheme III for defuzzifying output DWfuz to get DW. Linguistic variable name

Left bottom limit

Central bottom point

Right bottom point

LN MN ZR MP LP

1 DWmax DWmax/3 0 DWmax/3

DWmax DWmax/2 0 DWmax/2 DWmax

DWmax/3 0 DWmax/3 DWmax 1

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S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246

matrix which needs to be calculated in each iteration cycle. In this paper, one of the main contributions is in the formation of constant arrays [B1] and [B2] which is calculated once before starting the iterative loop. The use of sparse constant array reduces the memory and as it is an array, the calculations of its elements are simple. In the proposed FLF, [B1] and [B2] are arrays of lengths equal to (n  1) and (Npq), respectively. In order to obtain [B1] and [B2], two arrays [Y1] and [Y2], each of length n, are formed using the line data. If ‘fbi’ is the sending end bus number and ‘tbi’ is the receiving end bus number of the line i, then [Y1] and [Y2] have been calculated as

Y1ðfbi Þ ¼ Y1ðfbi Þ þ x1

for i ¼ 1; 2; 3; . . . ; Nline

i

Y1ðtbi Þ ¼ Y1ðfbi Þ 

xi þ2Yshi ðr 2 þx2 Þ i i tt2

Y2ðfbi Þ ¼ Y2ðfbi Þ þ Y2ðtbi Þ ¼ Y2ðtbi Þ þ

xi =ðr 2i

þ

ð10Þ

 for i ¼ 1; 2; 3; . . . ; Nline

ð11Þ

x2i Þ

This process will continue until all lines are included. If fbi (or tbi) appears in more than one, it will add the present second term of (10) and (11) to the previous [Y1] and [Y2] values accordingly. The elements of [B1] and [B2] are obtained using the following pseudo code.

rc = 0;cc = 0; for i = 1:n if bustype(i)=slack rc = rc + 1; B1(rc)=Y1(i); end if bustype(i)==PQ cc = cc + 1; B2(cc) = Y2(i);

% initialization of counters rc and cc % Applicable for all bus except slack bus % increase the counter rc % Form [B1] % Only applicable for load bus % increase the counter cc % Form [B2]

end end

If the ith bus is not slack bus, then the elements of [Y1] becomes the elements of [B1]. If the ith bus is a load bus or PQ bus, then the elements of [B2] will be the ith element of [Y2] matrix. These rules are compatible with (7). At every iteration, the fuzzified outputs DWfuz are directly proportional to fuzzified inputs DUfuz i.e. the crisp outputs Dd and D|V| are directly proportional to the crisp inputs DP and DQ, respectively. The maximum corrective action of fuzzified output DWmax is proportional to the corresponding fuzzified input DUmax. This has been shown using the pseudo code,

½DQ max ; q1 ¼ maxðabsðdelqÞÞ; ½DPmax ; p1 ¼ maxðabsðdelpÞÞ; 1 DjVjmax ¼ :DQ max ; B2ðq1Þ 1 Ddmax ¼ :DPmax ; B1ðp1Þ c out ¼ ½LN MN SN ZR SP MP LP g0; where ‘delp’ is the array of active power mismatches. The length of the array is (n  1); similarly ‘delq’ is the array of reactive power mismatches whose length is Npq; g0 is either the maximum corrective voltage angle (Ddmax) for P–d loop or, the maximum corrective voltage magnitude (D|V|max) for Q–V loop; abs or ‘||’ means the mag-

nitude; ‘q1’ and ‘p1’ are the indices corresponding to maximum reactive power mismatch (DQmax) and maximum active power mismatch (DPmax) for Q–V loop and P–d loop, respectively. B2(q1) is the value corresponding to DQmax and its reciprocal multiplied with DQmax provides the D|V|max for output. Similarly, B1(p1) is the value corresponding to DPmax and its reciprocal multiplied with DPmax provides the Ddmax for output. In Q–V loop, the corresponding fuzzified input DQmax is DUmax and the fuzzified output D|V|max is DWmax. In P–d loop, the corresponding fuzzified input DPmax is DUmax and the fuzzified output Ddmax is DWmax. Depending upon the ‘‘c_out’’, rule bases, the fuzzified inputs DU and DWmax, the corresponding output DW is to be found using the rule bases provided in Table 4 for Schemes I and II and Table 6 for Scheme III, respectively. Among the various defuzzification techniques, the centroid of area (COA) method is used to defuzzify the output DW to obtain the crisp values of DW i.e. Dd and D|V|. Then, the ith state vector is updated in rth iteration as

drþ1 ¼ dri þ Ddri i

ð12Þ

jV rþ1 j ¼ jV ri j þ DjV ri j i 5. Computational procedure

For easy implementation of the proposed FLC, the main computational steps are briefly explained in a sequence as follows: i. Read system data. ii. Form bus admittance matrix, B1 and B2 matrices. The iterative is started with either flat initialization or random initialization. iii. Initialize iteration count r = 0. KP = 1, KQ = 1. In this paper, two different tolerance criterions have been used. In one criterion, tolerance value (e) is set 0.001 and is compared old old old against DPold max and DQ max . DP max and DQ max are set to vey high 10 value like 10 , so that the iterative process does not meet the convergence criteria at the first iteration. In another criterion, Tolerance value (e) is set 0.0001and is compared against D|V|max and Ddmax. D|V|maxb and Ddmax are the the maximum difference in change in bus voltages and phase angles in two successive iterations, respectively. D|V|max and Ddmax are set to vey high value like 1010, so that the iterative process does not meet the convergence criteria at the first iteration. (marked # in Fig. 1). iv. Bus active power (Pcal) is to be calculated using the following relation.

Pcal ¼ real VðiÞ ej



di

( )! n X  Y ik  VðiÞ ej dk

ð13Þ

k¼1

v. While using the first tolerance criterion, check whether the current value of maximum active power mismatch, DP new max , has reached e limit or not. If not, then go to the step vi otherwise change KP to 0. Check if KQ is 0. If KQ = 0, then stop iteration and go to step xiii; else go to step viii. In the second tolerance criterion, check whether the current value of maximum change in phase angle mismatch, Ddmax, has reached e limit or not. If not, then go to the step vi otherwise change KP to 0. Check if KQ is 0. If KQ = 0, then stop iteration and go to step xiii; else go to step viii. vi. The value of Ddi is calculated using fuzzy logic and update di using (12). new vii. Calculate Pcal and find DP new max . Check whether DP max is less old than DP old or not. If yes, the D P is to be replaced by max max DP new max and go to step viii; else go to step vi.

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viii. Bus reactive power (Qcal) is to be calculated using the following relation.  j di

Q cal ¼ imag VðiÞ e

( )! n X  j dk  Y ik VðiÞ e

ð14Þ

k¼1

x. xi.

ix. While using the first tolerance criterion, check whether the current value of maximum reactive power mismatch, DQ new max , is reached e limit or not. If not, then go to the step x, otherwise change KQ to 0. Check if KP is 0. If KP = 0, then stop iteration and go to step xiii; else go to step xii. In the second tolerance criterion, check whether the current value of maximum change in voltage magnitude mismatch,

xii. xiii.

D|V|max, has reached e limit or not. If not, then go to the step vi otherwise change KP to 0. Check if KQ is 0. If KQ = 0, then stop iteration and go to step xiii; else go to step viii. The value of D|Vi| is calculated using fuzzy logic and update |Vi| using (12). new Calculate Q and find DQ new max . Check whether DQ max is less old old than DQ max or not. If yes, DQ max is to be replaced by DQ new max and go to step xii; else go to step x. Increase iteration count by 1 and go to step iv. Print results.

The whole computational scheme of the proposed FLC is shown in a flow chart in Fig. 7.

Fig. 7. Flow chart of the proposed FLC schemes (I, II and III).

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S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246 Table 8 Important data of IEEE standard test systems. IEEE test systems

Number of PV buses

Number of lines

Number of PQ buses

Total PD (in p.u.)

Total QD (in p.u.)

Number of transformer tapings

Number of shunt capacitances

Number of variables

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

0 4 5 6 53

7 20 41 80 186

4 9 24 50 64

1.65 2.59 2.834 12.518 36.78

0.400 0.735 1.262 3.364 14.38

0 3 4 15 9

0 1 2 3 14

8 22 53 106 181

Table 9 Comparison of times of convergence per iteration among results obtained using different membership functions at 100% loading with DPmax < e(=0.001), DQmax < e(=0.001) and flat initialization. IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus 11-Bus a

Time for convergence per iteration N–R method

SFPF scheme

Scheme I

Scheme II

Scheme III

0.02445 0.02535 0.03580 0.05510 0.06745 Da

0.00265 0.00439 0.01206 0.04575 0.04189 D

0.00248 0.00484 0.01122 0.04792 0.03421 0.00269

0.00232 0.00460 0.01064 0.04436 0.03168 0.00250

0.00267 0.00414 0.01255 0.04411 0.02411 0.00226

D = Diverged.

6. Simulation The performance of the proposed FLC has been evaluated by testing it on IEEE 5-bus, 14-bus, 30-bus, 57-bus and 118-bus test systems [46]. The simulation study is carried out by varying the load gradually at each bus in a similar fashion. A constant power factor is assumed, so that the reactive power follows the active counterpart. The change in total load is distributed amongst the generators according to their participation factors. In order to achieve the MLM point, load is increased from 100% system loading in steps of 0.001 p.u. until voltage instability occurs. A flat voltage start (or random voltage start) is used to initialize the iterative old schemes. A very high value (1010) is set for DPold max and DQ max in order to prevent the iterative loop to meet the tolerance criterion, e, in the first iteration. In the subsequent iterations, the values of old DPold max and DQ max will be updated according to the iterative process. In this paper, two different tolerance criterions have been used. In one case, the current values of maximum active and reactive power new mismatches, DP new max and DQ max , in each iteration cycle are checked whether both of them are simultaneously less than the tolerance limit (e = 0.001). If both values are smaller than e, then the convergence has been achieved, otherwise the iteration will continue. In another case, the tolerance criteria is chosen as the maximum difference in change in bus voltages (D|V|max) and phase angles (Ddmax) in two successive iterations. In each iteration cycle D|V|max

and Ddmax are checked whether both of them are simultaneously less than the tolerance limit (e = 0.0001). If both values are smaller than e, then the convergence has been achieved, otherwise the iteration will continue. The proposed FLC is also applied to 11-bus illconditioned systems under 100% system loading condition. The test results are presented in terms of various performance indicators to show the efficiency of the proposed method. The program was run in Intel(R) Core™ 2 Duo CPU E8400 @ 3.00 GHz. processor with 2 GB RAM. MATLAB version 7.8.0.347 (R2009a) was used as programming and simulation platform. 7. Results and discussions Although many important results have been found with the simulations as considered, for the sake of brevity the performance of the proposed FLC method has been presented for some of the important results. The important parameter values of the test systems are given in Table 8. Table 9 presents the comparisons of convergence times per iteration for load flow study among different schemes, presented in this paper. Times of convergence for Scheme I, Scheme II and Scheme III along with the SFPF scheme are also noted and presented in Table 9. The proposed FLC method takes lesser time than the N–R method for all IEEE test systems and successfully converges for the ill-conditioned 11-bus test system which cannot

Table 10 Comparison of numbers of iterations for convergence among results obtained using different membership functions at 100% loading with DPmax < e(=0.001), DQmax < e(=0.001) and flat initialization. IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus 11-Bus a

D = Diverged.

Number of iterations N–R method

SFPF scheme

Scheme I

Scheme II

Scheme III

2 2 2 3 3 Da

9 10 10 11 13 D

9 10 10 11 13 11

9 10 10 11 13 11

9 10 10 11 12 11

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be solved by the N–R method or the FDLF method. From the comparison, it has been found that the increase in range of memberships reduces the convergence time per iteration. For higher systems like IEEE standard 57-bus and 118-bus test systems, the use of proposed schemes (Scheme I, Scheme II and Scheme III) shows the better performance than the SFPF scheme. From Table 10, it is clear that for each test system, almost same number of iterations is required to converge for the above mentioned schemes. Furthermore, the Scheme III takes lesser time per iteration than

all the methods of Scheme I while solving the ill-conditioned 11bus test system. The convergence characteristics are shown in Figs. 8 and 9. From the figures, it is clear that the convergence characteristics are quadratic in nature like the N–R method and the FDLF method. In another scheme, the tolerance criteria is chosen as the maximum difference in change in bus voltage in two successive iterations (D|V|max) and a tolerance precession is adjusted as 0.0001. The comparisons of convergence times and number of iterations

Fig. 8. Convergence characteristic of DPmax.

Fig. 9. Convergence characteristic of DQmax.

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S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246 Table 11 Comparison of times for convergence among results obtained using different schemes of FLC at 100% loading with D|V|max < e(=0.0001), |Dd|max < e(=0.0001) and flat initialization. IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus 11-Bus

Time for convergence (in seconds) SFPF scheme

Scheme I

Scheme II

Scheme III

0.0264 0.0327 0.0620 0.2534 0.3795 D

0.0261 0.0435 0.1008 0.2733 0.1577 0.0203

0.0276 0.0374 0.0968 0.3633 0.1068 0.0174

0.0261 0.0297 0.0677 0.0622 0.1812 0.0156

Table 12 Comparison of numbers of iterations for convergence among results obtained using different schemes of FLC at 100% loading with D|V|max < e(=0.0001), |Dd|max < e(=0.0001) and flat initialization. IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus 11-Bus

Number of iterations SFPF scheme

Scheme I

Scheme II

Scheme III

8 7 6 9 10 D

8 8 7 9 7 8

8 7 8 9 7 7

8 7 11 8 9 7

using different schemes are shown in Tables 11 and 12. The comparison of results in Table 11 reveals that the choice of D|V|max as tolerance criterion gives faster convergence for all the IEEE standard test systems except for IEEE 5-bus test system. The SFPF scheme fails to yield any solution for ill-conditioned test system whereas the three proposed schemes have successfully yielded solutions. Table 12 shows the almost same number of iterations for the proposed schemes in each test system. The advantage of the FLC is the random start of iterative process which is not possible with any other iterative techniques like G–S method, N–R method and FDLF method. In the initialization process the starting voltages for PQ buses and the phase angles for all the buses are chosen randomly in two sets as: (i) between 0.9 p.u. and 1.1 p.u. and between 0 radian to 0.45 radian, respectively, and (ii) between 0.8 p.u. and 1.2 p.u. and 0 radian to 0.6 radian, respectively. With this random initialization, the FLC converges successfully for all the proposed FLC schemes (I, II and III). As the initialization is randomly chosen, the convergence time (T) and number of iterations (ITR) for convergence vary. Therefore, average iteration time (Taverage) and minimum iteration time (Tmin) are more appropriate for better performance analysis. These quantities are computed on the basis of 100 runs of each scheme for each test system. The results are shown in Tables 13 and 15. The minimum iteration number (ITRmin) and average iteration number

Table 13 Comparison of times for convergence among results obtained using different schemes of FLC at 100% loading with DPmax < e(=0.001), DQmax < e(=0.001) and random initialization (0:9 6 jVj 6 1:1and0:0 6 d 6 0:45). IEEE test systems

Time for convergence (in seconds) SFPF scheme

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus 11-Bus

Scheme I

Scheme II

Scheme III

Tmin

Taverage

Tmin

Taverage

Tmin

Taverage

Tmin

Taverage

0.0087 0.0115 0.0292 D D D

0.0151 0.0497 0.0636 D D D

0.0057 0.0137 0.0333 0.2174 0.3891 0.0212

0.0129 0.0447 0.0800 0.7891 0.8847 0.0286

0.0061 0.0178 0.0266 0.4926 0.3848 0.0193

0.0137 0.0435 0.0809 0.6271 0.5469 0.0284

0.0054 0.0102 0.0267 0.4695 0.2496 0.0073

0.0130 0.0371 0.0504 0.5841 0.3902 0.0262

Table 14 Comparison of numbers of iterations for convergence among results obtained using different schemes of FLC at 100% loading with DPmax < e(=0.001), DQmax < e(=0.001) and random initialization (0:9 6 jVj 6 1:1and0:0 6 d 6 0:45). IEEE test systems

Number of iterations SFPF scheme

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus 11-Bus

Scheme I

Scheme II

Scheme III

ITRmin

ITRavg

ITRmin

ITRavg

ITRmin

ITRavg

ITRmin

ITRavg

11 11 10 D D D

12 12 13 D D D

10 11 11 11 13 13

11 12 13 12 15 14

10 11 12 11 13 12

11 13 13 12 15 14

10 11 12 11 13 13

12 12 13 12 15 14

Table 15 Comparison of times for convergence among results obtained using different schemes of FLC at 100% loading with DPmax < e(=0.001), DQmax < e(=0.001) and random initialization (0:8 6 jVj 6 1:2and0:0 6 d 6 0:6). IEEE test systems

Time for convergence (in seconds) SFPF scheme

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus 11-Bus

Scheme I

Scheme II

Scheme III

Tmin

Taverage

Tmin

Taverage

Tmin

Taverage

Tmin

Taverage

0.0086 0.0193 0.0274 D D D

0.0144 0.0439 0.1582 D D D

0.0080 0.0179 0.0632 0.5547 0.4451 0.0183

0.0131 0.0439 0.1181 0. 8719 0.9781 0.0288

0.0059 0.0103 0.0258 0.4906 0.3760 0.0194

0.0125 0.0425 0.0776 0.7271 0.5448 0.0286

0.0084 0.0065 0.0273 0.4567 0.2186 0.0172

0.0127 0.0341 0.0512 0.5541 0.3945 0.0267

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Table 16 Comparison of numbers of iterations for convergence among results obtained using different schemes of FLC at 100% loading with DPmax < e(=0.001), DQmax < e(=0.001) and random initialization (0:8 6 jVj 6 1:2and0:0 6 d 6 0:6). IEEE test systems

Number of iterations SFPF scheme

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus 11-Bus

Scheme I

Scheme II

Scheme III

ITRmin

ITRavg

ITRmin

ITRavg

ITRmin

ITRavg

ITRmin

ITRavg

12 12 12 D D D

13 13 14 D D D

10 11 12 11 13 13

12 13 13 12 15 14

10 11 12 11 13 12

12 13 13 12 15 14

11 11 11 11 12 13

12 12 13 12 15 14

Table 17 MLM obtained by the N–R technique (by variation of active and reactive load demands). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

4.0912 4.2038 4.1331 14.1015 41.4069

0.9918 1.1930 1.8405 3.7895 16.1890

5.1232 5.3968 5.2506 18.1223 51.1499

1.2420 1.5315 2.3381 4.8701 19.9983

5.5362 6.4773 6.2374 20.9789 62.6327

1.3421 1.8382 2.7775 5.6377 24.4877

5.5536 10.5113 8.3648 23.6177 118.6339

1.3463 2.9829 3.7249 6.3469 46.3827

Table 18 MLM obtained by the SFPF scheme (by variation of active and reactive load demands). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

3.2538 5.0179 5.1630 14.3694 41.7821

0.7888 1.4240 2.2991 3.8615 16.3357

3.6521 5.5910 5.8964 18.7495 51.9775

0.8854 1.5866 2.6257 5.0386 20.3218

4.5905 6.7894 7.2261 21.5623 64.2179

1.1128 1.9267 3.2178 5.7945 25.1075

4.9259 10.7407 8.8018 23.7529 121.8485

1.1942 3.0480 3.9195 6.3832 47.6395

Table 19 MLM obtained by Scheme I (by variation of active and reactive load demands). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

3.7488 5.7446 6.0336 14.5835 42.4441

0.9088 1.6302 2.6868 3.9191 16.5945

4.6678 7.3919 8.0089 20.2917 54.1034

1.1316 2.0977 3.5664 5.4530 21.1530

5.1628 8.9018 9.2133 22.5261 64.7696

1.2516 2.5262 4.1028 6.0535 25.3232

5.3245 11.6550 10.0380 24.0972 129.0978

1.2908 3.3075 4.4700 6.4757 50.4738

Table 20 MLM obtained by Scheme II (by variation of active and reactive load demands). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

3.7496 5.7475 6.0237 14.5847 42.4441

0.9090 1.6310 2.6824 3.9194 16.5945

4.6685 7.3919 8.0126 20.2917 54.1071

1.1318 2.0977 3.5681 5.4530 21.1544

5.1628 8.9029 9.2204 22.5337 64.7696

1.2516 2.5265 4.1059 6.0555 25.3232

5.3245 11.6550 10.0380 24.0972 129.0978

1.2908 3.3075 4.4700 6.4757 50.4738

Table 21 MLM obtained by Scheme III (by variation of active and reactive load demands). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

PL (p.u.)

QL (p.u.)

4.1504 5.7498 6.0137 14.5872 42.4450

0.9092 1.6317 2.6780 3.9201 16.5945

4.6695 7.3932 8.0146 20.3167 54.1402

1.1320 2.0981 3.5689 5.4598 21.1674

5.1628 8.9070 9.2247 22.5512 64.7696

1.2516 2.5277 4.1078 6.0602 25.3232

5.3245 11.6550 10.0380 24.1597 129.0978

1.2908 3.3075 4.4700 6.4925 50.4738

S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246

Fig. 10. Loadability point versus voltage magnitude graph for different IEEE test systems using different algorithms.

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S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246

Table 22 MLM obtained by the N–R technique (varying only active powers of both load and generation). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

4.5573 4.6959 3.8812 13.9463 41.4695

4.6761 4.9226 4.1085 14.2449 42.8461

5.5230 4.9114 5.0703 16.5826 48.8843

5.6670 5.1485 5.3673 16.9376 50.5071

6.0098 6.1015 5.9015 20.2053 61.8787

6.1664 6.3960 6.2472 20.6379 63.9329

6.2537 10.3142 9.5160 22.3133 115.2722

6.4166 10.8119 10.0734 22.7910 119.0989

Table 23 MLM obtained by the SFPF scheme (varying only active powers of both load and generation). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

4.4979 4.7918 3.9843 14.1053 41.7269

4.6151 5.0230 4.2177 14.4073 43.1121

5.0957 5.1756 5.1120 16.8955 50.0907

5.2285 5.4254 5.4114 17.2573 51.7536

5.9603 6.2569 6.1282 20.7060 64.1517

6.1156 6.5589 6.4872 21.1493 66.2813

6.0392 10.9536 10.0859 22.8541 118.8509

6.1965 11.4823 10.6767 23.3434 122.7964

Table 24 MLM obtained by Scheme I (varying only active powers of both load and generation). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

4.4939 5.6169 5.5569 14.1416 42.1756

4.6111 5.8880 5.8824 14.4443 43.5757

5.5272 5.9940 6.1231 17.0045 54.4050

5.6712 6.2833 6.4818 17.3685 56.2111

5.9940 6.2642 6.1580 21.0277 66.3953

6.1502 6.5665 6.5187 21.4779 68.5994

6.0717 10.9762 10.0896 23.0068 119.6049

6.2299 11.5059 10.6806 23.4994 123.5755

Table 25 MLM obtained by Scheme II (varying only active powers of both load and generation). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

4.4966 5.6804 5.6731 14.1541 42.3853

4.6138 5.9545 6.0054 14.4571 43.7924

5.5275 6.1787 6.2181 18.5241 54.6588

5.6716 6.4769 6.5823 18.9207 56.4733

5.9928 6.2813 6.2835 21.1454 66.9543

6.1490 6.5844 6.6516 21.5981 69.1770

6.0801 11.0070 10.1712 23.6490 120.9731

6.2385 11.5382 10.7670 24.1553 124.9891

Table 26 MLM obtained by Scheme III (varying only active powers of both load and generation). IEEE test systems

5-Bus 14-Bus 30-Bus 57-Bus 118-Bus

Case 1

Case 2

Case 3

Case 4

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

PL (p.u.)

PG (p.u.)

4.5791 5.7322 5.9848 14.2918 42.7899

4.6984 6.0088 6.3354 14.5978 44.2104

5.5362 6.2046 6.2748 18.6381 54.8463

5.6805 6.5041 6.6423 19.0371 56.6671

5.9961 6.2864 6.2892 21.2581 67.2486

6.1524 6.5898 6.6576 21.7132 69.4810

6.0644 11.0588 10.2630 24.0521 121.8595

6.2225 11.5925 10.8642 24.5670 125.9049

Table 27 The highest ‘mul’ values obtained by different techniques for different IEEE test systems. IEEE test systems

N–R method

SFPF scheme

Scheme I

Scheme II

Scheme III

5-Bus 14 Bus 30 Bus 57 Bus 118 Bus

3.5478 1.4981 1.7264 1.0523 1.0320

3.5978 1.5506 1.7498 1.0882 1.0470

3.5027 1.7102 1.7592 1.0984 1.0484

3.4309 1.7519 1.7643 1.1052 1.0495

3.6329 1.8164 1.7804 1.1245 1.0523

S. Mallick et al. / Electrical Power and Energy Systems 52 (2013) 231–246

(ITRavg) for all the different IEEE test systems are shown in Tables 14 and 16. From Tables 13–16, it is clear that the SFPF scheme converges for small IEEE test systems, but it cannot give any solution of SLFE with random initialization for higher IEEE test systems and ill conditioned 11-bus test system. On the contrary, all the proposed schemes I, II and III have converged for all IEEE test systems including ill-conditioned 11-bus test system, with lesser average convergence times (Taverage) than those of the SFPF scheme. From those tables, it can be inferred that Scheme III yields the least Taverage for all the IEEE test systems as compared to Scheme I and Scheme II. The ITRmin and ITRavg for all the methods in each test system are almost same as shown in Tables 14 and 16. The SFPF scheme takes more ITRmin and ITRavg than those of Scheme III for all IEEE test systems, as shown in Table 16. The Maximum Load Margins (kmax or MLM) determined using the N–R method and the same determined for different fuzzy schemes under different voltage limits and phase angle limits (as specified in Table 1) are shown in Tables 17–21. From the comparison of tables, it is shown that though the N–R method gives the highest MLM for IEEE 5-bus system, for other IEEE test systems the fuzzy FLC schemes are superior to the N–R method. Among the FLC schemes, the SFPF scheme has the least MLM values. Scheme III yields the highest MLM values for Case 1, Case 2 and Case 3 for all IEEE test systems. For Case 4, all three proposed schemes i.e. Scheme I, Scheme II and Scheme III, have almost same high MLM values for all IEEE test systems. This is quite obvious. Maximum load margin (MLM) is the loading value at which voltage collapse occurs while the active and reactive loads are increased simultaneously during simulation. In Fig. 10, the maximum load (ML) points are shown for different test systems using SFPF, the proposed FLC schemes of FLF and the N–R method. In order to check the robustness of the proposed schemes, the active power demand is increased in steps of 0.0001 keeping reactive demand unchanged. This increase of active power is shared by generators according to their participation factors. This method is applied to the proposed schemes as well as the N–R method and the SFPF scheme. The comparisons of the MLM values are shown in Tables 22–26. The SFPF scheme and all the proposed schemes provide higher MLM values for all systems than the N–R method except IEEE 5 bus test system. Scheme I, Scheme II and Scheme III have higher MLM than the SFPF scheme for all the test systems. Moreover from the comparisons, it is clear that proposed Scheme III yields the best MLM values for all systems and all cases and the other schemes. Scheme II has higher MLM values than Scheme I. In this paper, one more test for the check of the robustness of the proposed schemes has been performed. The high r/x ratio causes the standard system to move towards ill-conditioned system. The r/x ratio has been increased in small steps of 0.0001 using a multiplier ‘mul’ for all the IEEE standard test systems until the systems fail to give solution. The ‘mul’ is used as

r=x ¼ ðr=xÞ mul;

ð15Þ

The multiplying factors, after which the test systems fail to converge, have been noted and shown in Table 27. From the comparison, it is clear that the proposed Scheme I and Scheme II can perform well for higher r/x ratios than the N–R method and the SFPF scheme. The Scheme III can successfully converge and yield the highest ‘mul’ values, which clearly indicate the robustness of the proposed scheme.

8. Conclusion In this paper, three new schemes of FLC have been proposed with new constant array formations. The proposed schemes are

245

tested on IEEE 5-bus, 14-bus, 30-bus, 57-bus and 118-bus test systems extensively along with ill-conditioned 11-bus test system for different normal operating conditions to solve the load flow problem as well as to get the maximum load margins, number of iterations and convergence times. The following new features of the proposed schemes are responsible for better performance as compared to the SFPF. i. Novel formation of the sparse constant array [B1] and [B2]. ii. No differentiation as well as formation of Jacobian matrix or any other matrix in the proposed methods. Various test systems are chosen to establish the superiority of the proposed schemes over the N–R method and the previous algorithm SFPF. The other highlights of the paper are: (i) Lesser computational time is achieved than conventional methods; (ii) Higher load margin is attained with practical security constraints; (iii) random initialization is possible with proposed schemes and (iv) The proposed schemes can converge easily with high r/x ratios where conventional methods fail to converge. Finally, it may be concluded that the proposed FLC schemes are efficient techniques in solving the nonlinear power flow equations and providing maximum load margins. Appendix A. Newton–Raphson load flow The Newton–Raphson (N–R) method solves a set of simultaneous nonlinear equations fm(xN) = 0 for m = 1,2,3,. . .,n and N = 1,2,3,. . .,n. ‘n’ is equal to the number of buses. At a given iteration point, each function fi(x) is approximated by its tangent hyper plane. Then, the linearized problem is constructed as the Jacobianmatrix (J) equation:

FðxÞ ¼ J Dx

ðA:1Þ

It is then solved for the correction Dx. The elements of the Jacobian matrix are defined in (7).

J ik ¼

@fi @xk

ðA:2Þ

For power flow applications in polar power mismatch version, (4) and (5) can be written for the convenience of presentation in the following partial forms.



#   " H N Dd DP ¼ DjVj DQ M L jVj

ðA:3Þ

In (A.3), slack bus power mismatches and voltage corrections are not included and likewise DQi and DVi for each generator (PV) bus are absent. The sub-matrices H, N, M and L represent the partial derivatives of (4) and (5) with respect to the relevant d values and V values. Appendix B. Fast decoupled load flow The fast decoupled load flow (FDLF) is an approximate version of the N–R method. The first assumption under the FDLF is that real power changes (DP) are less sensitive to changes in voltage magnitudes (D|V|) and mainly sensitive to changes in voltage angles (Dd). Similarly, the reactive power changes (DQ) are mainly sensitive to D|V| and less sensitive to changes in Dd. With these assumptions, (A.3) can be written in the forms:

DP ¼ B0 Dd jVj

ðB:1Þ

DQ ¼ B00 DjVj jVj

ðB:2Þ

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