intelligent tap changer duty cycle control for load voltage improvement

ELSEVIER Electric Power Systems Research 43 (1997) I-10

ELECTRIC POUER SYSTErnS RESEClRCH

Intelligent tap changer duty cycle control for load voltage improvement

C.S. Chang *, J.S. Huang Department of Electrical Engineering, National Unioersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

Received 18 July 1996

Abstract

Voltage stability of a power system is related to load characteristics and voltage control devices such as on-load tap-changing (OLTC) transformers. This paper discusses the need for enhancing the OLTC operation, and identifies five control requirements for intelligent control of the OLTC duty cycle. The paper also describes an adaptive fuzzy logic controller (AFLC) developed for satisfying these requirements. By simulations on a six-bus study system, the ALFC is shown to be an effective means of enhancing the OLTC operation. The five control requirements are fulfilled by preventing the reverse actions, suppressing voltage transients, reducing control errors, and ensuring fast response and robustness. 0 1997 Elsevier Science S.A.

Kqwords: Voltage stability; OLTC duty cycle; Tap-change affectivity; Target voltage; Dynamic load characteristics; Adaptive fuzzy logic control; Load voltage improvement

-

1. Introduction

Voltage stability has become one of the most impor- tant power system research areas. A lot of effort has been focused upon the cause and mechanism of voltage instability, and corresponding counter-measures [l-5]. Voltage stability is essentially a dynamic phenomenon and is affected by voltage control and load characteristics under voltage fluctuations. Several models have been developed to describe load behavior in the steady and dynamic states [6-S].

On-load tap changing (OLTC) transformers have been extensively employed to provide medium-term voltage and reactive power control. OLTC as related to the voltage stability region was studied in Refs. [1,9]. Beyond the stable region, further tap changing can not prevent a voltage collapse, and this is known as reverse actions [3]. The unstable region of tap changing should be identified and avoided. Refs. [l,l l] derived the conditions, the mechanism and the model which govern reverse actions.

The choice of duty cycle affects the OLTC performance [lo, 111. Older OLTCs usually have a small num-

* Corresponding author. Tel.: + 65 772 6543; fax: + 65 7779 1103; e-mail: [email protected].

0378-7796/97:$17.00 Q 1997 Elsevier Science S.A. All rights reserved. PI1 SO378-7796(97)01153-X

ber of steps and a large step size. Flexible on-line control may not be feasible with this type of tap changer. Modern OLTCs usually have a larger tapping range (typically + 11, + 16, + 22%) and a smaller step size (typically 1.25, 0.625”/0) [12], and most of them can complete operation within seconds. However, to prevent excessive tap changes and ‘hunting’ during voltage variations, the duty cycle can be delayed up to 1 min. The OLTC duty cycle is usually fixed, since it was believed that individual OLTC operation has little system-wide effect. This paper shows the need to vary the OLTC duty cycle for providing sufficient recognition of inter-OLTC and load dynamics.

When the OLTC duty cycle becomes too fast for the load response time, undesired voltage transients can occur during tap changing. On the other hand, a slow tap changing gives rise to sluggish voltage control. Voltage instability can propagate to the rest of the power system and cause cascaded voltage collapses. Therefore, it is essential that each OLTC operates satisfactorily with sufficient reactive margin. This paper addresses these issues and proposes and adaptive fuzzy logic controller (AFLC) for each OLTC. In addition to suppressing undesired voltage transients during tap changing, the paper identifies four other requirements for the AFLC: to prevent local voltage instability

2 C.S. Chang, J.S. Huang / Electric Power Systems Research 43 (1997) I-10

caused by reverse actions, to achieve fast response, to reduce control errors, and to ensure robustness. By fulfilling these five requirements, voltage stability of the power system is enhanced.

The rest of the paper is organized into five sections. Section 2 models the voltage transients following each

Fig. I. Single-OLTC study system.

“053

(a) timetsecond~

I - ” * ““‘1

(b)

0.96’ J 0 200 400 600 800 1000 1200 ,400 1600 ,800 2000

tlnwrecmd)

(f)

Fig. 2. (a) Tap positions during tap changing (tap changing rate cc voltage error). (b) Load voltage variations during tap changing (tap changing rate x voltage error). (c) Reactive power demand variations during tap changing (tap changing rate ic voltage error).

0.96 [ I __.--..__ target value __.. : . . . .

0.95

I r----- .------------.--::.:---- //-

1

curve 0.84

(a) ‘,a 0.85 0.9 0.95 tap bmtion

t .05 1.1 1.15 1.2

cl= A

VS S M L

,vE CPU)

T S M L

oo’., (c) Fig. 3. Stable and unstable regions of tap changing. (a) Voltage errors index. (b) Tap change affectivity index. (c) Load characteristics index.

tap change. By studying a single-OLTC model, Section 3 provides further insight into the effects of varying the duty cycle. Section 4 outlines the ALFC scheme for fulfilling the five control requirements as mentioned above. The six-bus-OLTC Ward and Hale system is used in Section 5 for verifying the proposed AFLC design. Section 6 contains the concluding remarks of the paper.

2. OLTC mathematical modeling

Simulation of OLTC operation involves modeling of the connected power system, the OLTC control mechanism, and dynamic load behaviors during tap changing.

2.1. System reactive power Jlobv equations

The system reactive power flow equations provide snap-shot simulations of the connected power system during tap changing. For a power system with N buses

C.S. Chang, J.S. Huang / Electric Power Systems Research 43 (1997) l-10

LIlQQQC.,, (cl Fig. 4. Membership functions of the inputs of the fuzzy logic controller.

and n OLTC transformers, the reactive power flow equations at the kth tap change can be expressed as:

Q( V(k), Q,, a(k)) = 0 (1)

where Q = [Q,, Q,,. .., Q,y]T represents the set of bus reactive powers, V(k) = [V,(k), V,(k),. . ., &(k)]’ is the vector of bus voltages, Qs = [es,, QsZ,. . ., Qs,V]T is the vector of reactive demands, and a(k) = b,(k), a,(k),...> a,(k)]= is the vector of tap positions. Linearising Eq. (1) around (V(k), a(k)) we have:

(2)

BY extracting the controlled voltages v, = [V,, I’, ,..., V,IT from Eq. (2), it follows [lo]:

V,(k + 1) = V,(k) + S(k)[a(k + 1) - a(k)] (3)

where S is an n x n square matrix evaluated from ZQ/dV and dQ/Za.

2.2. OLTC control mechanism

Each tap change is performed independently. When the voltage of the controlled bus varies beyond a specified range, tap changing is initiated automatically to correct the deviation. The control law can be expressed by the following discrete logic [10,13]:

ai(k + 1) = a,(k) + &J;( V,(k) - Vfin), i = 1, 2,.. ., n (44

Table 1 Rulebase 1

vs VE

vs s M L

S X X X X M VL L S vs L VL M vs vs

Table 2 Rulebase 2

LC TN

vs S M L VL X

vs vs vs vs vs vs X S vs vs S M L X M vs S M L VL X L vs S M VL VL X VL vs M L VL VL X

i

1, V(k)i- p > dV, A( V(k), - fl”) = 0, (V(k)iv\ I dVi (4b)

- 1, V(k)i- vf”< -av;

where R, is the step of tap-changer i, aV, is the threshold effecting the control action, and e” is the specified target voltage. The step size %i remains constant for all tap changes. The average tap-changing rate a = f &./ Td = f c is fixed by a constant tap-changing duty cycle Ta

Replacing each a,(k + 1) - a,(k) in Eq. (3) with Eqs. (4a) and (4b), we finally obtain the equation describing the dynamics of the controlled voltages:

Vi(k + 1) = I’,(k) + i S&( Vj(k) - y) j=1

(54

or in vector form:

V(k + 1) = V(k) + Af( V(k) - V’“) (5b)

In Eqs. (5a) and (5b), S, represents the (ij)th ele- ment of the matrix S and A = SA with A = diag(/i,, ,X2,. . ., A,). The above difference equations describe the OLTC dynamics. Eqs. (5a) and (5b) shows that each tap change is performed independently. The improvement of each controlled voltage Vi due to the variation of ai is SijAa,, and due to the interaction from other OLTC transformers is C;= r,,+, S,Aa,. Ref. [l l] derived the sufficient condition for each controlled voltage to reach its target value: the matrix A in Eq. (5b) must be diagonally dominant.

VS S M L VL X

-mm 3 (second)

5 15 25 35 45 55

Fig, 5. Membership function of intermediate and final output of the FLC: TN and TD.

C.S. Gang, J.S. Hung :‘Electric Powrr Systems Research 43 (1997) l- 10

I I two-stage fuzzy logic reasoning

decision mdhg logic

rule base 1 rule base 2

SCADA

identified load parameters

fuzzikatim VE

defuzzification L LC interface

Td

Fig. 6. AFLC block diagram.

2.3. Dynamic loud modeling

The dynamic load behaviors moderate the voltages at each connected bus during tap changing. The following dynamic load model is used:

gc(QDi, QDi> PI, V,) = 0. i = 1, 2 ,..., N (64

Q( V, Q,: 4k + 1)) = 0 (6b)

where g[(&, QDi, p!, Vj) = 0 is a first-order dynamic model used to describe the load performance during voltage transients [6-S] and QD = [Q,,,, Qb,,..., QD,V]T is the vector of dynamic reactive demands.

In the above load model, the load time constant can vary over a wide range from fractions of a second to tens of minutes. To avoid hunting and other undesir- able effects, the OLTC duty cycle should be adaptive in order to fulfil the five control requirements as mentioned earlier on. The following section represents two case studies on a single-OLTC power system, which highlight the needs for fulfilling the five control requirements.

Fig. 7. Multi-OLTC Ward-Hale system.

3. Voltage performance on a single-OLTC power system

A single-OLTC system is shown in Fig. 1. The load is connected to the controlled bus, and the rest of the power system is represented by a supply bus whose reactive reserve is restricted.

3.1, Voltage transients and control errors of single-OLTC operation

Fig. 2aac show the simulation results of the single- OLTC system. Initially, vs = 1.03 pu, I@” = 1 pu, the load voltage suffers a dip to 0.963 pu caused by a load increase. Tap changing is automatically initiated (Eqs. (4a) and (4b)). Fig. 2a-c show the simulations for three values of the OLTC duty cycle T, (Curve 1~ Td = 3T,; Curve 2- Td = To; Curve 3- Td = T43). T, is the time constant of the reactive demand connected to the system [6-81. Besides TQ, the other load parameters cP, cQ, x and p are respectively 1, 0.3, 1.2 and 0.8. Apart from inducing dynamic voltage variations, a small value of Td also causes control errors as shown in Curve 3 of Fig. 2c. On the contrary, a large Td leads to a sluggish system response (Curve 1 in Fig. 2aac).

3.2. Stable und unstable regions of single-OLTC operution

With a very heavy load before the voltage distur- bance, the problem of dynamic voltage variations and control errors can further lead to voltage instability. Fig. 3 shows the two voltage responses during tap changing for the single-OLTC system with a very large Td and a very small Td.

Curve 1 corresponds to Td >> T,, which returns the study system back to steady state before the next tap change. Tap changing is carried out continuously with an initial load voltage of 0.896 pu and an initial tap

C.S. Chang. J.S. Huang /Electric Powrr Systems Research 43 (1997) l-10

Table 3 Three case studies

Case Control” Reaction load Initial tap posi- Initial voltage

CPU) tion CPU) Target voltage

(Pul

Final voltage

CPU)

No. of tap-change opera- tionsb

I Q, = 1.80 Q2 = 2.70 Q3 = 0.00

II Q4 = 0.05

1 Q, = 8.30 Q2 = 0.18 Q, = 0.00

11 Q4 = 0.05

I Q, = 8.30 Q2 = 1.80 Q3 = 0.00

II Q4 = 0.05

a, = 0.900 az = 1.075

a, = 1.100 a2 = 1.025

a, = 1.100 a, = 1.025

C’, = 1.1206 V2 = 0.8932 V, = 1.0064 V4 = 0.9782

V, = 0.8978 v* = 0.9343 Vx = 1.0036 V4 = 0.9820

V, = 0.7385 T/z = 0.8701 v, = 0.8312 V4 = 0.8756

by” = vy = 1.0000

fl” = p””

= Loo;0

f.qn = V!j”

= 1.0000 vyn = qJ

= 1.0000

I”” - I - q’” = 0.95

vp = G” = 0.95

v, = 1.0007 v, = 0.9987 v3 = 1.0104 If‘, = 0.9456 v, = 1.0007 VI = 0.9987 v, = 1.0104 Vd = 0.9456

v, = 1.0022 Vz = 0.9968 V3 = 0.9615 V, = 0.9308 v, = 1.0003 v2 = 1.0004 v, = 0.9594 V1 = 0.9264

Voltage collapse

V, = 0.9467 V2 = 0.9458 VT = 0.7664 V4 = 0.7526

22

13

39

15

67

26

d Control I: fixed duty cycle, Control II: adaptive duty cycle. b A tap-change operation may involve either a single OLTC or both OLTCs.

position of 0.8. The target load voltage is 0.95 pu. For each point on Curve 1, the tap is updated by the constant step /. from Eq. (4a). As shown in Curve 1 of Fig. 3, the slow tap changer achieves the control target voltage.

On the contrary, Curve 2 is obtained by adopting a very fast r,( << To), not allowing the system to reach the steady state before the next tap change. Due to heavy fluctuations of reactive demand, a very fast tap- changing rate is seen to have caused instability (Curve 2) during tap changing. The load voltage cannot achieve its target value, which causes the tap changer to further increase the tap. Since no mechanism is in- cluded to curb such ‘uncontrolled’ tap changing, the system is finally pushed into instability as shown in Fig. 3.

Curve 2 of Fig. 3 has two regions of operation. In the stable operating region which is on the left-hand-side of the critical point a,, Av(ik)/Au(k) is positive. Beyond the critical point a,, a tap increase Au leads to a load voltage reduction, resulting in reverse actions in the unstable region of tap changing.

The above analysis can be extended to large systems. For each OLTC transformer, interactions from the rest of the system impose a limit of reactive reserve. With uncontrolled draining of reactive reserve, the load voltage will collapse when the tap is increased beyond

the critical point ~1,. This local instability can propagate to other parts of the system. In the following section, the AFLC is proposed to ensure satisfactory performance of each OLTC transformer and to improve stability of the entire system. It is known that tap changers at distribution and generation levels are oper- ated under slightly different control objectives. For global voltage stability enhancement, coordination of tap changers at different levels may be desired, which would be addressed in another paper.

4. Duty cycle adjustment by adaptive fuzzy logic control

In the previous sections, we have identified the needs for using adaptive OLTC duty cycle. A PID controller can be designed to perform the adjustment. However, tuning of a PID controller is difficult to cater for all the different operating conditions. Consequently, an ALFC is developed. The following desired control environ- ment is assumed: 1. Intelligent control tap-change duty cycle can be

implemented 2. Dynamic performance of loads can be determined

and modeled 3. On-line computers and control facilities are avail-

able

C.S. Chang, J.S. Huang/Elecrric Power Systems Research 43 (1997) l-10

1.02

1

0.99

0.96

Vl

(PU)

0.94

0.92

0.9

Fig. 8. OLTC performance for small reactive demand increase.

4.1. Requirements of the proposed AFLC

To recap, the objective of the proposed control scheme is to ensure stability of tap changing. In doing so, five control requirements have been identified during tap changing: 1. to prevent reverse actions 2. to achieve fast response 3. to suppress dynamic voltage variations 4. to reduce control errors 5. to ensure robustness

4.2. Choice of control variables

Three performance indices are used for representing the current operating conditions of the connected system and load characteristics, for adjusting the OLTC duty cycle: 1. control distance between the current operating point

and the target voltage; 2. tap-change affectivity for each tap-changer (as an

estimation of local voltage stability margin; 3. dynamic characteristics of the power system loads.

Using local information, the first index is readily available in the form of a voltage error:

VE, = 1 vfi” - I$@)\ (7)

Using the first index at successive time intervals, the second index is calculated by:

V&=VE,-,-VE, (8)

Dynamic characteristics of the load are described by the third index:

LC = co,\/3i - 21& (9)

where Coi, pi, TQi are parameters of the dynamic reactive demand located at bus i [6], with the coefficient Coi giving the magnitude of the reactive demand under unit voltage, the power exponent pi indicating the load type, and the time constant TQj reflecting the load response rate. pi is either smaller or equal to 2. pi is 2 for a pure reactance or static load and is 0 for a constant power load. As an assumption of our system design, these load parameters are identifiable.

For each OLTC, the above three quantities VS, VE and LC are used to represent the current state of the

C.S. Chang, J.S. Huang / Electric Power Systems Research 43 (1997) I-10

1.02

1

0.96

0.96

VI

(PU)

0.94

0.92

0.9

0.68 I

200 I I I I

4co 600 600 loo0 1

time (second) 0

Fig. 9. (See ouerleaf for legend.)

connected load. They are expressed in membership functions in the form shown in Fig. 4.

4.3. Fuzzy rulebase design for the OLTC duty cycle control

The AFLC is designed to fulfil the five control requirements as listed in Section 4.1, and the following rules of thumb have been adopted: 1. When the tap-change affectivity is detected smaller

than a preset value, any further tap changing must be prohibited.

2. When the tap-changer is working with a large affectivity a small Td is preferred, and vice versa.

3. A fast tap changing (small Td) is performed when the load voltage is far away from the target value. With the operating point approaching to the target, the tap changing is gradually slowed down by in- creasing Td.

4. The duty cycle Td is adjusted against load parameter changes. When the load connected to the OLTC bus has a large and slowly decayed dynamic component (a large index as given by Eq. (9)), a large Td is adopted, and vice versa.

In short, Rule 1 curbs the reverse actions. Rules 2 and 3 make trade-offs between fast tap changing and voltage is far away from the target value, a fast tap changing is preferred so as to speed up the control, because the power system can now sustain a large impact caused by tap changes. When the power system becomes weaker and/or the controlled voltage gets closer to the target, the tap changing will be slowed down to suppress the dynamics and prevent the possi- ble control errors due to th over-control. The final rule 4 improves robustness of the AFLC against load parameter changes.

From the above general rules, two rulebases are designed. By assuming an ‘average’ load type and condition, rulebase 1 (Table 1) estimates an intermediate value of the duty cycle TN. To account for a change of load characteristics, rulebase 2 (Table 2) modifies the first estimate of the duty cycle TN by mapping TN*LN -+TD using a scaling factor LN. In the two tables, the symbols VS, S, M, L and VL represent respectively the linguistic levels of very small, small, medium, large and very large. The symbol X means the respective tap changer is prohibited from further operation. Fig. 5 shows the membership functions of TN and TD.

C.S. Chung. J.S. Huang/Electric Power Systems Research 4.3 (1997) I- IO

1

0.98

0.98

0.94

0.92

0.9

&

0.88

0.88

0.84

0.82

0.8 0

l-

l- .7

I I 1 I I

target of V, ;

target of VZ

I I I , I

0.75 0.8 0.85 0.9 0.95 1

VI (PM

Fig. 9. OLTC performance for large reactive demand increase. (a) Fixed-duty-time OLTC. (b) Tap change with ALFC.

Using these two separate rulebases, both the computer memory and the processing time would be greatly reduced as compared with the use of one general rulebase containing VS, VE and LC as the input variables. Fig. 6 gives the schematic layout of the AFLC. De- fuzzification of the control output (TJ is performed with the center of gravity method. The final crisp value is the duty cycle T, for the current operating cycle. It is seen from the two rulebases that whenever the tap change affectivity (an estimation of the local stability margin) is detected smaller than the preset value (see Fig. 4(a)), any further tap changing will be prohibited.

5. Results and discussions

The six-bus two-OLTC Ward-HALE system [IO,1 I. 141 is used to verify the AFLC. Fig. 7 shows the one-line diagram of the system, which contains the two OLTC controlled voltages V, and V,. Although the test system used in the paper is a small example with special structure, the control philosophy proposed can be applied to tap-changers of a large-scale power system with complicated configuration. Using the dynamic load

model of Ref. [6], all the loads are assumed to be constant P and Q with time constants of 15 s (mid-term voltage stability is concerned). The nominal value of T,, is 8 s, which is adjusted by the AFLC within a range from 5 to 55 s.

Altogether, three case studies are presented to demonstrate the performance of the ALFC.

Case 1 shows the tap-position and voltage oscillations caused by inappropriately fast tap changing. Fol- lowing the step reactive demands as shown in Column 3 of Table 3, the two generators increase their excita- tions to Ys = 1.10 pu and V, = 1.07 pu. Initially, both OLTC controlled voltages V, and Vz have not been well regulated (Column 5 of Table 3). According to Eqs. (4a) and (4b), tap changing is activated. With a fixed duty cycle, both the V, and V, undergo a large number of oscillations before reaching their target values (Fig. 8). With the AFLC, both V, and VI change monotonically towards their target values smoothly with a much lesser number of tap changing operations (Fig. 8, column 8 of Table 3).

In contrast to Case 1, heavier reactive demands are imposed in Case 2. The generator excitation at Bus-6 is further increased to V6 = 1.10 pu with Vs = 1.1 pu.

C.S. Chug, J.S. Huang /Electric Power Systems Research 43 (1997) I- 10

0.98

0.96

0.94

0.92

0.9

&

0.88

0.86

0.84

0.82

0.0 0

I-

.7

I I I I ,

I

0.75 I

0.8 I

0.85 VI CPU)

I I

0.9 0.95

Fig. IO. OLTC performance for very large reactive demand increase and inadequate generator excitation

Following the generator excitation increase, tap chang- adopts a two-stage reasoning which greatly reduces the ing is performed. With a fixed duty cycle, V, overshoots processing time and the computer memory as needed significantly (Fig. 9). With the AFLC, excursions of V, for a single-rulebase scheme. The simulation results are greatly reduced. A similar response is found in VI. verify the performance of the proposed ALFC.

With even heavier reactive demands than in Case 2, the generator voltages in Case 3 are increased by V, = 1.05 pu and V, = 1 .l pu The simulation shows a voltage collapse under the fixed-duty-cycle tap changing (Fig. 10). By using the AFLC, both the V, and Vz achieve their low target values in a stable operation.

The developed control scheme is applied to a rela- tively simple two-OLTC power system. However, the study system is of sufficient complexity to demonstrate the feasibility for applying the proposed control to large-scale power systems.

6. Conclusions

By considering the single-OLTC dynamics, an ALFC has been developed to meet the OLTC requirements by preventing reverse actions, suppressing extensive voltage dynamics, eliminating control errors, achieving fast response, and ensuring robust performance. Using the ALFC, the OLTC duty cycle is adjusted in accor- dance with the current operating states of the power system and the connected load characteristics. By doing this, stable tap changing is achieved, and the operating range of the OLTC is extended. The proposed AFLC

Acknowledgements

The authors would like to thank Dr. K.S. Lock and Professor A.C. Liew for their encouragement and sup- port. They would also like to thank Mr. C. Duan for the use of his computer program for dealing with non-linear loads in network solution.

References

[l] C. Liu. K.T. Vu, Analysis of tap-changer dynamics and con- struction of voltage stability regions. IEEE TrdnS. CAS 36 (4) (1989) 575-590.

10 C.S. Chang, J.S. Huang /Eiectric Power Sysiems Research 43 (1997) l-10

[2] N. Yorino, H. Sasaki, Y. Masuda, Y. Tamura, M. Kitagawa, A. Oshimo, An investigation of voltage instability problems, IEEE Trans. Power Syst. 7 (2) (1992) 600-611.

[3] H. Ohtsuki, A. Sasaki, Y. Masuda? Reverse action of on-load tap changer in association with voltage collapse, IEEE Trans. Power Syst. 6 (1) (1991) 300-306.

[4] W. Taylor, Power System Voltage Stability, McGraw-Hill, 1994.

[5] B.H. Lee. K.Y. Lee, A study on voltage collapse mechanism in electric power system, IEEE Trans. Power Syst. 6 (3) (1991) 966974.

[6] J. Hill, Nonlinear dynamic load models with recovery for voltage stability studies, IEEE Trans. Power Syst. 8 (1) (1993) 1666176.

[7] D. Karlsson, D.J. Hill, Modeling and identification of nonlinear dynamic loads in power systems, IEEE Trans. Power Syst. 9 (1) (1944) 157-166.

[8] W. Xu, Y. Mansour, Voltage stability analysis using generic

dynamic load models, IEEE Trans. Power Syst. 9 (1) (1994) 479-493.

[9] Y.Y. Hong, H.Y. Wang, Investigation of the voltage stability region involving on-load tap changers, Electr. Power Syst. Res. 32 (1995) 45554.

[IO] K. Lim, Decentralized control of under load tap-changer in power systems, Int. Conf. Power Engineering, Nanyang Techno- logical University, Singapore, March 1993, pp. 31-36.

[I I] J. Medanic, M. I.-Spong, J. Christensen, Discrete models of slow voltage dynamics for under load tap-changing transformer coordination, IEEE Trans. Power Syst. PWRS-2 (4) (1987) 8733882.

[12] A. Laughton and M.G. Say, Electrical Engineer’s Reference Book, Butterworth Heinemann, 1994.

[13] M. Llic, New approaches to voltage monitoring and control, IEEE Cont. Syst. Mag. January (1989) 5-l 1.

[14] A. Pai, Computer Techniques in Power System Analysis. Tata McGraw-Hill, 1979.

intelligent tap changer duty cycle control for load voltage improvement

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