Three-phase active power line conditioner planning

Y.-Y. Hong

Y.-T.Chen Y.-L. HSU

Indexing terms: Active power.filters, APLC, Harmonic reduction, Optimisation, Three-phase

Abstract: The active power line conditioner (APLC) is considered to be a very efficient type of active power filter that compensates the voltage waveform distortion caused by nonlinear loads in power systems. A new approach is based on a previously presented two-level algorithm, the three-phase optimal harmonic power flow (TOHPF), to investigate the determination of locations and sizes of three-phase ( 3 4 APLCs. The objective is to minimise the new APLC investment cost, the 34 voltage distortion, and the system 34 MW losses while satisfying the 345 power flow equations, 345 harmonic power flow equations, 3q5 security constraints, and the harmonic standard. The master level of TOHPF determines the compensatoritap settings and the new APLC locationsisizes; the slave level of TOHPF includes subproblems which involve cases of fundamental and harmonic frequencies separately. Test results for an 1 %bus distribution system show the applicability of the proposed method.

1 Introduction

The power quality of a power system is influenced by harmonics generated from the power electronic circuits. Active power filters are considered to be the most effi- cient devices for reducing voltage distortions caused by harmonics [ 1-10]. However, installation of active power filters in a power system is a very sophisticated prob- lem; factors involved include the number of nonlinear loads (single or multiple harmonic sources), harmonic standard, locations, size and investment cost of active power filters, existing controller settings, network struc- ture, and balancediunbalanced linearinonlinear loads

The active power line conditioner (APLC) is pres- ently considered a very efficient active power filter for reducing harmonics [5-lo]. Grady and Chang pre- sented methodologies [ b l 01 based on optimisation methods to investigate the installation of one or more APLCs in a power system to reduce the voltage distor- tion caused by harmonics. Only one APLC can be

[5-lo].

0 IEE, 1998 IEE Proceedings online no 19981941 Paper first received 26th August 1997 and in revised form 27th January 1998 The authors are with the Department of Electncal Engineering, Chung Yuan University, Chung Li 320, Taiwan

IEE Proc.-Gener. Tvansm. Dbtrib., Vol. 145, No. 3, May 1998

identified from a set of candidate buses to reduce the voltage distortion via two subproblems in [8]; moreo- ver, the solution may only be near optimal. In [9], the installed locations for multi-APLCs were determined by a set of re-evaluated indices for all candidate buses because the smallest feasible size of one APLC is con- sidered at each iteration. Chang in [9, 101 also admitted that the algorithm presented in [9] does not necessarily obtain the optimal solution (locations and sizes of APLC) if the multiple-harmonic solution procedure is invoked. Nonlinear software based on the generalised reduced gradient algorithm is used to find the optimal size and location of one APLC to reduce harmonics in [lo]. Although the solution is optimal, only one loca- tion can be installed with the APLC in [lo]; moreover, only a continuous size (not discrete) is obtained because the generalised reduced-gradient algorithm can deal only with the APLC size as a continuous variable. Chang and Grady proposed a new algorithm consider- ing multi-APLC [ll], however, it does not make the most of the existing compensators before APLC plan- ning. The method presented in [5] has the same defi- ciencies as presented in [l 11.

On the other hand, the power system is generally unbalanced, especially the distribution system in which harmonic pollution is severe. The imbalance may be likely to be caused by untransposed transmission lines, unbalanced loads, electrified railroads, and asymmetri- cal faults, etc. In this paper, the harmonic factor, an important but seldom noticed imbalance, is addressed.

To improve the methods reported in [5-111, an enhanced method based on the authors’ previously pre- sented three-phase optimal harmonic power flow (TOHPF) [12] is proposed to determine the locations and sizes of multiple APLCs for reducing the harmonic voltage distortions caused by multiple harmonics in three-phase unbalanced systems. TOHPF is designed to co-ordinate control settings for cases involving both fundamental frequency and harmonics simultaneously. The motivation for designing a TOHPF is the fact of the contradictory effect in controller settings between the fundamental and harmonic cases [12, 131. The gen- eralised Benders decomposition theory (GBDT) [ 141 is used in designing the TOHPF. The proposed method has the following main features: (i) The APLC investment cost included in the objective function is minimised. (ii) The marginal cost for each APLC candidate bus is generated from a nonlinear software package. The identified locations determined by the marginal costs for installation may be varied in each iteration to obtain optimal locations; however, the previously iden-

281

tified locations cannot be altered for the later iterations in Chang’s and Grady’s methods [8, 91. (iii) Although the injection currents from APLCs are continuous, the sizes of APLCs are discrete. The reason is that the DC source in the APLC is implemented by capacitors or inductors [lM] which are discrete devices. (iv) The proposed algorithm based on our previous study [12] makes the most of existing controllers, e.g. 34 compensators and 34 taps, to reduce 34 voltage dis- tortions for obtaining a minimal APLC cost. Existing controller settings also ensure the 34 system security constraints and the 34 power flow equations for the fundamental frequency. (v) Settings of compensators and taps are discrete while the system states, e.g. bus voltages and harmonic volt- ages, are continuous. (vi) 34 models for the system components are consid- ered to study the imbalance.

2 Problem formulation

(4)

Qc E Q tap E T (7)

IJ E Ia ( 8 ) Fo denotes the system three-phase (34) MW losses, F, represents the square of the total 34 harmonic distor- tion (TTHD) for all buses and all harmonics, FJ is the 34 investment cost of APLCs. The symbols p and q are the weighting factors. If engineers are interested only in the minimisation of MW losses, the weighting factor p is zero and only the harmonic standard is required to be satisfied. Also, the value of q is greater than that of p, i.e. the installed amounts of APLCs barely satisfy the harmonic standard. The subscripts ‘Oih’ represent the cases for fundamental/harmonic frequencies, respectively; the set H represents the harmonic orders (..., h, ..., H ) concerned. The vector Ho signifies the 34 power flow equations; the vector Hh represents the 34 harmonic power flow equations; the vector Go includes the system 34 security constraints; the vector Gh includes 34 inequality constraints for the harmonic standard. The vector of the state variable xo includes 34 voltage magnitudes for PQ buses and angles for all buses except the reference bus; the vector of the state variables xh consists of 34 harmonic voltage magni- tudes and angles. The vectors of the 34 generator volt- age magnitudes (Vg), 34 compensators (Qc), 34 transformer taps (tap) and 34 APLC injection current (IJ) are the vectors of control variables. The symbols Q, T and Ia are the sets including the 34 discrete varia- bles for Qc, tap and IJ, respectively. The subscript J represents the set of candidate buses ( ..., j , ... J) for

282

installing 34 APLCs. Assume that the resonance will not occur in the sets of Q and T (the systems were well planned); moreover, the power factors can be satisfied in the set of Q. The detailed formulation and descrip- tion of eqns. 2-5 is referred to [12].

A 34 active power filter can be implemented to be three individual single-phase APLCs or a three-phase APLC. This paper uses the model for three individual single-phase APLCs because this type is more well developed than the other. More specifically, for one phase of a candidate bus j ,

r i 0.5 r 1 0 . 5

(9) where I$ is the total 14 APLC injection current for har- monic order h at candidate bus j , and Gh‘ and Ghi are the corresponding 14 real and imaginary parts of the APLC injection currents for the harmonic order h at candidate bus j . Note that 4 is a discrete variable but I jh , Ijh’ and Ijhi are continuous variables.

Solving eqns. 1-8 is time consuming and difficult, because the elements of xo, xh, Vg, lib, ljh‘ and Ghi are continuous variables while those of Qc, tap and I j are discrete variables; there are ( H + 1) sets of equality and inequality constraints; and eqns. 2-5 are coupling. The formulation of eqns. 1-8 is a large-scale mixed-integer programming problem, and a direct method for solving the continuous and discrete variables simultaneously is unacceptable because of the computation cost. To cope with these difficulties, an extended method based on the author’s previous work, TOHPF [12] using GBDT, is proposed in this paper. A general description of the GBDT is given in Section 3.1. Some important steps in applying the algorithm based on the GBDT are pro- vided in Section 3.2.

3 Proposed method

3. I General description The generalised Benders decomposition theory provides an algorithm, including master and slave levels, to solve a mixed-integer programming problem. The upper level, i.e. the master level, determines the control variables. The effectiveness of these control variables are verified at the lower level, called the slave level, by their marginal costs. The marginal costs of the con- straints for the corresponding control variables and other information are delivered to the master level for determining the next new values of the control varia- bles. The master and slave levels interact in this way until convergence. The GBDT becomes more attractive when a large number of constraints coupled at the slave level can be solved separately in many subprob- lems with fixed control variables [14].

On the basis of the GBDT, eqns. 1-8 can be solved at the master and slave levels. The master level deter- mines the control settings of Qc, tap, +, and 4. The slave level verifies the effect of fixed Qc, tap and ijh determined in the preceding master level to obtain their marginal costs from all cases for the next master-level solution process. The elements of Vg are used as con- trol variables only in the base cases with the fundamen- tal frequency. The master and slave levels are solved iteratively until convergence. Fig. 1 illustrates the inter- action of these two levels. The problem formulations

IEE Proc -Gener Tvansm Distvrb , Vol 145, No 3, May 1998

and methods of solutions for both of the master level and the slave level are provided in the Appendix (Sec- tions 7.1 and 7.2).

min MW losses s.t. power flow eqns.

and security constraints with Qc and tap fixed but

Vg varying

master Level minimises APLC investment cost and deals with vectors of Qc,

tap, I j h for al l harmonics and I,.

..'

and tap

value and maroinal

min sqrd. h-th

harmonic standa,rd and

harmonic power flow eqns.

THD Si. h- th ..

Fig. 1 Interactions of two levels

min sqrd. H-th

harmonic standard and

harmonic power flow eqns.

THD S.t. H- th

3.2 Algorithm

3.2.1 Convergence criterion According to the GBDT, the objective value 2

(eqn. 10) as shown in Section 7.1 is the lower bound for the original problem eqns. 1-8 because the con- straints are relaxed at the master level. The sum of all objective functions for all subproblems at the slave level is referred to as the upper bound for the original problem eqns. 1-8 because the control variables Qc, tap, and Ijh are fixed at the slave level. If the upper bound approximates the lower bound in an acceptable tolerance, the iterative solutions are convergent. Detailed information concerning the convergence crite- rion is available in [12, 141.

3.2.2 Solution steps: The original problem eqns. 1- 8 should be reformulated by eqns. 10-15 and eqns. 16- 21 as shown in the Appendix (Sections 7.1 and 7.2) when the GBDT is used. The steps in applying the algorithm for the solution of this problem are described as follows: Step 1: Estimate the initial Qc, tap, Gh and 4. Let the lower bound = -9999.9. Step 2a: Solve the subproblem eqns. 16-18 for deter- mining the Fo, Vg, xo and the marginal costs corre- sponding to Qc and tap. Step 26: Solve the subproblems eqns. 19-21 for deter- mining FA, xh and the marginal costs corresponding to Qc, tap and rjh (for every h E H). Step 2c: Add Fo and Fh (all h E H) for obtaining the upper bound. Step 3: If the upper bound is close to the lower bound, output the solution and stop; otherwise go to step 4. Step 4: Generate a cut like eqn. 11 to the master level if all subproblems have optimal solutions, or generate a cut like eqn. 12 to the master level if any one of the subproblems has no solution. Step 5: Solve the master level eqns. 10-15 for obtaining the lower bound Qc, tap, rjh and 4. Step 6: Send the new Qc, tap and h,, determined in step 5 to the slave level. Go to step 2.

IEE Proc.-Gener. Transm. Distrib., Vol. 145, No. 3, May 1998

4 Simulation results

Harmonics exist generally in distribution systems and industrial power systems. In distribution systems the total distributed load on each phase of a line section is often lumped half-and-half at the two end nodes of the line section for simplicity [15]. This simplification reduces the dimension of the system studied. Also, the harmonic effect attenuates very quickly with local char- acteristics [16, 171. For these reasons, only small sys- tems are used for studying the harmonic power flows. Moreover, when the models of three-phase systems are fully considered, the systems are further reduced to a few nodes (fewer than 10) for special observations [18, 191, e.g. imbalance and three-phase transformer con- nections.

In this Section a metal factory composed of 18 plants (buses) with 17 main transformers is used to serve as an example for showing the applicability of the proposed method. The system parameters are reported in [13]. A one-line diagram for this power system is shown in Fig. 2. The most serious harmonic pollution in this system arises from six-pulse ACiDC power converters which are included in many plants. These harmonic sources are at buses 4, 6-8, 10-14, and 16 as shown in Fig. 3. Generally, harmonics below the 26th are con- cerned. For a six-pulse harmonic source, this means there are nine subproblems. Without loss of generality, the first two harmonics are considered because the dominant harmonics in the system are of orders five and seven.

from uti l i ty ,6, kV 1 I

I

18 Q 161133kV e

33/11

Fig.2 One-line diugram for 18-bus system

0.OL

2 0.03

0.02

0.01

0 & 6 7 8 10 11 12 13 14 16

bus

0 5th 7th Fig.3 tions

Fqth and seventh harmonic currents ofphuse a at harmonic loca-

The simulation results address two points: the unbal- anced harmonics caused by nonlinear loads, and the

283

different three-phase transformer connections at load buses. Buses 4, 6, 8, 9, 10, 12, 16 and 17 are considered to be candidate buses for installing APLCs. The magni- tudes of the harmonic voltages for the fifth and seventh are constrained within 0.03 and 0.01 p.u., respectively, and the THD for each phase of each bus is restricted within a range of 3%. The voltages of the base case with the fundamental frequency are constrained within [0.95, 1.05lp.u. The unit sizes of APLCs and shunt capacitors are 0.01 and O.Olp.u., respectively, and the step size for taps is 0.01. The mutual impedances between phases are assumed to be 5% of the self impedances for all of the transmission lines and trans- formers.

0.4 ' 0.3

0.2 0.1

0

2

6 a 10 16 17 bus

C I A aB C Fig. 4 tions

Three-phase capacitors at buses with Y-Y transformer connec-

4. I Three-phase transformer with Y-Y connection It is assumed that the system is balanced except that the nonlinear loads are unbalanced; phases a have har- monic currents at those nonlinear loads but phases b and c do not. The 34 transformers at the load buses are assumed to be grounded-Y to grounded-Y connections.

0 ' 0 3 [ 0.02

I ? a

0.01

0 1 2 3 4 5

C I A e a B C Fig. 5 Ffth harmonic voltages with APLCs

F 0.010

0.008

0.006

0-004 :

0.002

0 1 2 3 4 5

6 7

There are seven master-slave iterations required for converging to an optimal solution by the method pro- posed in this paper. The APLCs are required at the phases a of buses 6 and 16 with 0.06 and 0.04p.u, respectively; zero is required at the other buses. The identified locations are reasonable because buses 6 and 16 are observed to have larger harmonic currents as shown in Fig. 3. All of the 3q5 settings of the trans- former taps are balanced in the final solution; however, the capacitors at phases a generally have larger amounts than those in phases b and c to help reduce harmonics as shown in Fig. 4.

As discussed in our previous paper [12], the system voltages at phase a will be influenced directly if the 3q5 transformers at the load sides are grounded-Y to grounded-Y connections and only phases a have har- monics. The system voltages at phases b and c will have less impact under this circumstance, as shown in Sec- tion 7.2. Figs. 5 and 6 illustrate the 34 fifth and seventh harmonic voltages with installation of APLCs. The fifth and seventh harmonic voltages are found to be reduced within the limits because of the installation of APLCs .

4.2 Three-phase transformer with A-Y connection The condition for simulation is the same as that in Sec- tion 4.1 except that the 3q5 transformers at the load sides are assumed to be A (load) to grounded-Y (sys- tem) connections. There are also seven master-slave iterations required for converging to an optimal solu- tion by the method proposed in this paper. APLCs are required at buses 6 and 16 with 0.06 and 0.02p.u.,

8 9 IO 1 1 12 13 14 15 bus

n

6 7 8 9 IO 1 1 I2 13 I4 15 bus

16 17 18

16 17 IS

C ] A D B C Fig.6 Seventh harmonlc voltages with APLCs

284 IEE Proc -Gener Transm Distrib , Vol 145, No 3, May 1998

respectively, for the corresponding phase a; none is required at the other buses. The identified locations are reasonable because buses 6 and 16 are found to have larger harmonic currents as shown in Fig. 3 . All of the 34 settings of the transformer taps are also balanced in the final solution; the capacitors amounts are generally balanced except those at buses 6 and 8 as shown in Fig. 7.

0.03

0.02

? a

0.01

0.L 2 0.3

0 .2 a

0 .I 0

-

-

-

6 8 10 16 17 bus

C l A m B .c Fig.7 tions

Three-phase capacitors at buses with A-Y transformer connec-

As discussed in [12], not only the system voltages at phase a but also those at phase c will be influenced directly if the 34 transformers are A-to-grounded-Y connections and only phases a have harmonics. The system voltages at phases b will have a less impact under this circumstance, as shown in Section 1.3. Figs. 8 and 9 illustrate the 34 fifth and seventh har- monic voltages with installation of APLCs. The fifth and seventh harmonic voltages are found to be reduced within the limits because of the installation of APLCs.

Note that the total 34 Qc amount (5.04p.u.) for this case is more than that (4.80p.u.) for the case in Section 4.1 because the voltage distortions are more system- wide in the viewpoint of all phases for this case. Also note that the total APLC (0.08p.u.) installed for this case is less than that (0.1p.u.) for the case in Section 4.1 because of more Qc amounts for this case.

5 Conclusions

A new method based on three-phase optimal harmonic power flow incorporating with the generalised Benders decomposition theory has been proposed for determi- nation of the locations and sizes of active power line conditioners in this paper. The master level minimises APLC investment cost and determines the control set- tings of compensators, taps and locations/sizes of APLCs from the viewpoint of co-ordination among the base case and harmonic cases. The slave level mini- mises the 3q5 MW losses and 3q5 voltage distortions while satisfying the 34 power flow equations, 34 har- monic power flow equations, 345 security constraints, and harmonic standard. The compensators, taps and APLCS are treated as discrete variables. The simula- tion results address the unbalanced harmonic currents and the different transformer connections. The test results for an 18-bus system show that the proposed method is efficient in dealing with the determination of the locations/sizes for new APLCs in unbalanced power systems.

0 k 1 18 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17

bus

OA 0 5 Mc Fig.8 Ffth harmonic voltages with APLCs

. 0.06 o'ol ? a

0.02

0 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17

bus

1 18

.c Fig.9 Seventh harmonic voltages with APLCs

IEE Proc.-Geneu. Transm Distvih., Vol. 145, No. 3, May 1998 285

~

6

1

2

3

4

5

6

7

8

9

References

AKAGI, H., NABAE, A., and ATOH, S.: ‘Control strategy of active power filter using multiple voltage-source PWM convert- ers’, IEEE Trans., 1986, IA-22, (3), pp. 460465 PENG, F.Z., AGAKI, H., and NABAE, A.: ‘A study of active power filters using quard-series voltage-source PWM converters for harmonic compensation’, IEEE Trans., 1990, PE-5, (l), pp. 9-1 5 CHOE. G.H.. and PARK. M.H.: ‘A new iniection method for AC harmonic elimination ’by active power f;ter’, IEEE Trans.,

GRADY. W.M.. SAMOTYJ. M.J.. and NOYOLA, A.H.: ‘Sur- 1988, IE-35, (I), pp. 141-147

vev of active nower line conditionine methodoloeies’. IEEE “ I

T&ns., 1990, PWRD-5, (3), pp. 1536-1G1 HONG, Y.Y., and CHANG, Y.C.: ‘Determination of locations and sizes for active power line conditioners to reduce harmonics in power systems’, IEEE Trans., 1996, PWRD-11, (3), pp. 1610- 1617 GRADY, W.M., SAMOTYJ, M.J., and NOYOLA, A.H.: ‘Mini- mizing network harmonic voltage distortion with an active power line conditioner’, IEEE Trans., 1991, PWRD-6, (4), pp. 1690- 1697 GRADY, W.M., SAMOTYJ, M.J., and NOYOLA, A.H.: ‘The application of network objective functions for actively minimizing the imnact of voltaee harmonics in nower svstems’. IEEE Trans.. 1992, PWRD-7, (3x pp 1379-1386 CHANG, W K , GRADY, W M , and SAMOTYJ, M J ‘Meet- ing IEEE-519 harmonic voltage and voltage distortion constraints wsh an active power line-conditioner;, IEEE Trans., 1994, PWRD-9, (3), pp. 1531-1537 CI-IANG, W.K., GRADY, W.M., and SAMOTYJ, M.J.: ‘A practical method for siting and sizing multiple active power line conditioners in a power system’. Presented at the IEEE confer- ence on Transmission and distribution, Chicago, IL, April 1994

10 CHANG, W.K., GRADY, W.M., and VERDE, P.: ‘Determining the optimal current injection and placement of an active power line conditioner for several harmonic-related network correction strategies’. Presented at the sixth international conference on Har- monics in power systems, Bologna, Italy, September 1994

11 CHANG, G.W.K., and GRADY, W.M.: ‘Minimizing harmonic voltage distortion with multiple current constrained active power line conditioners’. Proceedings of the IEEE PES 1996 summer meeting, Denver, CO, July 28-August 1 1996,

12 HONG, Y.Y.: ‘Three-phase optimal harmonic power flow’, IEE Proc., Gener. Transm. Distrib., 1996, 143, (4), pp. 321-328

13 YAN, Y.H., CHEN, C.S., MOO, C.S., and HSU, C.T.: ‘Har- monic analysis for industrial customers’, IEEE Trans., 1994, IA- 30, (2), pp. 462468

14 GEOFFRION, A.M.: ‘Generalized Benders decomposition’, J. Optim. Theory Appl., 1972, 10, (4), pp. 237-262

15 CHEN, C.S., and SHIRMOHAMMADI, D.: ‘A three-phase power flow method for real-time distribution system analysis’. Presented at the IEEE PES 1994 summer meeting, San Francisco, CA, USA, 1994

16 : ‘Modeling and simulation of the propagation of harmonics in electric power networks. Parts I and 11’, IEEE Trans., 1996, PWRD-11, (l), pp. 452474 (IEEE Task Force on Harmonics Modeling and Simulation)

17 FARACH, J.E., GRADY, W.M., and ARAPOSTATHIS, A.: ‘An optimal procedure for placing sensors and estimating the locations of harmonic sources in power systems’, IEEE Trans.,

18 TANAKA, T., and AKAGI, H.: ‘A new method of harmonic 1993, PWRD-8, (3), pp. 1303-1310

power detection based on the instantaneous active power in three- phase circuit’, IEEE Trans., 1995, PWRD-10, (4), pp. 1737-1742

19 OLEJNICZAK, K., and HEYDT, G.T.: ‘Basic mechanisms of generation and ‘flow of harmonic signals in balanced and unbal- anced three-phase power systems’, IEEE Trans., 1989, P W R D 4 , (4). nn. 2162-2170 , ,, I I

20 GILL, P.E., MURRAY, W., and WRIGHT, M.H.: ‘Practical optimization’ (Academic Press, New York, 1981)

21 ARORA, J.S., TANEDAR, P.B., and TSENG, C.H.: ‘User’s manual for Drogram IDESIGN’. Universitv of IOWA. technical report ODL-85.70, 1985

7 endix

This appendix provides the formulations and the meth- ods of solutions for both of the master and the slave levels.

7.7 Formulation and method of solution for master level Let w = [Qc,tap] and U = [Qc,tap, IJ]. According to the GBDT the formulation of the problem at the master level is as follows:

min Z (10)

symbols 4, Ah, ,q, and are vectors of the marginal costs (Lagrange multipliers); the superscript t signifies the transpose of the vectors. The variables with aster- isks ‘*’ are known and determined at the slave level. A linear ‘cut’ (constraint) such as eqn. 11 or eqn. 12 is generated to accumulate at the master level from the slave level when the slave level has (or does not have) an optimal solution with U determined from the preced- ing solution at the master level. Actually, there are many cuts like eqn. 11 or eqn. 12 at the master level. The number of cuts depends on the convergence crite- rion, details of which are specified in [12]. Cuts are accumulated one at a time for each master-slave itera- tion. Once one cut like eqn. 11 or eqn. 12 is sent to the master level, U is solved at the master level. The control settings are automatically determined by the marginal costs in all cuts. The marginal costs are solved at the slave level via the Lagrangian function [20]. The dis- crete values for Qc, tap, and I J are constant at the slave level. According to GBDT, the objective function 2 is the lower bound of the original Fo + pF, + vF, formu- lated in eqn. 1.

The master level remains a mixed-integer program- ming problem. It is time consuming to search exhaus- tively for the optimal solution to a problem like eqns. 10-1 5. An efficient method is therefore required to solve the master level. A modified heuristic approach based on [20], pp. 281-284, is adopted in this paper to handle the discrete variables, Qc, tap and I J and the continuous variables 4 h simultaneously. This approach employs a nonlinear programming method to solve the problem. The general-purpose nonlinear programming package IDESIGN using a recursive quadratic pro- gramming algorithm [21] is adopted in this paper. When the optimal (continuous) solution is obtained the next stage of the solution process is to fix one continu- ous variable whichever is the closest to the nearest dis- crete value, to its nearest discrete value. Any other variable with a discrete value may be set in this man- ner. The problem is then solved again, minimising with respect to the remaining continuous variables of Qc, tap and I J using the previous optimal value as the initial estimate of the new solution. This process continues until all Qc, tap and I J become discrete in a master-level

286 IEE ProcGener. Tvansm. Distrib., Vol 145, No. 3, May 1998

solution process. The values of all Qc/tap and the sizes/ locations of 4a re solved again with a new accumulated linear constraint like eqn. 11 or eqn. 12 in the next master-level solution process. The values of Qc, tap and 4 are highly dependent on the marginal costs in eqn. 11 or eqn. 12. More specifically, the location of 4 is determined mathematically by its corresponding mar- ginal cost via IDESIGN; the size of 4 is also deter- mined mathematically by the same method for obtaining a continuous value at the same time and then is heuristically discretised. A nonzero value of r j means candidate bus j has been identified. These discrete val- ues may change until convergence. The solution at the master level can always be obtained. Divergence is never encountered in the present experience.

7.2 Slave level The slave level includes ( H + 1) subproblems: one base case with the fundamental frequency and H subprob- lems with consideration of harmonics. The formula- tions for the base case and the harmonic cases are different.

7.2. I Formulation and method of solution for base case:

min Fo(x0, V,) w*) (16)

s.t. Ho(xo,Vg, w*) = 0 (17)

where the variables with asterisk are constants deter- mined at the preceding master level. According to GBDT, the constants are formulated as equality con- straints in eqn. 17 to obtain the marginal costs. In other words, the elemental values of Qc and tap deter- mined at the master level are verified at the slave level, and the effectiveness of Qc and tap is reflected by the marginal cost. Note that the formulation expressed by eqns. 16-18 is an optimisation problem with pure con- tinuous variables (xo and Vg). The general-purpose nonlinear programming package IDESIGN is adopted to solve eqns. 16-18. This subproblem is the same as that in [12].

7.2.2 Formulation and method of solution for harmonic cases: The H harmonic cases can be solved independently when the vector of U is fixed. For harmonic order h

min P F h ( X h ) (19)

s.t. H ~ ( x o , x ~ , u * ) = 0 (20)

Gh(Xh) I O (21) where the elements of U* are constants determined at the preceding master level and those of xo are obtained from eqns. 16-18. According to GBDT, the constants in U* are formulated as equality constraints in eqn. 20 to obtain the marginal costs. The reason is the same as that described in Section 7.2.1. eqns. 19-21 constitute an optimisation problem of zero control variables.

For this problem only the marginal costs of elements of Qc, tap and Gh are concerned, although the state varia- bles X h can also be obtained. Furthermore, the formula- tion expressed by eqns. 19-21 is an optimisation problem with pure continuous variables xh. The nonlin- ear programming package IDESIGN is used to solve eqns. 19-21.

7.3 Simplified models This Section provides the simplified models of the 34 grounded-Y to grounded-Y and the 341 A-to-grounded- Y transformer connections as shown in Figs. 10 and 1 1, respectively. For detailed derivation and formula- tion refer to [12]. The simplified model means that the 34 coupling effect is small enough and can be ignored. Note that these coupling effects are considered in the simulation results.

vaTT vA + to system ___)

to load & c--

vb7r---is vB & f

vc7-----r vc -L -L -

Fig. 10 Simplified grounded- Y-to-grounded- Y transformer model

to load

to system ___)

' h g Fig. 1 1 Simplified delta-to-grounded- Y transformer model

The rectangles shown in Figs. 10 and 11 could be inductive or capacitive depending on the tap setting. As shown in Fig. 10, the three phases are decoupled, i.e. the harmonic injection current at phase a will have a direct impact on phase A of the system. On the other hand, the harmonic injection current at phase a will influence the voltages at phases A and C of the system as shown in Fig. 11.

IEE ProcGener. Transm. Distrib., Vol. 145, No. 3, May 1998 287