k l k u Impedance from PCC Harmonic Order

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1 OPTIMAL DESIGN OF TUNED PASSIVE FILTERS Jos e Maria Maza Manuel Burgos Carlos Izquierdo maza@esi.us.es mburgos@cica.es mitchell@cica.es Department of Electrical Engineering University of Sevilla Sevilla, Spain Abstract - A new method for designing passive lters is presented in this paper. The lters not only reduce the harmonic distortion but also maximize the power factor of the facility. First, it is shown that the objective function is nonconvex due to the resonances created by harmonics. Then, an interior point algorithm is used to solve this minimization problem. Two additional restrictions are added to the original problem in order to consider some practical issues: parameter changes and fundamental reactive power. Finally, the proposed method is applied to a numerical example. Keywords - power quality, harmonics, passive tuned lters, reactive power. INTRODUCTION ASa result of wide-ranging addition of power electronic equipment in modern industry, harmonic levels are a matter of concern nowadays. In this sense, the harmonics currents that the customers or the electronic equipment can produce are limited by standards in order to maintain the electromagnetic compatibility (EMC) of the public distribution system [], []. The harmonic currents can be mitigated by two methods: improving the design of the polluting loads (increasing the number of pulses, introducing bigger choke reactors, etc.) or installing lters in the power system. Traditionally, passive lters have been used for these purposes, but nowadays the focus are on active lters. In spite of this actual situation, passive lters are preferred sometimes due to economic considerations. The state of the art on the design procedures of passive lters shows that most of them are not very well established. Although several papers related to this topic exist, the proposed procedures are based on trial and error methods or on minimization techniques with a lot of simpli cations. This paper presents a new method for the optimal design of passive lters. First, it is shown that the objective function used to design the lters is non-convex. Then, the proposed method for minimizing it, based on an interior point algorithm, is explained. In order to improve the proposed method of design, some restrictions have to be included. Finally, the method is applied to an example and some results are presented. THE OBJECTIVE FUNCTION The proposed method of design is based on the minimization of an objective function. Previous works on this area minimize the current harmonic distortion [9], the current and voltage harmonic distortion [4], the RMS value of the current [], or the cost of the lters [8]. The proposed method takes into account the relation between harmonics and power factor when passive lters are used for mitigation purposes. Fig. shows the one-line diagram of a simple facility with non-linear loads producing current harmonics and fed by an utility with distorted voltage. The lters are composed by reactive elements (reactors and capacitors) which modify the reactive power demanded by the facility. In consequence, the lters can be used not only to reduce the harmonic distortion of the system but also to maximize the power factor. The objective function to minimize is related to the power factor of the facility. In order to achieve a reduction of harmonic distortion, the lters are usually tuned to xed harmonics, corresponding to the major harmonics in the system, i.e. fth, seventh, eleventh and thirteenth. Figure :One-line scheme of a simple distribution system. The classical de nition of reactive power and power factor cannot be applied in non-sinusoidal systems. Although multiple approaches to this problem exist, the theory used in this work was presented by Ferrero and Superti-Furga [5]. These authors propose a formulation in Park domain, well-suited for three-phase systems in absence of zero-sequence components. In such situation, a three phase magnitude can be represented by a complex quantity called Park vector. In case of periodic signals, the complex Fourier transform can be applied to the Park vector: +X y(t) =y d + jy q = Y ke jk!t This formulation has the following advantages: () ffl A three-phase system in absence of zero sequence components can be represented as a single-phase one.

2 ffl Positive and negative sequence of a harmonic are treated as different harmonics. Positive indexes of () are related to the positive sequence of harmonics, while negative indexes are related to the conjugate of the negative sequence. In this domain, the current demanded by the facility can be decomposed into orthogonal components []. When passive lters are included, only the orthogonal components of this current with respect to the load voltage can be modi ed, as shown in Fig. ; the lower these components, the higher the facility power factor. In this gure, the harmonic k of the system current I sk is decomposed into two orthogonal components: active, I sak, and reactive, I srk. Figure :Orthogonal components of the system current. Once the tuned frequency of a lter has been chosen, the lter is characterized by this tuned frequency and its capacitance. Then, the proposed objective function is the following: min C i I sr (C i)= P k I srk (C i) () Each of these harmonic components of the system current, I srk, can be written as a function of the load voltage, the linear load, non-linear load and the installed lters. The linear load and the lters are represented by their equivalent admittances and the non-linear load by a current source. On the other hand, the introduction of passive lters in the system generates two related problems: ffl A passive lter below its tuned frequency behaves like a capacitor which creates a resonance with the system impedance in the point of common coupling (PCC). A resonance can be de ned as a maximum that is produced in one harmonic of the system current due to the installed lters. Then, the resonance for harmonic k can be written ck = k> () where I sk is the harmonic k of the system current and B ck the susceptance of the lters for harmonic k. In case of distorted source voltage and load current, eq. () leads to a second order equation on the susceptances of the load and the lters: H(B lk + B ck ) + L(B lk + B ck )+M = (4) where H, L and M are constant values depending on system parameters and B lk is the susceptance of the linear load. If A k is the solution of (4), the resonance produced on harmonic k due to the installation of N passive lters with tuned frequencies k hi veri es: NX i k! (k=k hi ) C i +(B lk A k )= (5) In consequence, a resonance for the harmonic k is a linear function on the capacitance of each lter, i.e., a hyperplane which de nes a parallel harmonic resonance. ffl From an electrical point of view, the described resonances are dangerous because of the harmonic ampli cation. Mathematically, a resonance is a maximum that is produced in one harmonic of the system current due to the installed lters, as shown in (). This current has as maxima as resonances appear in the system, that is, the number of existing harmonics below the tuned frequency of the lters. Then, the objective function is non-convex, existing multiple minima contained in regions bounded by resonances (maxima of the system current). All the local minima of these regions have to be computed to obtain the global minimum. Previous works, [9], [], [8] do not mention this problem; other works state the non-convexity of the objective function, but do not solve the problem properly [4]. The problem of looking for a the local minimum inside a region bounded by resonance hyperplanes is a constrained one that can be formulated as: min C i I rs (C i) (6) subject to f (C i )? (7) where f (C i ) corresponds to the linear boundaries, resonance hyperplanes that de ne the region where the local minimum is to be computed. This minimization problem has to be solved for each region de ned by the existing resonances in the system. OPTIMIZATION PROCESS The optimization of the lters can be performed in two steps: ffl Compute all the feasible regions de ned by the resonances. Before optimizing the objective function, the regions where the local minima have to be found must be known. ffl Minimize the objective function inside each of these regions. The numerical method used in this task has to guarantee that the successive points obtained during the optimization process are within the corresponding feasible region. The numerical method applied to this minimization problem is an Interior Point Algorithm [7], [], []. These steps are described brieλy in the following subsections.

3 . Feasible regions Inside a feasible region, capacitance of the passive lters have to be positive. In a real case, harmonics below and between the tuned frequencies of the passive lters exist. If the problem involves m resonance hyperplanes and N passive lters, a system of m inequalities with N unknowns has to be solved. Introducing slack variables to turn inequalities into equalities results in a system of m equations with N +m unknowns. Thus, a feasible solution of a linear programming problem have to be found. Setting N unknowns to zero, the linear problem has a unique solution x =[x B ; ]. In consequence, the region is feasible if all components of x B are positive, but if one of the m components of this vector is negative, other unknowns must be set to zero. To assure that a region is not feasible, the number of linear systems to be solved is the binomial m+n coef cient N. If the region is feasible, the number linear systems that must be solved is always less than this number.. The Interior Point Algorithm A Primal-Dual Logarithmic Barrier (PDLB) Interior Point Technique [7], [], [] is applied to the problem given by eq. (6). Adding auxiliary and slack variables, eq. (6) turns into: min C i I rs (C i) (8) subject to x f (C i )= x + s = where all the components of the vectors x and s, auxiliary and slack variables respectively, must be positive. The logarithm-barrier Lagrangian function of the above optimization problem can be de ned as follows: positive, the corresponding Lagrange multipliers, ^ff, must also be positive as indicated by (4). The procedure to solve the optimization problem can be summarized as follows. First, a starting point is assumed that satis es the positivity condition on the slack variables, and a barrier parameter μ () > that causes the objective function logarithmic terms to dominate over the value of the original objective. Second, the rst-order optimality equations are solved by only one iteration of the Newton's method. Third, all the variables are updated. Fourth, the barrier parameter is appropriately reduced. This iterative process is repeated until primal and dual feasibility is achieved within acceptable accuracy, and a stopping criterion is satis ed. 4 ADDITIONAL RESTRICTIONS The solution of the problem set up by (8) for each of the feasible regions determined by the resonances provides an optimized lter from the power factor point of view. However, this method does not take into account two aspects that are related with some practical issues of passive lters: ffl The installation of passive lters introduces some resonances in the power system. The behaviour of a passive lter below its tuned frequency is like a capacitor, generating resonances with the system impedance from the PCC. These resonances are always located on harmonics below and close to the tuned frequency of the lters, as is clearly noticed in a PCC impedance versus frequency diagram. Fig. shows an example of this diagram for two passive lters tuned at fth and seventh harmonics. L = I r (C i)+ff t (x f (C i )) + ^ff t ( x + s) μ X i ln s i (9) 8 k l where ff and ^ff are vectors of Lagrange multipliers, and μ is a positive penalty factor that decreases as the number of iterations increases. The rst-order optimality conditions lead to the following equations: Impedance from PCC 6 4 k i r i = x f (C i)= ^ff = x s = = ff ^ff = = =) S ^ff μe= (4) where S is a diagonal matrix of slack variables, and e is avector of one's. As μ and the slack variables must be Harmonic Order Figure :Impedance from the PCC versus frequency in case of two tuned lters. The impedance for the tuned frequencies equals zero for ideal lters, and reaches the maximum for the resonances, harmonics k l and k u in the illustrated example. One of the problems related with passive lters is that parameter changes can move the resonances on harmonics k l and k u to the tuned frequencies or close to them, creating harmful harmonic ampli cations in the system. Typical parameter changes that may appear in the system are vari-

4 ations on source impedance due to system recon- gurations or changes on the tuned frequency of the lters due to loss of capacitance produced by ageing of capacitors. ffl The proposed method maximizes the power factor of the facility considering the harmonics. But, in spite of the harmonic content of current and voltage, the system operation is performed as if it was a sinusoidal one. In this sense, it would be desirable that, once the lters are installed, the facility does not inject fundamental reactive power in the power system. But, the formulated problem does not consider this practical issue. The proposed method can be improved by adding some restrictions in order to avoid these shortcomings. These additional restrictions will be analyzed in the following subsections. 4. Parameter changes In order to avoid the harmful resonances that can be created in the system due to parameter changes, the distance between the tuned frequencies and the resonance harmonics have to be high enough. That is, the location of resonance harmonics, k l and k u,have to be limited below two predetermined values, namely k llim and k ulim.to understand how including these restrictions in the problem is necessary to analyze the resonance hyperplanes. The resonances located on harmonics k l and k u, shown in Fig., can be formulated by a linear equation (5). In this example with two passive lters, the resonances are straight lines linking the capacitance of the lters. These resonance lines are represented in Fig. 4. The resonance lines corresponding to harmonics k l and k s have negative and positive slopes, respectively. the resonances introduced by the passive lters have tobe maintained below the limits k llim and k ulim in order to immunize the lters against parameter changes, the feasible region where the solution has to be looked for is shown in Fig. 5. This region is determined by the following equations: k llim!c u (k llim =k hu ) + k llim!c l (k llim =k hl ) + B lkl + B skl > (5) k ulim!c u k (k ulim =k hu ) + ulim!c l (k ulim =k +B lku+b hl ) sku < (6) where B lkl, B lku, B skl and B sku are the susceptances of the linear components of the load and the source admitance for the harmonic limits k llim and k ulim. These harmonics does not exist in the system and can be considered as ctitious ones. Their related resonance hyperplanes are also ctitious, and are introduced to take into account possible parameter changes. Figure 5:Feasible region determined by the ctitious resonances added. 4. Fundamental reactive power The restriction that correspond to fundamental reactive power demanded by the facility can be written in terms of a current constraint. If fundamental reactive power cannot be injected to the utility, the rst harmonic of the system current has a negative reactive component. This condition can be written as a function of the system parameters and yields to another linear equation of the capacitance of the lters. I sr < =) (B l + B c (C i ))U I gr < (7) Figure 4:Resonance lines in case of two tuned lters. Resonance lines with negative slope correspond to harmonics below the tuned frequency of both lters, while resonance lines with positive slope correspond to harmonics between the tuned frequency of the lters. The intersection of these lines determines the capacitance of each lter. Once the topology of the resonance lines has been determined as a function of the harmonic number with respect to the tuned harmonic of the lters, it is interesting to know the region where these resonance lines are placed for other harmonics. Fig. 4 shows the location of resonance lines for harmonics bigger than k l and k u. If where I gr is the fundamental reactive current demanded by the non-linear load. As the fundamental harmonic is below the tuned frequency of the lters, the feasible region is the one shown in Fig. 6. Figure 6:Feasible region determined by the fundamental reactive power.

5 The constraints introduced to consider the problems related with parameter changes and fundamental reactive power are both linear and can be treated in the same way as resonance restrictions described in section. In consequence, the problem is that of eq. (8) and (9), but f (C i ) includes now the additional restrictions described in this section. 5 EXAMPLE OF APPLICATION The proposed method has been applied to the industrial distribution system shown in Fig.. The facility has linear and non-linear loads and the utility voltage is distorted. Table shows the characteristics of the transformers. Transformer Power Voltages ffl cc X/R (MVA) (kv) (%) (p.u.) T 66/ 5 T / Table : Transformer characteristics. A scaled model of this simple distribution system has been built in the laboratory in order to validate the data obtained from the simulations after the optimization process for calculating the lters. Table shows the parameters corresponding to the real and the scaled system; the impedances of the transformers in the real system are referred to their high voltage side. The base magnitudes in the real system are 66 kv and MVA which correspond to 8 V and 5 kva in the scaled one. Parameters Real system Scaled system U s (kv) 66.8 System (Ω) j T (Ω) 8.54+j4.7.8+j.944 Line (Ω).67+j.65.6+j.7 T (Ω) 4.7+j.55.+j.566 Table : Impedances of the real and the scaled systems. The linear load is composed by kw of resistances and kvar of reactors. The non-linear load is a 5 kva AC adjustable speed drive (ASD). The utility voltage of the scaled system is obtained from the low voltage distribution system of the laboratories. This voltage is unbalanced at the fundamental harmonic and mainly distorted with fth and seventh harmonics. The harmonics of the voltage Park vector are shown in Table. Sequence Harm. Positive Negative Mod. (V) Ph. (rad) Mod. (V) Ph. (rad) Table : Voltage harmonics. The ASD is composed by an uncontrolled six-pulse recti er, a DC-link and a PWM inverter. The current harmonics of this polluting load depend on the load torque of the motor. For this application the ASD is in continuous current operation as shown in Fig. 7. The unbalanced of the load voltage can be appreciated in the different amplitude of the pulses. A phase current (A) B phase current (A) C phase current(a) Angle (rad) Angle (rad) Angle (rad) Figure 7:Phase currents of the ASD. In order to reduce the harmonic distortion and to improve the power factor of the installation, passive lters have to be installed. The lters are designed applying the proposed method: the reactive component of the system current considering all the feasible regions determined by resonances and the additional restrictions for the parameter changes and fundamental reactive power problems is minimized. The tuned frequency of the passive lters as well as the harmonic limits where the resonances can be located, k llim and k ulim,have to be previously determined: ffl The lters are tuned to the major harmonics in the system, that is, fth, seventh, eleventh, etc. ffl The secure region where the resonances can be located k lim. These limits depend on the desired degree of immunization of the designed lters against parameter changes. High immunization implies harmonic limits k lim far enough from the corresponding tuned harmonics of the lters. But it is important to advise that a relationship between these limits and the reactive power restriction exists. It could be possible that after imposing severe limits, the fundamental reactive power restriction cannot be veri ed. This fact can be graphically shown after installing two passive lters in the proposed distribution system. Fig. 8 shows all the feasible regions of the problem determined by the resonances and the additional restrictions for k llim = 4:5 and k ulim = 6:5. In this gure, the arrows indicate the feasible regions determined by the additional restrictions, while the restrictions determined by harmonic resonance lines do not impose a region, only separate the plane into two possible valid regions. In this case, a unique feasible region exists where the local minimum have to be looked for. But if the resonances for the ctitious harmonics are located

6 C5 (F) x 4 farther from the tuned frequencies, i.e. k llim =4: and k ulim =6:, Fig. 9 shows that there is no feasible region and the problem has no solution. 6.5º harmonic Fundamental reactive power º harmonic resonance 4º harmonic resonance 4.5º harmonic perform the experimental validation of the numerical analysis. Table 5 shows the characteristics of the lters. Tuned C L harmonic (μf) (mh) Table 5: Characteristics of the installed lters. Fig. shows the PCC impedance versus frequency in case of installing the lters. It is clearly shown that the resonances comply with the additional restrictions introduced by the ctitious resonances. In fact, the resonances produced below the 5 th and 7 th harmonics are located in the imposed limits, that is 4.5 and 6.5, respectively C7 (F) x 4 Figure 8:Feasible regions determined by all the restrictions. k llim = 4:5, k ulim =6: x 4.5 º harmonic resonance Impedance from the PCC º harmonic 4 C5 (F).5 Fundamental reactive power 4º harmonic resonance 4.º harmonic C7 (F) x 4 Figure 9:Feasible regions determined by all the restrictions. k llim = 4:, k ulim =6:. In consequence, a compromise between the additional restrictions has to be achieved. For this numerical example, the ctitious resonances have been set to k llim =4:5, k ulim =6:5. The number of tuned lters depends on the desired harmonic reduction. Table 4 gives the global results of the compensation values for two and three lters tuned to the major harmonics: fth, seventh and eleventh. The RMS value of the system current is referred to the previous one and the total harmonic distortion of the system current and load voltage are speci ed in percentage. Magnitude Before lters lters RMS current (p.u.).9.9 THD u (%) THD i (%) Table 4: Global results of the compensation. Table 4 shows that the global compensation results are similar in case of installing two or three lters for this application. The three- lters option was selected in order to Harmonic order Figure : Impedance from the PCC versus frequency. Figs. and show the comparison between experimental and theoretical harmonics for the system current, the good agreement validating the simulation results in an experimental way. It is important to note that the theoretical results have been obtained considering the variations of the harmonic current injected by the ASD due to the load voltage changes produced by the lters; i.e., the non-linear load has not been replaced by a constant current source. The model for the non-linear load considers the changes produced in the current due to variations of the load voltage [6]. Current (A).5.5 Theoretical results Experimental results Harmonic order Figure : Positive sequence harmonics of system current.

7 Current (A).5 Theoretical results Experimental results REFERENCES [] G. Blanchon, J.-C. Dodu, A. Renaud, and M. Bouhtou. Implementation of a primal-dual interior-point method applied to the plannig of reactive power compensation devices. In Power Systems Computation Conference, pages 87 86, 996. [] CENELEC. EN56. Voltage characteristics of electricity supplied by public distribution systems, January Harmonic order Figure : Negative sequence harmonics of system current. 6 CONCLUSIONS This paper has presented a new method for designing passive lters. The design method minimizes the reactive component of the current demanded by the facility, that is, the power factor of the facility is maximized. It has been proven that the objective function is non-convex due to the presence of harmonics below the tuned frequency of the lters. These harmonics create resonances which are maxima of the objective function which can be formulated as linear functions. The resonance hyperplanes are the boundaries of the feasible regions where the local minima have to be looked for. The minimization process has two steps: rst, all the feasible regions are determined by applying a linear programming technique; then, the local minima are looked for inside each region by applying an interior point technique. The proposed method have been improved by adding two restrictions to take into account possible parameter changes and the injection of fundamental reactive power. These restrictions are linear equations of the capacitance of the lters, and are introduced easily in the formulated problem. It has been also proved that a relation between these two additional restrictions exists, being not possible sometimes to achieve a high immunization and compensation of the fundamental reactivepower. In consequence, the advantages of the proposed method with respect to previous and classical approaches can be summarized as follows: the non-convex behaviour of the objective function, the reactive power at the fundamental harmonic and the immunization of the lters against the parameter changes are dealt with. Finally, an application of the method to a simple distribution system has been presented. A scaled distribution system has been built in order to perform an experimental validation of the theoretical results. These theoretical results have been obtained with a model of the non-linear load that considers the variations of the non-linear current due to the load voltage changes when the lters are installed. These lters reduce the harmonic content about 4% and fully compensate the fundamental reactive power in the proposed example. [] L.S. Czarnecki. An orthogonal decomposition of the current of non-sinusoidal voltage sources applied to non-linear loads. International Journal on Circuit Theory and Applications, :5 9, 98. [4] L.S. Czarnecki and H. Ginn. Effectiveness of resonant harmonic lters and its improvement. In IEEE WinterMeeting,. [5] A. Ferrero and G. Superti-Furga. A new approach to the de nition of power components in three-phase systems under nonsinusoidal conditions. IEEE Transactions on Instrumentation and Measurement, 4(): , June 99. [6] J. Garcia Mayordomo, L.F. Beites, and L. Zabala. A contribution for modeling controlled and uncontrolled ac/dc converters in harmonic power Λows. IEEE Transactions on Power Delivery, (4):5 58, October 998. [7] J.L. Martinez Ramos, A. Gomez Expsito and V.H. Quintana. Reactive-power optimization by interiorpoint methods: Implementation issues. In Power Systems Computation Conference, pages , 996. [8] C. Kawann and A.E. Emanuel. Passive shunt harmonic lters for low and medium voltage: A cost comparison study. IEEE Transactions on Power Systems, (4):85 8, November 996. [9] K.P. Lin, M.H. Lin, and T.P. Lin. An advanced computer code for single-tuned harmonic lter design. IEEE Transactions on Industry Applications, 4(4):64 648, July/August 998. [] P. Mattavelli. Design aspects of harmonic lters for high-power ac/dc converters. In IEEE Winter- Meeting,. [] IEEE Industry Applications Society and Power Engineering Society. IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems, 99. [] S. Wei, H. Sasaki, J. Kubokawa, and R. Yokoyama. An interior point nonlinear programming for optimal power Λow problems with a novel data structure. IEEE Transactions on Power Systems, ():87 877, August 998.