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Compensation based on active power filters – The cost minimization
Applied Mathematics and Computation xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Compensation based on active power filters – The cost minimization _ ⇑, Dariusz Grabowski, Marian Pasko, Michał Lewandowski Marcin Macia˛zek Faculty of Electrical Engineering, Silesian University of Technology, ul. Akademicka 10, 44-100 Gliwice, Poland
a r t i c l e
i n f o
Keywords: Power quality Power system harmonics Optimization Active power filters
a b s t r a c t The paper deals with the problem of power quality and its solution. Passive or active compensators are usually used to reduce distortions of voltage and current waveforms. Active power filters seem to be the best choice if the effect of compensation is the only criterion. However, the financial cost of such a solution put the potential investors off. The paper shows that the cost could be reduced considerably if the optimization of active power filter allocation, as well as their nominal currents, is carried out. Two cases have been considered – active power filters with a classical control algorithm, which operates locally and aims at obtaining sinusoidal shape of waveforms in the point of installation, and active power filters with a modified control algorithm, which aims at reduction of waveform distortions in all nodes of an analyzed power system. Goal functions and optimization problems for both cases have been defined. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction There are more and more electrical and electronic devices available in the market which require high quality power supply. On the other hand, the number of devices, which cause power quality deterioration due to nonlinear behavior influencing the supplying power system, increases too. Moreover, dynamical changes of nonlinear loads during their normal operation make elimination of interferences impossible using passive compensators. Nowadays, active power filters (APF) [1,2] are the only solution which is able to improve power quality in contemporary power systems with distributed generation [3,4] and dynamical changes of supplying current and voltage distortions. Many publications on optimization of APF sizing and allocation can be found in technical literature worldwide. However, the proposed solutions are first of all based on minimization of the filter RMS currents or THD coefficients without direct application of economic criterion [5–7]. High cost of APFs leads to optimization of the system from the economical point of view, which has been presented in the paper and could lead to a wider application of APFs in the future. The paper shows that the minimization of APF sizes is not enough to obtain the cheapest possible solution. In this sense the proposed approach can be treated as an added value to the power quality field.
⇑ Corresponding author. _ E-mail addresses: [email protected] (M. Macia˛zek), [email protected] (D. Grabowski), [email protected] (M. Pasko), michal. [email protected] (M. Lewandowski). http://dx.doi.org/10.1016/j.amc.2015.01.001 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
2
2. Active power filters One of the distinctive features of active power filters is that they realize the function of a controlled voltage or current source. In this paper, parallel APFs, injecting additional current to the system, have been used. The current generated by the APF results in elimination of some unwanted components, e.g. higher harmonics in power system voltages and currents. APFs could have many applications depending on their structure:
filtering of current higher harmonics, filtering of voltage higher harmonics, voltage symmetrization, elimination of voltage flicker, load balancing, reactive power compensation for fundamental frequency.
A parallel APF is usually realized with the help of a three-phase power inverter followed by coils what is equivalent to a three-phase current source. In most of the applications the inverter is controlled by a tracking proportional–integral controller (PI) and the reference current is generated on the base of a given power theory [2]. In order to accomplish the task with the minimum delay (real-time operation) very fast VLSI (Very-Large-Scale Integration) systems should be used in the feedback control loop. Therefore, signal processors are the most popular solution. The idea of the compensation using a parallel APF has been presented in Fig. 1. Although APFs are the best solution for elimination of nonlinear load influence on power systems, the high cost of such compensation discourage potential investors from the APF application. It seems that optimization algorithms can bring about reduction of the compensation cost by such APF selection which ensures minimum costs while achieving the compensation goal [8]. 3. System under test An exemplary power system presented in Fig. 2 has been used to validate the developed optimization algorithms. It is a large power system described in [9] and it has been used to supply an expansive winter sport area. It includes buses supplying DC motors driven by six-pulse line-commutated adjustable speed drives. These nonlinear loads are the sources of higher harmonics in the power system under consideration. This system has been implemented as an example in PCFLO software [9]. This system has been chosen to be used for testing of the proposed solutions because: there are 37 nonlinear loads distributed in the system which are the source of higher harmonics, distances between buses exceed 1000 m, full data about the system, including line parameters, are available.
Fig. 1. Block diagram of a power system with a parallel APF.
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
3
Fig. 2. Block diagram of the power system under consideration.
4. Optimization problem Minimization of the higher harmonic compensator cost requires defining a goal function and constraints [10]. In this case the decision-making variables include current phasors for all considered harmonics and for all compensators placed in the system because the APF cost depends on its nominal current. Moreover, it has been assumed that the power system can be represented by a linear model when performing the frequency analysis and furthermore nonlinear loads are regarded as current sources for each harmonic. So the impedance matrix describing the power system for each harmonic can be determined separately. Currents and voltages in the system have been calculated using PCFLO software [9]. The optimization problem has been defined assuming that the APF connected to a system bus can be treated as an ideal current source injecting higher harmonics to the system. Such a source can be described by the Fourier series (phase index has been omitted in order to simplify notation, index k refers to the compensator): k
iw ðtÞ ¼
H pffiffiffi X 2Re Ikwh ejhx0 t ;
ð1Þ
h¼2
where H – number of harmonics generated by the compensator, x0 – fundamental frequency, Ikwh – phasor of an hth harmonic of the compensator located in the bus w:
k Ikwh ¼ jIkwh jej/wh ¼ Re Ikwh þ jIm Ikwh ;
ð2Þ
jIkwh j, /kwh – RMS value and phase of an hth harmonic of the compensator located in the bus w, Reð Ikwh Þ,Imð Ikwh Þ – real and imaginary part of an hth harmonic of the compensator located in the bus w. During the development the goal function, two cases have been considered: minimization of APF sizes is carried out assuming modified APF control algorithm which enables generating arbitrary current with given amplitude and phase spectrum (goal function f1) [8,11], minimization of APF sizes is performed assuming standard APF control algorithm based on instantaneous power theory [2] – minimization of the cost requires a compromise between minimum cost and minimum waveform distortions (goal function f2). 4.1. Goal function f1 The goal function f1 enables to take into account economical criterion by minimization of the compensation cost. Additionally, it has been assumed that the compensator price is a function g() of its nominal current:
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
4
min f 1 ðxÞ ¼ x
W X gð jIkw jÞ; fReðIkwh Þ;ImðIkwh Þ;T½wgw¼1
min
ð3Þ
where W – total number of APFs, T – vector defining allocation of APFs, dim(T) = W, jIkw j – RMS value of the APF current located in bus T [w]:
jIkw j
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u H n o n o2 uX 2 : ¼t Re Ikwh þ Im Ikwh
ð4Þ
h¼2
The function g(), which reflects the cost of a single compensator, is usually discontinuous – some intervals of currents correspond to different costs being a result of an available series of APF nominal currents. An exemplary function g(), developed on the base of price lists of commercially available APFs, has been shown in Fig. 3. The function g() is both discontinuous and nonlinear. It limits the range of algorithms which can be applied to solve the optimization problem. Solution to (3) enables to answer the question: how many compensators and in which buses must be used in order to limit the maximum THDV coefficient and minimize the cost of this solution. The step shape of the function g() is a result of the available series of APF nominal currents and depends on the pricing policy of the APF manufacturer. Solutions to the problem for different functions g() may lead to completely different results – it could be especially important when comparing the offers of several manufacturers which could include APFs with different prices and nominal parameters. The discontinuity is the main disadvantage of the goal function f1. Thus, solution to the optimization task cannot be reached using gradient methods. This problem can be overcome using the interpolation of the function g() (for example by means of the spline functions). The other problem may arise if the goal function f1 has a few local minima for which the function values are similar and which correspond to solutions with different numbers and sizes of compensators. In order to force solutions with lower number of compensators, an additional coefficient depending on fixed cost caused by installation and maintenance expenses can be introduced. In this case the function g() should represent the relation between total costs (purchase price + installation and maintenance costs) and its size. As a result, solutions with lower number of compensators, and hence a lower fixed cost, will be automatically preferred. Classical optimization methods have some significant restrictions. Analytical methods search for the solution locally and require existence and continuity of goal functions. On the other hand, enumerative methods, either deterministic or random, are usually not too effective. Even in the case of small scale problems like the 17-bus power system shown in [11,12] a classical combinatorial algorithm leads to computation times of a few days for just two compensators. Therefore, besides application of classical optimization methods, the problem of compensator sizing and allocation represented by (3) has been solved using genetic algorithms. In this case optimization of compensator allocation requires only setting the maximum number of compensators and indices of buses to which they can be connected. The genetic algorithm finds the buses in which compensators should be connected as well as their parameters. Computing complexity of the analyzed problem increases along with the number of buses to which compensators can be connected and with the number of compensators. Computation time is longer than in the case of the classical APF control algorithm, but much more shorter than in the case of combinatorial approach. It allows to seek optimal solutions even for large scale power systems. The detailed description of the applied procedure and the solution obtained by means of the genetic algorithm for the 17bus power system including APF allocation and sizing for 1, 2 and 3 compensators, have been presented in [11].
100000
price (€)
80000 60000 40000 20000 0 0
100 200 300 400 500 600 700 800 900 nominal current (A)
Fig. 3. Exemplary relation between compensator price and its nominal current g().
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
5
4.2. Goal function f2 It enables to find the minimum compensation cost and realizes a compromise between THDV coefficient improvement and financial expenses necessary to reach this improvement for a classical APF control algorithm. In this case the same function g(), which reflects the relation between the cost of a single compensator and its nominal current, can be taken into account: ½iþ1 maxw THDV w max THDV ½i P w w ; min f 2 ðxÞ ¼ min P W ½iþ1 W k k ½i x i g jI j g jI j w w w w
THDV w 6 THDV max
for w ¼ 1; . . . ; W;
ð5Þ
where i – index pointing at successive solutions obtained using compensators with a standard control algorithm sorted in ascending order in accordance with the maximum value of THDVw (w = 1, . . ., W), jIkw j½i – RMS value of the compensator current obtained for the solution number i, maxw THDV ½i w – maximum value of the THDV coefficient for all buses of the system reached if a given APF arrangement is used (the arrangement is pointed out by the index i in the sorted set of solutions), PW k ½i w gðjIw j Þ – cost of all connected APFs for the solution pointed out by the index i, THDV w 6 THDV max – constraint which enables finding a compromise between compensation cost and quality, namely increasing the value of THDVmax makes the compensation cost lower, but at the same time the supplying network waveform distortion level becomes higher; the acceptable value of THDVmax depends strongly on sensitivity of loads installed in the network under consideration and so it should be set individually.
(a)
0.1
2 bus voltage 1383
0
0
-2 0
0.005
0.010
0.015
i 1383-1384( t ), pu
u1384 ( t ), pu
line current1383-1384
-0.1 0.020
t, s
(b)
2
0.1 bus voltage 1383
0
0
-2 0
0.005
0.010
0.015
i 1383-1384 ( t ), pu
u 1384 ( t ), pu
line current1383-1384
-0.1 0.020
t, s
(c)
2
0.1 bus voltage 1383
0
0
-2 0
0.005
0.010
0.015
i 1383-1384 ( t ), pu
u 1384 ( t ), pu
line current1383-1384
-0.1 0.020
t, s Fig. 4. Exemplary voltage (bus 1348) and current (line 1383–1384) waveforms: (a) no compensation, (b) Variant II, (c) Variant III.
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
relative cost of APF (%)
6
THDVmax=9.58
THDVmax=0.75
THDVmax=5
THDVmax=6.61
No APF
Variant I
Variant II
Variant III
variant of compensation Fig. 5. Comparison of the relative cost for three variants of compensation – goal function f2.
4.3. Simulation results for goal function f2 In this case optimization of the compensator allocation requires analysis of all combinations of APF connection points in the power system. The aim is reached by application of the combinatorial approach and simultaneous evaluation of the voltage distortion level and its economical impact. As a result the required number as well as the location (bus numbers) and sizes of APFs have been determined. Some exemplary current and voltage waveforms before and after compensation using different strategies have been shown in Fig. 4. The Variant II denotes solution obtained with the help of the goal function f2, for which requirements imposed by the IEEE standards are fulfilled, i.e. the maximum THDV coefficient is not greater than 5%. This state of the system has been reached applying 11 APFs located in buses 42, 199, 60, 196, 1189, 1190, 1354, 1376, 1383, 1319 and 1323 – the system layout is described in [9]. The Variant III corresponds to the solution which enables to achieve the best price/effect ratio using the goal function f2. This state has been reached applying 6 APFs located in buses 42, 60, 196, 1354, 1376 and 1383. The computation complexity depends on the number of buses to which the compensators can be connected, but the computation time is relatively short and it is possible to apply this approach for large scale power systems. The optimum solution ensures a compromise between voltage distortion reduction and the economic cost of the compensation. In order to follow this approach the maximum values of the THDV coefficients before and after compensation have been compared keeping in mind the cost of the compensators (see Fig. 3) necessary to reach this state. The final results have been shown in Fig. 5. Variant I represents results obtained for total compensation (relative cost equal to 100%), i.e. APFs connected to all buses with nonlinear loads. The solution, for which the IEEE standards are fulfilled (Variant II), is more than 50% cheaper comparing to the total compensation (Variant I). The last case presented in Fig. 5 corresponds to the solution which enables to achieve the best price/effect ratio (Variant III). It is 80% cheaper than the total compensation. 5. Conclusions The optimization of APF nominal currents and allocation has been carried out taking into account the requirements for voltage distortions (THDV) and reactive power compensation for the fundamental frequency. The research results presented in this paper show that using a classical APF control algorithm, it is possible to achieve a given voltage distortion level (around 6.6% in the analyzed exemplary power system) with considerably lower financial costs (around 80% for the test system) in comparison with the total compensation case. On the other hand, experiments carried out for a few test systems lead to a conclusion that the cost could be noticeably lower if the modified APF control algorithm is applied. However, the modified control algorithm requires information about voltage distortions in all system buses and is significantly more complex than the classical one which requires only local data collected in each bus with the APF. The complexity of the method with the modified control algorithm is also evident if optimization procedure computing times are compared. They are a few times longer for the method based on the modified algorithm, so at the moment it cannot be applied for on-line control. However, it enables analysis of the system in order to find the best buses for APF allocation and the obtained results can be a reference point for the other methods. The buses for APF allocation must be selected very carefully, because the results of the research show that the wrong selection of the buses may lead to a significant rise of the financial cost. So the application of an optimization approach for this task should be an important step of the design procedure. Acknowledgment This research was supported by Polish Ministry of Science and Higher Education under the project number N N510 257338
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
7
References [1] H. Akagi, Modern active filters and traditional passive filters, Bull. Polish Acad. Sci. Tech. Sci. 54 (3) (2006) 255–269. _ [2] M. Macia˛zek, Power theories applications to control active compensators, in: Power Theories for Improved Power Quality, Power Systems Series, pp. 49–116, 2012. [3] T. Vaimann, J. Niitsoo, T. Kivipõld, T. Lehtla, Power quality issues in dispersed generation and smart grids, J. Electron. Electr. Eng. 18 (8) (2012) 23–26. [4] S. Biricik, O.C. Ozerdem, S. Redif, M.O.I. Kmail, Performance improvement of active power filter under distorted and unbalanced grid voltage conditions, J. Electron. Electr. Eng. 19 (1) (2013) 35–39. [5] A. Moradifar, H.R. Soleymanpour, A fuzzy based solution for allocation and sizing of multiple active power filters, J. Power Electron. 12 (5) (2012) 830– 841. [6] R. Keypour, H. Seifi, A. Yazdian-Varjani, Genetic based algorithm for active power filter allocation and sizing, Electr. Power Syst. Res. 71 (1) (2004) 41– 49. [7] Y.-Y. Hong, Y.-K. Chang, Determination of locations and sizes for active power line conditioners to reduce harmonics in power systems, IEEE Trans. Power Delivery 11 (3) (1996) 1610–1617. _ M. Pasko, Sizing of active power filters using some optimization strategies, Int. J. Comput. Math. Electr. Electron. Eng. [8] D. Grabowski, M. Macia˛zek, COMPEL 32 (4) (2013) 1326–1336. [9] W.M. Grady, PCFLO v6 Users manual, [Online]. Available:
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Compensation based on active power filters – The cost minimization _ ⇑, Dariusz Grabowski, Marian Pasko, Michał Lewandowski Marcin Macia˛zek Faculty of Electrical Engineering, Silesian University of Technology, ul. Akademicka 10, 44-100 Gliwice, Poland
a r t i c l e
i n f o
Keywords: Power quality Power system harmonics Optimization Active power filters
a b s t r a c t The paper deals with the problem of power quality and its solution. Passive or active compensators are usually used to reduce distortions of voltage and current waveforms. Active power filters seem to be the best choice if the effect of compensation is the only criterion. However, the financial cost of such a solution put the potential investors off. The paper shows that the cost could be reduced considerably if the optimization of active power filter allocation, as well as their nominal currents, is carried out. Two cases have been considered – active power filters with a classical control algorithm, which operates locally and aims at obtaining sinusoidal shape of waveforms in the point of installation, and active power filters with a modified control algorithm, which aims at reduction of waveform distortions in all nodes of an analyzed power system. Goal functions and optimization problems for both cases have been defined. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction There are more and more electrical and electronic devices available in the market which require high quality power supply. On the other hand, the number of devices, which cause power quality deterioration due to nonlinear behavior influencing the supplying power system, increases too. Moreover, dynamical changes of nonlinear loads during their normal operation make elimination of interferences impossible using passive compensators. Nowadays, active power filters (APF) [1,2] are the only solution which is able to improve power quality in contemporary power systems with distributed generation [3,4] and dynamical changes of supplying current and voltage distortions. Many publications on optimization of APF sizing and allocation can be found in technical literature worldwide. However, the proposed solutions are first of all based on minimization of the filter RMS currents or THD coefficients without direct application of economic criterion [5–7]. High cost of APFs leads to optimization of the system from the economical point of view, which has been presented in the paper and could lead to a wider application of APFs in the future. The paper shows that the minimization of APF sizes is not enough to obtain the cheapest possible solution. In this sense the proposed approach can be treated as an added value to the power quality field.
⇑ Corresponding author. _ E-mail addresses: [email protected] (M. Macia˛zek), [email protected] (D. Grabowski), [email protected] (M. Pasko), michal. [email protected] (M. Lewandowski). http://dx.doi.org/10.1016/j.amc.2015.01.001 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
2
2. Active power filters One of the distinctive features of active power filters is that they realize the function of a controlled voltage or current source. In this paper, parallel APFs, injecting additional current to the system, have been used. The current generated by the APF results in elimination of some unwanted components, e.g. higher harmonics in power system voltages and currents. APFs could have many applications depending on their structure:
filtering of current higher harmonics, filtering of voltage higher harmonics, voltage symmetrization, elimination of voltage flicker, load balancing, reactive power compensation for fundamental frequency.
A parallel APF is usually realized with the help of a three-phase power inverter followed by coils what is equivalent to a three-phase current source. In most of the applications the inverter is controlled by a tracking proportional–integral controller (PI) and the reference current is generated on the base of a given power theory [2]. In order to accomplish the task with the minimum delay (real-time operation) very fast VLSI (Very-Large-Scale Integration) systems should be used in the feedback control loop. Therefore, signal processors are the most popular solution. The idea of the compensation using a parallel APF has been presented in Fig. 1. Although APFs are the best solution for elimination of nonlinear load influence on power systems, the high cost of such compensation discourage potential investors from the APF application. It seems that optimization algorithms can bring about reduction of the compensation cost by such APF selection which ensures minimum costs while achieving the compensation goal [8]. 3. System under test An exemplary power system presented in Fig. 2 has been used to validate the developed optimization algorithms. It is a large power system described in [9] and it has been used to supply an expansive winter sport area. It includes buses supplying DC motors driven by six-pulse line-commutated adjustable speed drives. These nonlinear loads are the sources of higher harmonics in the power system under consideration. This system has been implemented as an example in PCFLO software [9]. This system has been chosen to be used for testing of the proposed solutions because: there are 37 nonlinear loads distributed in the system which are the source of higher harmonics, distances between buses exceed 1000 m, full data about the system, including line parameters, are available.
Fig. 1. Block diagram of a power system with a parallel APF.
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
3
Fig. 2. Block diagram of the power system under consideration.
4. Optimization problem Minimization of the higher harmonic compensator cost requires defining a goal function and constraints [10]. In this case the decision-making variables include current phasors for all considered harmonics and for all compensators placed in the system because the APF cost depends on its nominal current. Moreover, it has been assumed that the power system can be represented by a linear model when performing the frequency analysis and furthermore nonlinear loads are regarded as current sources for each harmonic. So the impedance matrix describing the power system for each harmonic can be determined separately. Currents and voltages in the system have been calculated using PCFLO software [9]. The optimization problem has been defined assuming that the APF connected to a system bus can be treated as an ideal current source injecting higher harmonics to the system. Such a source can be described by the Fourier series (phase index has been omitted in order to simplify notation, index k refers to the compensator): k
iw ðtÞ ¼
H pffiffiffi X 2Re Ikwh ejhx0 t ;
ð1Þ
h¼2
where H – number of harmonics generated by the compensator, x0 – fundamental frequency, Ikwh – phasor of an hth harmonic of the compensator located in the bus w:
k Ikwh ¼ jIkwh jej/wh ¼ Re Ikwh þ jIm Ikwh ;
ð2Þ
jIkwh j, /kwh – RMS value and phase of an hth harmonic of the compensator located in the bus w, Reð Ikwh Þ,Imð Ikwh Þ – real and imaginary part of an hth harmonic of the compensator located in the bus w. During the development the goal function, two cases have been considered: minimization of APF sizes is carried out assuming modified APF control algorithm which enables generating arbitrary current with given amplitude and phase spectrum (goal function f1) [8,11], minimization of APF sizes is performed assuming standard APF control algorithm based on instantaneous power theory [2] – minimization of the cost requires a compromise between minimum cost and minimum waveform distortions (goal function f2). 4.1. Goal function f1 The goal function f1 enables to take into account economical criterion by minimization of the compensation cost. Additionally, it has been assumed that the compensator price is a function g() of its nominal current:
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001
M. Macia˛z_ ek et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx
4
min f 1 ðxÞ ¼ x
W X gð jIkw jÞ; fReðIkwh Þ;ImðIkwh Þ;T½wgw¼1
min
ð3Þ
where W – total number of APFs, T – vector defining allocation of APFs, dim(T) = W, jIkw j – RMS value of the APF current located in bus T [w]:
jIkw j
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u H n o n o2 uX 2 : ¼t Re Ikwh þ Im Ikwh
ð4Þ
h¼2
The function g(), which reflects the cost of a single compensator, is usually discontinuous – some intervals of currents correspond to different costs being a result of an available series of APF nominal currents. An exemplary function g(), developed on the base of price lists of commercially available APFs, has been shown in Fig. 3. The function g() is both discontinuous and nonlinear. It limits the range of algorithms which can be applied to solve the optimization problem. Solution to (3) enables to answer the question: how many compensators and in which buses must be used in order to limit the maximum THDV coefficient and minimize the cost of this solution. The step shape of the function g() is a result of the available series of APF nominal currents and depends on the pricing policy of the APF manufacturer. Solutions to the problem for different functions g() may lead to completely different results – it could be especially important when comparing the offers of several manufacturers which could include APFs with different prices and nominal parameters. The discontinuity is the main disadvantage of the goal function f1. Thus, solution to the optimization task cannot be reached using gradient methods. This problem can be overcome using the interpolation of the function g() (for example by means of the spline functions). The other problem may arise if the goal function f1 has a few local minima for which the function values are similar and which correspond to solutions with different numbers and sizes of compensators. In order to force solutions with lower number of compensators, an additional coefficient depending on fixed cost caused by installation and maintenance expenses can be introduced. In this case the function g() should represent the relation between total costs (purchase price + installation and maintenance costs) and its size. As a result, solutions with lower number of compensators, and hence a lower fixed cost, will be automatically preferred. Classical optimization methods have some significant restrictions. Analytical methods search for the solution locally and require existence and continuity of goal functions. On the other hand, enumerative methods, either deterministic or random, are usually not too effective. Even in the case of small scale problems like the 17-bus power system shown in [11,12] a classical combinatorial algorithm leads to computation times of a few days for just two compensators. Therefore, besides application of classical optimization methods, the problem of compensator sizing and allocation represented by (3) has been solved using genetic algorithms. In this case optimization of compensator allocation requires only setting the maximum number of compensators and indices of buses to which they can be connected. The genetic algorithm finds the buses in which compensators should be connected as well as their parameters. Computing complexity of the analyzed problem increases along with the number of buses to which compensators can be connected and with the number of compensators. Computation time is longer than in the case of the classical APF control algorithm, but much more shorter than in the case of combinatorial approach. It allows to seek optimal solutions even for large scale power systems. The detailed description of the applied procedure and the solution obtained by means of the genetic algorithm for the 17bus power system including APF allocation and sizing for 1, 2 and 3 compensators, have been presented in [11].
100000
price (€)
80000 60000 40000 20000 0 0
100 200 300 400 500 600 700 800 900 nominal current (A)
Fig. 3. Exemplary relation between compensator price and its nominal current g().
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4.2. Goal function f2 It enables to find the minimum compensation cost and realizes a compromise between THDV coefficient improvement and financial expenses necessary to reach this improvement for a classical APF control algorithm. In this case the same function g(), which reflects the relation between the cost of a single compensator and its nominal current, can be taken into account: ½iþ1 maxw THDV w max THDV ½i P w w ; min f 2 ðxÞ ¼ min P W ½iþ1 W k k ½i x i g jI j g jI j w w w w
THDV w 6 THDV max
for w ¼ 1; . . . ; W;
ð5Þ
where i – index pointing at successive solutions obtained using compensators with a standard control algorithm sorted in ascending order in accordance with the maximum value of THDVw (w = 1, . . ., W), jIkw j½i – RMS value of the compensator current obtained for the solution number i, maxw THDV ½i w – maximum value of the THDV coefficient for all buses of the system reached if a given APF arrangement is used (the arrangement is pointed out by the index i in the sorted set of solutions), PW k ½i w gðjIw j Þ – cost of all connected APFs for the solution pointed out by the index i, THDV w 6 THDV max – constraint which enables finding a compromise between compensation cost and quality, namely increasing the value of THDVmax makes the compensation cost lower, but at the same time the supplying network waveform distortion level becomes higher; the acceptable value of THDVmax depends strongly on sensitivity of loads installed in the network under consideration and so it should be set individually.
(a)
0.1
2 bus voltage 1383
0
0
-2 0
0.005
0.010
0.015
i 1383-1384( t ), pu
u1384 ( t ), pu
line current1383-1384
-0.1 0.020
t, s
(b)
2
0.1 bus voltage 1383
0
0
-2 0
0.005
0.010
0.015
i 1383-1384 ( t ), pu
u 1384 ( t ), pu
line current1383-1384
-0.1 0.020
t, s
(c)
2
0.1 bus voltage 1383
0
0
-2 0
0.005
0.010
0.015
i 1383-1384 ( t ), pu
u 1384 ( t ), pu
line current1383-1384
-0.1 0.020
t, s Fig. 4. Exemplary voltage (bus 1348) and current (line 1383–1384) waveforms: (a) no compensation, (b) Variant II, (c) Variant III.
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relative cost of APF (%)
6
THDVmax=9.58
THDVmax=0.75
THDVmax=5
THDVmax=6.61
No APF
Variant I
Variant II
Variant III
variant of compensation Fig. 5. Comparison of the relative cost for three variants of compensation – goal function f2.
4.3. Simulation results for goal function f2 In this case optimization of the compensator allocation requires analysis of all combinations of APF connection points in the power system. The aim is reached by application of the combinatorial approach and simultaneous evaluation of the voltage distortion level and its economical impact. As a result the required number as well as the location (bus numbers) and sizes of APFs have been determined. Some exemplary current and voltage waveforms before and after compensation using different strategies have been shown in Fig. 4. The Variant II denotes solution obtained with the help of the goal function f2, for which requirements imposed by the IEEE standards are fulfilled, i.e. the maximum THDV coefficient is not greater than 5%. This state of the system has been reached applying 11 APFs located in buses 42, 199, 60, 196, 1189, 1190, 1354, 1376, 1383, 1319 and 1323 – the system layout is described in [9]. The Variant III corresponds to the solution which enables to achieve the best price/effect ratio using the goal function f2. This state has been reached applying 6 APFs located in buses 42, 60, 196, 1354, 1376 and 1383. The computation complexity depends on the number of buses to which the compensators can be connected, but the computation time is relatively short and it is possible to apply this approach for large scale power systems. The optimum solution ensures a compromise between voltage distortion reduction and the economic cost of the compensation. In order to follow this approach the maximum values of the THDV coefficients before and after compensation have been compared keeping in mind the cost of the compensators (see Fig. 3) necessary to reach this state. The final results have been shown in Fig. 5. Variant I represents results obtained for total compensation (relative cost equal to 100%), i.e. APFs connected to all buses with nonlinear loads. The solution, for which the IEEE standards are fulfilled (Variant II), is more than 50% cheaper comparing to the total compensation (Variant I). The last case presented in Fig. 5 corresponds to the solution which enables to achieve the best price/effect ratio (Variant III). It is 80% cheaper than the total compensation. 5. Conclusions The optimization of APF nominal currents and allocation has been carried out taking into account the requirements for voltage distortions (THDV) and reactive power compensation for the fundamental frequency. The research results presented in this paper show that using a classical APF control algorithm, it is possible to achieve a given voltage distortion level (around 6.6% in the analyzed exemplary power system) with considerably lower financial costs (around 80% for the test system) in comparison with the total compensation case. On the other hand, experiments carried out for a few test systems lead to a conclusion that the cost could be noticeably lower if the modified APF control algorithm is applied. However, the modified control algorithm requires information about voltage distortions in all system buses and is significantly more complex than the classical one which requires only local data collected in each bus with the APF. The complexity of the method with the modified control algorithm is also evident if optimization procedure computing times are compared. They are a few times longer for the method based on the modified algorithm, so at the moment it cannot be applied for on-line control. However, it enables analysis of the system in order to find the best buses for APF allocation and the obtained results can be a reference point for the other methods. The buses for APF allocation must be selected very carefully, because the results of the research show that the wrong selection of the buses may lead to a significant rise of the financial cost. So the application of an optimization approach for this task should be an important step of the design procedure. Acknowledgment This research was supported by Polish Ministry of Science and Higher Education under the project number N N510 257338
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References [1] H. Akagi, Modern active filters and traditional passive filters, Bull. Polish Acad. Sci. Tech. Sci. 54 (3) (2006) 255–269. _ [2] M. Macia˛zek, Power theories applications to control active compensators, in: Power Theories for Improved Power Quality, Power Systems Series, pp. 49–116, 2012. [3] T. Vaimann, J. Niitsoo, T. Kivipõld, T. Lehtla, Power quality issues in dispersed generation and smart grids, J. Electron. Electr. Eng. 18 (8) (2012) 23–26. [4] S. Biricik, O.C. Ozerdem, S. Redif, M.O.I. Kmail, Performance improvement of active power filter under distorted and unbalanced grid voltage conditions, J. Electron. Electr. Eng. 19 (1) (2013) 35–39. [5] A. Moradifar, H.R. Soleymanpour, A fuzzy based solution for allocation and sizing of multiple active power filters, J. Power Electron. 12 (5) (2012) 830– 841. [6] R. Keypour, H. Seifi, A. Yazdian-Varjani, Genetic based algorithm for active power filter allocation and sizing, Electr. Power Syst. Res. 71 (1) (2004) 41– 49. [7] Y.-Y. Hong, Y.-K. Chang, Determination of locations and sizes for active power line conditioners to reduce harmonics in power systems, IEEE Trans. Power Delivery 11 (3) (1996) 1610–1617. _ M. Pasko, Sizing of active power filters using some optimization strategies, Int. J. Comput. Math. Electr. Electron. Eng. [8] D. Grabowski, M. Macia˛zek, COMPEL 32 (4) (2013) 1326–1336. [9] W.M. Grady, PCFLO v6 Users manual, [Online]. Available:
_ Please cite this article in press as: M. Macia˛zek et al., Compensation based on active power filters – The cost minimization, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.001