Download 4. Test Example

Authors:
Jandecy Cabral Leite, Institute of Technology and Education Galileo of Amazonia (ITEGAM), Manaus, Brazil.
(e-mail: [email protected]) .
Ignacio Pérez Abril, Central University of Las Villas (UCLV), Cuba (e-mail: [email protected]).
Maria Emilia de Lima Tostes, Universidad Federal do Pará, Brazil (e-mail: [email protected]).
Roberto Celio Limão de Oliveira, Universidad Federal do Pará, Brazil. (e-mail: [email protected])
Title:
Multi-objective optimization of passive filters in industrial power systems
Abstract:
Passive shunt harmonic filters are a convenient means of power quality preservation as well as effective reactive
power compensators. The present paper considers a new multi-objective formulation for the multiple passive filters
selection and sizing problem that includes the maximization of the NPV of the compensation project. While
previous contributions solve the multi-objective problem by the minimization of a single objective function
composite to several sub-objectives, the present paper uses the Non-dominated Sorting Genetic Algorithm
(NSGA – II) for solving the problem. The developed optimization program is capable of determining a set of
optimal solutions which contain the type and the design parameters of the needed filters. The program can select a
proper filter configuration for each of the compensated buses. The effectiveness of the proposed procedure is tested
by solving a practical example.
Keywords:
Passive filters, harmonics, reactive power, optimization, genetic algorithms
1. Introduction
Passive shunt harmonic filters are a convenient means of power quality preservation as well as effective reactive
power compensators.
Their optimal planning has been solved by various optimization methods such as: sequential quadratic programming
[1], simulated annealing [2], interactive fuzzy procedure [3], differential evolution [4], neural network approach [5],
particle swarm optimization [6], genetic algorithm [7][8][9].
The problem formulation for the optimization of passive filters has been addressed using various approaches. The
employed problem formulations can be classified as: single-objective or multi-objective formulations.
All the formulations include a proper set of constraints that assures that the voltage and current distortion indexes
comply with power quality standards. However, the single-objective formulations such as: the minimization of
filters cost [2], or the maximization of the Net Present Value (NPV) that result from the filters installation benefits
[1], do not search for the reduction of these distortion indexes below the limits stated by power quality standards.
Commonly, the multi-objective problem formulations include various objectives to be minimized, such as:
a) The Total Demand Distortion (TDD) of the current at the Point of Common Coupling (PCC) [3-8].
b) The Total Harmonic Distortion (THD) of the voltage at every bus of the system [3-5, 7, 10].
c) The investment cost of the filters [3, 6, 10].
d) The active losses of the filters [4, 5, 7].
However, none of the multi-objective formulations consider the reactive power compensation benefits produced by
filters as an objective, and the amount of fundamental reactive power of the filters is normally treated as a constraint
to be fulfilled.
In an industrial facility the minimum required fundamental reactive power is imposed by the effect of the power
factor clause on the power bill [11][12]. However, this amount must be considered as a minimum limit to install.
The reactive power compensation benefits on the industrial power bill consider the annual benefits of: the power
factor increase, the power demand decrease and the energy losses reduction in the system components. The present
value of the evaluation of these benefits on N years, decreased by the investment cost of the filters is the NPV of the
compensation project.
The present paper considers a new multi-objective formulation for the multiple passive filters selection and sizing
that includes the maximization of the NPV of the compensation project by passive filters.
While previous contributions [4, 5, 7, 8, 10] solve the multi-objective problem by the minimization of a single
objective function composite to several sub-objectives, the present paper use the Non-dominated Sorting Genetic
Algorithm (NSGA – II) [13] for solving the problem.
The developed optimization program is capable of determining a set of optimal solutions for the multi-objective
optimization problem which state the type and the design parameters of the needed filters. The program can select a
proper filter configuration for each of the compensated buses.
The effectiveness of the proposed procedure is tested by solving a practical example.
2. Problem Formulation
Taken into account that various shunt passive filters can be placed in a single bus, in this paper will be define a
harmonic compensator HC as an array of multiple shunt filters, were each filter can be a single-tuned or a high-pass
damped filter.
Instead of defining a single HC configuration for all the selected buses, this approach considers four possible filters
configurations [14] that can be selected by the optimization program.
1. Several single-tuned filters to eliminate the low order harmonics.
2. A single high-pass second order filter for the entire spectrum of harmonics.
3. Several single-tuned filters to eliminate the low order harmonics and a high-pass second order filter for the
higher order harmonics.
4. Two high-pass second order filters for the lower and the higher order harmonics respectively.
The type 1 configuration (fig. 1) is composed by an array of m (m ≤ 4) single-tuned filters tuned to h1, h2, h3, h4, to
eliminate the most important voltage harmonics of low order (h ≤ 13) in the selected bus. The quality factor of each
branch is selected from (10 ≤ Q ≤ 50).
Single-tuned filters are not tuned exactly to the intended harmonic h. They are, instead, detuned to a lower frequency
value of about 0.95h. This detuning can prevent the harmonic component to be amplified due to the resonance that
arises in the lower vicinity of the driving point impedance after the single-tuned filter is installed [14, 15].
The type 2 configuration (fig. 2) is composed by a single high-pass second order filter to reduce a wide band of
harmonics. The filter is tuned at a frequency h selected in the range of the harmonic sources of the circuit. The
quality factor of the filter is selected from (0.5 ≤ Q ≤ 10).
The type 3 configuration (fig. 3) joins a type 1 configuration to eliminate the lower order frequencies h1, h2, h3, h4,
and a high-pass second order filter for the higher frequencies.
The high-pass second order filter is tuned at a frequency h5 selected above the maximum frequency filtered by the
single-tuned filters. The quality factor of the second order filter is selected from (0.5 ≤ Q ≤ 10).
The type 4 configuration is composed by two high-pass second order filters. The first one is tuned at a frequency h1
selected from the lower order frequencies (h ≤ 13) with a quality factor selected form (10 ≤ Q ≤ 50). The second
filter is tuned at a frequency h2 selected from the higher frequencies range (h > 13) with a quality factor selected
from (0.5 ≤ Q ≤ 10).
The parameters to be optimized for each HC are represented by the Table 1:
Cfg and m are integer values that determine the type of HC configuration and the number of single-tuned filters for
the HC 1 and 3 configurations.
Qc is a real value bounded from 0.5 to 1.5 times the maximum reactive power that is supplied form the bus where
the compensator is placed.
This reactive power Qc is divided between all the filters of the HC, proportional to the corresponding distribution
factors for reactive power, that are bounded in the range 1 to 10.
If the vector x represents the array of all the HC that will be optimized for certain problem, the proper sizing of the
elements of x implies the selection of the configuration and design parameters of all the HC to be placed.
2.1 Objective functions
To determine the economic effect of the reactive power compensation, all the typical daily load scenarios (L) must
be considered for calculating energy consumption and power factor.
For every scenario, the total active (PT) and reactive (QT) power supplied by the power source, as well as the active
and reactive losses in every circuit element (including the filters) can be calculated using a fundamental frequency
power flow program [14]. Using the calculated PT and QT values for each one of the typical scenarios, the maximum
demand of active and reactive power, as well as the active and reactive energy consumption of the facility can be
estimated for a characteristic work day. Thus, the monthly and annual power bill can be estimated if a number of
typical work days are assumed for the facility. This aggregation method can be more or less exact depending on the
load characterization of the facility.
The annual compensation benefits for the L typical scenarios are determined as the difference between: Cost(0) (the
calculated annual electrical consumption cost of the base case) and Cost(x) (the calculated annual consumption cost
after the installation of the x harmonic filters).
The filters investment cost I(x) depends on: the capacitor, reactor and resistor costs, the protection and switches cost,
and the enclosure cost. The capacitor, reactor and resistor costs are a linear function of its power for each voltage
level. Other components of cost can be assumed proportional to the filter reactive power [16].
Considering an evaluation period of N years with an interest rate i, the NPV of the harmonic compensation project is
calculated as:
N
NPV ( x)   I ( x)   (Cost (0)  Cost ( x)) /(1  i ) k
(1)
k 1
As the first objective the present paper formulates the minimization of the –NPV(x).
min f1 ( x)   NPV ( x)
(2)
Passive harmonic filters are primarily harmonic compensation devices designed to avoid the distorted current
circulation through the system elements and to reduce the voltage distortion at all buses of the system.
In order to evaluate the effect of the filters in the distortion indexes, all the possible system scenarios W must be
evaluated, which include the L typical load scenarios as well as other set of special load and system conditions.
These special conditions may include variations in the power source impedance, different operating modes for
harmonic producing loads, filters detuning, etc.
For each considered scenario k, the total distortion of the current at the PCC (TDDk) and the total voltage distortion
at each system bus i (THDk,i) can be calculated by a harmonic penetration program [14, 17].
TDDk 
I
hH
THDk ,i 
2
k ,h
V
hH
2
k ,i , h
IL
(3)
Vnomi
(4)
Where H is the set of the considered harmonics, IL is the maximum current demand at PCC [17] and Vnomi is the
nominal voltage at bus i.
As the second and third objectives of the filter installation project, the present paper formulates the minimization of
the distortion indexes (3) and (4) for the worst scenario, that is:
(5)
min f 2 ( x)  max kW TDDk ( x)
min f 3 ( x)  max kW THDk ,i ( x)
(6)
iU
Where the U set represents all the system buses.
2.2 Constraints
The optimization problem considers: the power quality constraints for current demand at PCC and for voltages at
every system buses, as well as the maximum stress constraints for filters components.
The rms voltage at every bus i of the distribution system for every probable scenario k must lie between the
minimum and maximum limits Vmini, Vmaxi respectively:

V min i  

V
hH
k ,i , h
2

 V max i

ikUW
(7)
The total harmonic distortion of the voltage must be lower than the limit of the IEEE-519 (THDLIMIT) for every bus i
and for every probable scenario k:
THD 
k ,i kW
iU
 THDLIMIT
(8)
The individual harmonic distortion of each harmonic h of the voltage must be lower than the limit of the IEEE-519
(IHDLIMIT(h)) for every bus i and for every probable scenario k:
V
Vnomi ikUW  IHDLIMIT (h)
k ,i ,h
(9)
hH
The total demand distortion must be lower than the limit of the IEEE-519 (TDDLIMIT) for every probable scenario k:
TDDk kW  TDDLIMIT
(10)
The individual harmonic demand distortion of each harmonic h must be lower than the IEEE-519 limit (IDDLIMIT(h))
for every probable scenario k:
I
k ,h
I L kW  IDDLIMIT (h)
hH
(11)
The IEEE Std. 18 [18] states the maximum limits for the power capacitor voltage, current and reactive power. Thus,
a set of constraints must assure the capacitor stress be lower than these limits.
The rms voltage at every capacitor c must be lower than the 110% of the capacitor rated voltage Vnomc for every
probable scenario k:



V
2
k ,c ,h
hH

  1.1Vnomc
ckCW
(12)
Where C represents the filter capacitors set.
The peak voltage at every capacitor must be lower than the 120% of the capacitor rated peak voltage for every
probable scenario k:
Vpeak 
k ,c kW
cC
 1.2 2Vnomc
(13)
The rms current at every capacitor must be lower than the 135% of the capacitor rated current Inomc for every
probable scenario k:



I
hH
2
k ,c , h

 1.35 Inomc

ckCW
(14)
The total reactive power generated by every capacitor must be lower than the 135% of the capacitor rated reactive
power Qnomc for every probable scenario k:


  Qk ,c,h   1.35Qnomc
hH
ckCW
(15)
To search for a feasible solution for the optimization problem, an additional objective function to minimize is
defined, which is evaluated as the quadratic sum of all constraints violations, that is:
min f 4 
 r  r
ri rLIMIT
i
LIMIT
2
(16)
Where ri and rLIMIT represent the calculated and the limit value for each of the (7-15) mismatched constraints.
3. Optimization procedure
The formulated problem is a highly non-linear discrete optimization problem whose solution requires a special
multi-objective optimization algorithm such as the NSGA-II. This algorithm is capable of obtaining the set of all
non-dominated solutions known as Pareto Frontier, which depicts the optimal tradeoffs that exist between competing
objectives [19].
A Matlab implementation of the NSGA-II algorithm [20] has been used for solving the presented problem. As is
common to genetic algorithms, only a function to evaluate the objectives for a particular solution is needed to
implement the optimization program. On the other hand, this implementation has been developed to work directly
with bounded real and integers variables.
3.1 Evaluate-objectives function
The NSGA-II evaluate-objectives function, evaluates all the fi declared objectives for a certain individual (solution
vector x) of the searched population.
function f = evaluate_objectives(x, data);
(17)
The parameter data represents all the needed data to evaluate the performance of the power system for a certain
compensation solution.
The main algorithm of the referred function includes the following steps:
1) Decoding the x solution vector and creating the corresponding set of filters to be placed on the candidate
system buses.
2) Calculating the filters investment cost.
3) For all the W possible system scenarios, the industrial power system operation is analyzed by a fundamental
frequency power flow and a harmonics penetration program.
4) With the calculated results for the L typical load scenarios, the power bill cost (including losses in all the
system components and the added filters) is estimated and the reactive power compensation objective (2) is
evaluated.
5) The voltages and current distortion indexes are calculated for every possible system scenario. The harmonics
compensation objectives (5, 6) are calculated.
6) The power quality constraints (7-11) and capacitor stress constraints (12-15) are checked.
3.2 Main optimization algorithm
The main optimization algorithm for solving the presented problem is the following:
1) Analyze the industrial power system base case (evaluate the initial cost and all the initial harmonics indexes).
2) Optimize x by the NSGA-II.
4) Select a solution from the NSGA-II final population.
4. Test Example
As a test example (see Appendix) for the presented methodology, the industrial power system depicted in fig. 5 has
been used.
Three typical load scenarios (1, 2, 3) of 6, 10, and 8 hours/day respectively are considered for evaluating the annual
power bill of the industrial facility for 12 months of 30 days.
Two special scenarios (4, 5) are pessimistic conditions with minimal linear load, maximum harmonics injections and
reduced short circuit capacity at the PCC. These two later scenarios add a drift from nominal values of the
capacitance (C) and inductance (L) for all the selected filters: +5%C and +2.5%L in scenario 4 and +10%C and
+5%L in scenario 5.
The buses (N4, N8 and N10) are selected as candidates for the filters placement.
The compensation project is evaluated for a five years period of time with a 10% of interest rate.
4.1 Initial results (base case)
The example initial annual consumption cost is estimated and the maximum TDD and THD are found (Table 2).
On the other hand, violations to IEEE 519 THD and individual harmonic voltage limit are found at several buses.
4.2 NSGA-II results
After 100 generations, the genetic algorithm produces a population of 300 solutions for the example. Only the
feasible ones are shown in the fig. 6.
In order to select a single solution for the problem, and considering that the IEEE 519 voltage distortion limits are
violated in the base case, the obtained solutions were sorted in ascending order of their three objective functions
values: first the maximum THD, then the maximum TDD and finally the –NPV. The results of this sorting are shown
in fig. 7.
The Table 3 shows the corresponding HC parameters for the solution of minimal maximum THD.
Once the selected compensators are placed, the maximum TDD and THD objectives are decreased to 36% and 33%
respectively and a good NPV value is obtained (Table 4).
To evaluate the effect of possible variations of the C and L parameters on the performance of the selected filters for
the considered scenarios, three detuning conditions for the filters were analyzed: (a). nominal filters (the selected L
and C values), (b). +10% and +5% drift for C and L respectively and (c). – 5% drift for L. The results of the analysis
are shown in fig. 8, were the obtained maximum TDD and maximum THD for these three detuning conditions are
compared with the base case results (without compensators).
As can be seen (fig. 8), the selected solution reduces de TDD and THD maximum values for all analyzed system
scenarios and every detuning conditions. In general, the maximum TDD is lowered to (35% - 37%) and the
maximum THD is lowered to (33% - 38%) with respect to base case.
Finally, a frequency scan was performed on several buses of the test system for the previously presented detuning
conditions showing that the most harmful condition occur in bus N8 for the scenario 5 (fig. 9). However, the
impedance peak occurs at a frequency not presented in the harmonic spectrum of the non-linear loads (Table A.10).
5. Conclusion
This paper presents a NSGA-II optimization algorithm application which is capable of determining a set of good
solutions for the harmonics compensation problem. As any genetic algorithm, NSGA-II does not assure the
optimality of the solutions.
Instead of using a predetermined configuration, the algorithm can select between four predefined types a
configuration for each HC that produces a good overall solution for the problem. This feature cannot guarantee the
selection of the optimal configuration in each case. On the other hand, the user can reduce the set of configurations
available for a certain bus or even specify the configuration for it.
Several typical and special scenarios of load and system conditions can be analyzed for obtaining a solution with
good performance in all the scenarios. This cannot guarantee that the solution will have a good performance in a
scenario not analyzed. So, additional analysis could be required to evaluate the benefits of a certain solution.
The method allows for the needed constraint set to be formulated such that IEEE-519 harmonic distortion limits can
be complied with, as well as filter stress constraints are checked.
Acknowledgements
The authors acknowledge the Institute of Technology Galileo of Amazon (ITEGAM) and the Amazonas Research
Foundation (FAPEAM) for the financial support as doctorate grant for performing this work.
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[20]
Aravind
Seshadri,
NSGA-II
source
code
available
in
http://www.mathworks.com/matlabcentral/fileexchange/10429-nsga-ii-a-multi-objective-optimizationalgorithm/content/NSGA-II/