Multiobjective Environmental/Economic Dispatch considering Prohibited
Operating Zones
Robert T. F. Ah King*
Department of Electrical and Electronic Engineering
Faculty of Engineering
University of Mauritius
Reduit
Mauritius
Email: r.ahking@uom.ac.mu
Harry C. S. Rughooputh
Department of Electrical and Electronic Engineering
Faculty of Engineering
University of Mauritius
Reduit
Mauritius
Email: r.rughooputh@uom.ac.mu
Kalyanmoy Deb
Kanpur Genetic Algorithms Laboratory
Department of Mechanical Engineering
Indian Institute of Technology
Kanpur, PIN 208 016
India
Email: deb@iitk.ac.in
Abstract
A thermal or hydro generating unit may have prohibited operating zones due to physical
limitations of power plant components caused, for example, by vibrations in a shaft bearing
which are amplified in a certain operating region. In this case, the unit can only operate
above or below the prohibited operating zone. The effect of prohibited operating zones on
the multiobjective environmental/economic dispatch problem is investigated in this paper.
The IEEE 30-bus system with transmission losses has been considered with prohibited
operating zones and simulation performed using NSGA-II for two constraint handling
methods, namely penalty parameterless and decoder-based approaches. Simulation results
show
that
both
methods
can
equally handle
this
particular
scenario
in
the
environmental/economic dispatch problem. Different number of prohibited operating zones
was considered for the six generators in the system yielding discontinuous and non-convex
non-dominated fronts. A knee can also be obtained in some cases and the best compromise
solution would be this particular solution.
Keywords: Power Systems, Environmental/Economic Dispatch, Prohibited Operating Zones,
Nondominated Sorting Genetic Algorithm.
*For correspondences and reprints
1. Introduction
Economic dispatch aims at finding the minimum fuel cost of generating units for a particular
load demand on the power system. Due to rising environmental concerns, utilities can no
longer focus on this sole objective. On the other hand, emission dispatch minimizes the
emissions from the power generating plant. Thus, this has given rise to the multiobjective
environmental/economic dispatch or economic emission dispatch
problem which
simultaneously considers both cost and emission objectives.
In practice, a thermal or hydro generating unit may have prohibited operating zone(s) due to
physical limitations of power plant components (e.g. vibrations in a shaft bearing are
amplified in a certain operating region (Lee and Breipohl 1993). The unit can only operate
above or below the prohibited operating zone. Prohibited operating zones will cause the
decision space to be non-convex yielding a non-convex optimization problem which cannot
be solved by classical methods such as the Lagrangian Relaxation (LR) method. The nonconvex economic dispatch problem could be solved by a dynamic programming approach but
its computational requirements are quite large even for a small system. Lee and Breipohl
(1993) and Fan and McDonald (1994) have proposed methods based on conventional LR
approach for solving the reserve constrained economic dispatch problem with prohibited
operating zone(s) after decomposing the non-convex decision space. A genetic algorithm
was used by Orero and Irving (1996) to solve the non-convex economic dispatch problem.
Su and Chiou (1997) considered an enhanced Hopfield neural network to solve the economic
dispatch with prohibited operating zones problem. Moreover, Jayabarathi et al. (1999) used
an evolutionary programming approach whilst Gaing (2003) presented a particle swarm
optimization to handle the problem under consideration. All of the above mentioned studies
have considered the economic dispatch problem with regards to prohibited operating zones.
More recently, evolutionary algorithms have emerged as an effective tool for multiobjective
optimization and have been applied to the problem at hand (Abido 2006, Ah King et al.
2005).
In this paper, the multiobjective dispatch problem comprising of simultaneous
minimization of fuel cost and emissions with consideration of prohibited operating zones by
the elitist Nondominated Sorting Genetic Algorithm (NSGA-II) is studied.
This work
represents a first attempt to study the effect of prohibited operating zones on the
multiobjective environmental/economic dispatch problem with focus on the best compromise
solution which represents the operating point which the power system operator shall consider
in practice.
2. Environmental/Economic Dispatch Problem
Environmental/economic dispatch is a multiobjective problem with conflicting objectives
because pollution minimization is conflicting with minimum cost of generation.
The
environmental/economic dispatch involves the simultaneous optimization of fuel cost and
emission objectives which are conflicting ones. The problem is formulated as described
below.
2.1
Fuel Cost Objective
The classical economic dispatch problem of finding the optimal combination of power
generation which minimizes the total fuel cost while satisfying the total required demand can
be mathematically stated as follows (Wood and Wollenberg 1996):
n
F   ai  bi PGi  ciPGi2
(1)
i 1
where F is total fuel cost ($/hr); ai, bi, ci are fuel cost coefficients of generator i; PGi: is real
power generated by generator i (MW); and n is number of generators.
2.2
Emission Objective
The minimum emission dispatch as opposed to the above classical economic dispatch
minimizes the NOx emission objective which can be modeled using second order polynomial
functions (Talaq et al. 1994):
NOx Emission Objective
n
E NO x   (a iN PGi2  biN PGi  ciN )
(2)
i 1
Unit of E NO X is ton/hr.
The optimization problem is bounded by the following constraints:
2.3
Power Balance Constraint
This is represented by an equality constraint:
n
P P
i 1
i
D
 PL  0
(3)
where PD is total load (MW), and PL is transmission losses (MW).
To calculate the transmission losses, a loadflow or powerflow solution has to be carried out
and this involves an iterative process using Newton-Raphson method. In this case, the
equality constraints on real and reactive power at each bus as follows (Abido 2006):
N
PGi  PDi  Vi V j [Gij cos( i   j )  Bij sin(  i   j )]  0
(4)
j 1
N
QGi  QDi  Vi V j [Gij sin(  i   j )  Bij cos( i   j )]  0
(5)
j 1
where Vi and V j are voltages at bus i and j ;  i and  j are voltage angles at bus i and j ;
QGi and Q Di are reactive power generated and reactive power demand at bus i ; and G ij and
Bij are real and imaginary components of YBUS .
Then, the real power loss in transmission lines can be calculated as:
NL
PL   g k [Vi 2  V j2  2ViV j cos( i   j )]
(6)
k 1
where NL is the number of transmission lines and g k is the conductance of the k th line that
connects bus i to bus j .
2.4
Power Generation Limits Constraint
The power generated PGi by each generator should lie between its minimum and maximum
limits, i.e.,
PGi min  PGi  PGi max
(7)
where PGi min is minimum power generated; and PGi max is maximum power generated.
2.5
Prohibited Operating Zones
Many studies have simplified the economic dispatch problem by assuming that the generation
output of a unit can be adjusted instantaneously to meet its share of the load. However, in
practice, the operating range of all on-line units is restricted by their ramp rate limits for
forcing the units operation continually between two adjacent specific operation period.
Consequently, the inequality constraints due to ramp rate limits are given (Chen and Chang
1995):
1) if generation increases
Pi  Pi 0  URi
(8)
Pi  Pi 0  DRi
(9)
2) if generation decreases
where Pi is the current power output and Pi 0 is the previous output power. URi is the upramp
limit of the i-th generator (MW/hr) and DRi is the downramp limit of the i-th generator
(MW/hr). Practically, DRi is greater than URi .
The feasible operating zones are described by the following (Lee and Breipohl 1993):
Pi min  Pi  Pi ,l1
Pi ,uj 1  Pi  Pi ,l j ,
j  2,3,..., ni
(10)
Pi ,uni  Pi  Pi max
where j is the number of prohibited zones of unit i.
Combining (1), (3), (8), (9) and (10), the economic dispatch problem when considering ramp
rate limits and prohibited operating zones is modified as (Gaing 2003):
n
Minimize:
F   f i ( Pi )
i 1
n
Subject to:
P  P
i 1
i
D
 PL
max( Pi min , Pi 0  DRi )  Pi  min( Pi max , Pi 0  URi ) and
 Pi min  Pi  Pi ,1l

Pi   Pi ,uj 1  Pi  Pi ,l j ,
 u
max
 Pi ,ni  Pi  Pi
(11)
j  2,3,..., ni , i  l ,..., n
The generator cost function (input-output curve) is obtained from data points taken during
"heat run" tests, when input and output data is measured as the unit is slowly varied through
its operating region (Noyola et al. 1990).
Wire drawing effects, occurring as each steam
admission valve in a turbine starts to open, produce a rippling effect on the unit input-output
curve. These "valve-points" are illustrated in Figure 1.
Figure 1: Valve-point effects on the input-output curve (source: IEEE Committee Report
1971)
To model the effects of valve-points, a recurring rectified sinusoid contribution was added to
the input-output equation. The result is (Walter and Sheble 1993):
Fˆi ( PGi )  Fi ( PGi )  ei sin( f i ( Pi min  Pi )
(12)
3. Constraint Handling
Any real world optimization problem will be subjected to constraints in the form of equality
or non-equality constraints. These constraints are taken into account by including additional
terms in the objective function thus causing solutions having constraint violation to have
larger objective function values and thus lower fitnesses.
A common method for handling constraint is by adding an exterior term (Deb, 1995) which
penalizes infeasible solutions to the objective function:
J
K
j 1
k 1
(13)
F ( x)  f ( x)   R j g j ( x )   rk hk ( x) ,
where R j and rk are user-defined penalty parameters. The operator
denotes the absolute
value of the operand if the operand is negative while it returns a value of zero if the operand
is non-negative.
3.1
Penalty Parameterless Approach
In an attempt to circumvent problems associated with methods based on penalty functions
such as artificial locally optimal solutions, Deb proposed a method which does not need any
penalty parameter. The transformed objective function is (Deb 2000):
f ( x), if x is feasible;


J
K
F ( x)  
f max   g j ( x )   hk ( x) , otherwise.

j 1
k 1
(14)
f max is the objective function value of the worst feasible solution in the population.
Figure 2 clearly shows how the transformed objective function is obtained and that an
unfeasibe solution always has an objective function which is larger than the worst feasible
solution.
Figure 2: Constraint handling with the penalty parameterless method (Deb 2000)
3.2
Decoder-based Method
A decoder-based approach for solving constrained numerical optimization problems was
proposed by Koziel and Michalewicz (1998). Their method uses a homomorphous mapping
between n-dimensional cube and a feasible search space as described below.
The
homomorphous mapping  transform the n-dimensional cube [1,1]n into the feasible region
F where F can be concave or consists of disjoint (non-convex) regions.
The search space S is defined as a Cartesian product of domains of variables, l (i)  xi  u (i) ,
for 1  i  n , whereas a feasible part of F of the search space is defined by problem specific
constraints g j ( x)  0 , for j  1,..., q . Then, any boundary point s of the search space S
defines a line segment L between r0 and s . The one-to-one mapping g :[1,1]n  S can be
defined as
g ( y )  x , where xi  yi
u (i )  l (i ) u (i )  l (i )

, for i  1,..., n .
2
2
(15)
A line segment L between any reference point r0  F and a point s at the boundary of the
search space S , is defined as
L(r0 , s)  r0  t.(s  r0 ) , for 0  t  1 .
(16)
Thus, a one-to-one mapping  :[1,1]n  F for convex feasible search spaces F can be
established as follows:
r  ymax .t0 .( g ( y / ymax )  r0 ) if
if
 r0
 ( y)   0
y0
(17)
y0
where r0  F is a reference point and ymax  maxin1 yi
If we have k intervals of feasibility [t1 , t2 ],...,[t2 k 1 , t2 k ] instead of [0, t0 ] , an additional
mapping  which transforms the interval [0,1] into sum of intervals [t2i 1 , t2i ] :
(t2i 1 , t2i ] .
(18)
(t2i 1 , t2i ]  (0,1]
(19)
 : (0,1] 
k
i 1
The reverse mapping  is defined as:
:
k
i 1
as follows:
 (t )  (t  t2i 1   j 1 d j ) / d ,
i 1
where d j  t2 j  t2 j 1 , d   j 1 d j and t2i 1  t  t2i .
k
(20)
Thus, the mapping  is reverse of  :
 (a)  t2 j 1  d j
a   (t2 j 1 )
(21)
 (t2 j )   (t2 j 1 )
where j is the smallest index such that a   (t2 j ) .
The mapping  that transforms the constrained optimization problem into an unconstrained
one for every feasible set F is given by:
r  t0 .( g ( y / ymax )  r0 ) if
if
 r0
y0
 ( y)   0
(22)
y0
where r0  F is a reference point, ymax  maxin1 yi and t0   ( ymax ) .
A function i of one independent variable t (for a fixed reference r0  F and the boundary
point s of S ) can be used to represent each of the m constraints gi ( x)  0 as follows:
i (t )  gi ( L(r0 , s))  gi (r0  t.(s  r0 )) , for 0  t  1 and i  1,..., m .
By
partitioning
the
interval
[0,1]
into
v
subintervals
(23)
[v j 1 , v j ] ,
where
v j  v j 1  1/ v (1  j  v) so that equations i (t )  0 have at most one solution in every
subinterval so that the points of intersection can be determined by a binary search.
4. Decision Making
Out of the non-dominated solutions constituting the Pareto optimal front, the decision maker
may have to select one solution. Mathematically, every Pareto optimal point is an equally
acceptable solution of the multiobjective optimization problem (Miettinen 1999). To select
this operating point calls for the information not contained in the objective function and this
detail differentiate a multiobjective optimization problem from a single objective
optimization problem according to Miettinen. In this work, fuzzy set theory is used to select
this operating point and the method is described as follows.
4.1
Best Compromise Solution using Fuzzy Set Theory
A multiobjective optimization algorithm generates the non-dominated set of solutions known
as the Pareto-optimal solutions. The decision maker who is in this context the power system
operator may have imprecise or fuzzy goals for each objective function. To aid the operator
in selecting an operating point from the obtained set of Pareto-optimal solutions, fuzzy set
theory is applied to each objective functions to obtain a fuzzy membership function  f i as
follows (Dhillon et al. 1993):
1

f i max  f i

 fi  
max
min
 fi  fi
0
f i  f i min
f i min  f i  f i max
(24)
f i  f i max
The best non-dominated solution can be found when Equation (25) is a maximum where the
normalized sum of membership function values for all objectives is highest.
N
k 

M
i 1
N
k
fi
 
k 1 i 1
(25)
k
fi
where M is the number of non-dominated solutions and N is the number of objective
functions.
5. Simulation Results
Previous studies (Abido 2006, Ah King et al. 2005) have presented results for the IEEE 30bus system to define a clear Pareto front for this test system. For ease of reference, the
nondominated solutions obtained in Ah King et al. (2005) are reproduced in Figure 3 together
with best objective and compromise values in Table 1.
0.22
NSGA-II
epsilon-constraint
NOx Emission (ton/hr)
0.215
0.21
0.205
0.2
0.195
0.19
605
610
615
620
625
630
Fuel Cost ($/hr)
635
640
645
Figure 3: Nondominated solutions for original system without Prohibited Operating Zones
(Ah King et al. 2005)
Table 1: Best fuel cost, best NOx emission and best compromise for original system without
Prohibited Operating Zones
Best fuel cost
PG1
PG2
PG3
PG4
PG5
PG6
Cost
Emission
0.1182
0.3148
0.5910
0.9710
0.5172
0.3548
607.801
0.21891
Best NOX emission
0.4141
0.4602
0.5429
0.4011
0.5422
0.5045
0.19419
644.133
Best compromise
solution
0.2697
0.3645
0.5545
0.6951
0.5619
0.4186
616.502
0.20089
In this study, the IEEE 30-bus system with transmission losses has been considered with
prohibited operating zones and simulation performed using NSGA-II for two constraint
handling methods, namely penalty parameterless (Deb 2000) and decoder-based (Koziel and
Michalewicz 1998) approaches described in Section 3. As discussed in Section 2.5, the ramp
rate limit restricts the generation level of a unit to lower and upper values and is relevant
during the transition period of generator scheduling.
This constraint is relaxed in the
simulations since it is transitional.
A population size of 100 individuals has been chosen and 500 generations of NSGA-II were
completed for each simulation run. Different number of prohibited operating zones were
considered for the six generators in the system yielding discontinuous and non-convex nondominated fronts as compared to previous studies (Abido 2006, Ah King et al. 2005) which
did not consider prohibited operating zones.
Figures 3 to 8 report the most interesting fronts obtained for 6 cases. The corresponding
prohibited operating zones for the respective cases are given in Tables 2, 4, 6, 8, 10 and 12.
The corresponding best fuel cost, best NOx emission and best compromise solution are given
in Tables 3, 5, 7, 9, 11 and 13 respectively.
Results reveal that both the parameter-less and decoder-based constraint handling techniques
are effective as NSGA-II achieves the same results with these two different methods.
As seen from the nondominated solutions obtained, the extreme solutions (minimum (best)
cost and minimum (best) emission solutions) are preserved in the cases considered as long as
the optimal solution does not require power generation in prohibited operating zones.
However, it does matter for instance for other solutions, more particular for the best
compromise solution which may be any solution along the front. Starting from Case POZ 1
to Case POZ 3, as more prohibited operating zones are included, the number of
discontinuities in the front increases. As expected, when generation (PG2) is not in the
prohibited operating zones, the shape of the nondominated front is not affected as confirmed
by Case POZ 5 from Case POZ 4 as opposed to Case POZ 6 (PG4).
According to Branke et al. (2004), the most interesting solutions of the Pareto-optimal front
are solutions where a small improvement in one objective would lead to a large deterioration
in at least one other objective. These solutions are sometimes also called “knees”. It is worth
noting that in the cases where prohibited operating zones are considered, a knee is obtained
and the best compromise solution obtained using fuzzy set theory (Equation (25)) is not
necessarily the knee solution as seen from Cases POZ 1, POZ 4 and 5.
Table 2: Prohibited operating zones for Case POZ 1
Generator
Number
1
2
3
4
5
6
Prohibited Operating Zones
(p.u.)
None
[0.30,0.45]
[0.55,0.60]
[0.60,0.80]
[0.50,0.55]
[0.30,0.55]
Figure 4: Nondominated solutions for Case POZ 1
Table 3: Best fuel cost, best NOx emission and best compromise for Case POZ 1
Best fuel cost
PG1
PG2
PG3
PG4
PG5
PG6
Cost
Emission
0.1199
0.3000
0.6086
0.9877
0.5512
0.2997
608.169
0.22135
Best NOX emission
0.4017
0.4547
0.5316
0.3768
0.5501
0.5505
646.692
0.19428
Best compromise
solution
0.2342
0.4502
0.5368
0.5995
0.4936
0.5506
625.361
0.19840
Table 4: Prohibited operating zones for Case POZ 2
Generator
Number
1
2
3
4
5
6
Prohibited Operating Zones
(p.u.)
None
[0.30,0.45]
[0.55,0.60]
[0.40,0.95]
[0.50,0.55]
[0.30,0.55]
Figure 5: Nondominated solutions for Case POZ 2
Table 5: Best fuel cost, best NOx emission and best compromise for Case POZ 2
Best fuel cost
PG1
PG2
PG3
PG4
PG5
PG6
Cost
Emission
0.1204
0.2999
0.6101
0.9871
0.5503
0.2992
608.175
0.22132
Best NOX emission
0.4041
0.4500
0.5316
0.3796
0.5502
0.5500
646.429
0.19428
Best compromise
solution
0.2273
0.2997
0.5414
0.9501
0.5504
0.2989
609.655
0.21558
Table 6: Prohibited operating zones for Case POZ 3
Generator
Number
1
2
3
4
5
6
Prohibited Operating Zones (p.u.)
[0.10,0.20] [0.30,0.40]
[0.30,0.45]
[0.55,0.60] [0.65,0.75] [0.90,1.00]
[0.40,0.50] [0.60,0.70] [0.80,0.95]
[0.50,0.55]
[0.35,0.45] [0.50,0.55]
Figure 6: Nondominated solutions for Case POZ 3
Table 7: Best fuel cost, best NOx emission and best compromise for Case POZ 3
Best fuel cost
PG1
PG2
PG3
PG4
PG5
PG6
Cost
Emission
0.0992
0.2999
0.6314
0.9887
0.4974
0.3499
607.884
0.22146
Best NOX emission
0.4124
0.4569
0.5488
0.3955
0.5527
0.4983
644.119
0.19421
Best compromise
solution
0.2756
0.2996
0.5482
0.7210
0.5684
0.4522
615.940
0.20238
Table 8: Prohibited operating zones for Case POZ 4
Generator
Number
1
2
3
4
5
6
Prohibited Operating Zones
(p.u.)
[0.10,0.40]
None
[0.50,0.60]
None
[0.50,0.56]
None
Figure 8: Nondominated solutions for Case POZ 4
Table 9: Best fuel cost, best NOx emission and best compromise for Case POZ 4
Best fuel cost
PG1
PG2
PG3
PG4
PG5
PG6
Cost
Emission
0.0965
0.2927
0.6121
0.9954
0.4829
0.3877
608.019
0.22145
Best NOX emission
0.4151
0.4692
0.4998
0.4057
0.5600
0.5162
644.899
0.19430
Best compromise
solution
0.4012
0.3719
0.5000
0.7063
0.4813
0.4057
622.496
0.20020
Table 10: Prohibited operating zones for Case POZ 5
Generator
Number
1
2
3
4
5
6
Prohibited Operating Zones
(p.u.)
[0.10,0.40]
[0.25,0.26] [0.35,0.36]
[0.50,0.60]
None
[0.50,0.56]
None
Figure 8: Nondominated solutions for Case POZ 5
Table 11: Best fuel cost, best NOx emission and best compromise for Case POZ 5
Best fuel cost
PG1
PG2
PG3
PG4
PG5
PG6
Cost
Emission
0.1000
0.3233
0.6055
0.9814
0.4884
0.3684
607.898
0.22005
Best NOX emission
0.4205
0.4682
0.4999
0.3993
0.5605
0.5179
645.725
0.19430
Best compromise
solution
0.4021
0.3485
0.4995
0.7051
0.4971
0.4142
622.307
0.20030
Table 12: Prohibited operating zones for Case POZ 6
Generator
Number
1
2
3
4
5
6
Prohibited Operating Zones
(p.u.)
[0.10,0.40]
None
[0.50,0.60]
[0.40,0.60]
[0.50,0.56]
None
Figure 9: Nondominated solutions for Case POZ 6
Table 13: Best fuel cost, best NOx emission and best compromise for Case POZ 6
Best fuel cost
Best NOX emission Best compromise
solution
PG1
0.0995
0.4039
0.4005
PG2
0.3216
0.4654
0.3549
PG3
0.6049
0.6004
0.4992
PG4
1.0057
0.3956
0.6865
PG5
0.4833
0.4997
0.4938
PG6
0.3522
0.4985
0.4315
Cost
607.920
643.906
623.233
Emission
0.22189
0.19445
0.19952
6. Conclusions
The multiobjective environmental/economic dispatch problem with prohibited operating
zones has been addresses for the first time. Simulations performed using NSGA-II for two
constraint handling methods, namely penalty parameterless and decoder-based approaches on
the IEEE 30-bus system with prohibited operating zones. Results reveal that both constraint
handling
methods
can
equally
deal
with
this
particular
scenario
in
the
environmental/economic dispatch problem. Six cases of prohibited operating zones on the
different generators in the system were presented and it has been shown that the nondominated fronts are discontinuous and non-convex in such cases.
The extreme solutions (minimum (best) cost and minimum (best) emission solutions) are
preserved in the cases considered as long as the optimal solution does not require power
generation in prohibited operating zones. However, the best compromise solution can be
affected when prohibited operating zones are considered. It is also interesting to note that the
best compromise solution obtained using fuzzy set theory is not necessarily the knee solution.
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