Solving Unit Commitment Problem Using Chemo-tactic PSO–DE Optimization Algorithm Combined with Lagrange Relaxation

ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010

Solving Unit Commitment Problem Using
Chemo-tactic PSO–DE Optimization Algorithm
Combined with Lagrange Relaxation
P.Praveena1, K.Vaisakh2, and S.Rama Mohana Rao3
1
Andhra University/Department of Electrical Engineering, Visakhapatnam, India
Email: nambaripraveena@yahoo.co.in
2
3
Email: { vaisakh_k@yahoo.co.in, ramu_sanchana@yahoo.com}

Abstract—This paper presents Chemo-tactic PSO-DE higher dimension problem the problem size increases
(CPSO-DE) optimization algorithm combined with rapidly with the number of generators and this requires
Lagrange Relaxation method (LR) for solving Unit enormous computation time and large memory space.
Commitment (UC) problem. The proposed approach Branch-and-bound and mixed integer linear
employs Chemo-tactic PSO-DE algorithm for optimal
settings of Lagrange multipliers. It provides high-quality
programming method also requires large computation
performance and reaches global solution and is a hybrid time and memory space [3], [4]. Lagrange relaxation
heuristic algorithm based on Bacterial Foraging for UC problem is advanced than dynamic
Optimization (BFO), Particle Swarm Optimization (PSO) programming due to its faster computational time. The
and Differential Evolution (DE). The feasibility of the solution to Lagrange Relaxation for UC problem
proposed method is demonstrated for 10-unit, 20-unit, depends on the updating of Lagrange multipliers; hence
and 40-unit systems respectively. The test results are it suffers from solution quality problem. This paper
compared with those obtained by Lagrangian relaxation proposes a new hybrid heuristic method for solving UC
(LR), genetic algorithm (GA), evolutionary programming
problem. The proposed method is developed in such
(EP), and genetic algorithm based on unit characteristic
classification (GAUC), enhanced adaptive Lagrangian way that a Chemo-tactic PSO-DE optimization
relaxation (ELR), integer-coded genetic algorithm technique is applied to update Lagrange multipliers and
(ICGA) and hybrid particle swarm optimization (HPSO) this improves the performance of LR method. To
in terms of solution quality. Simulation results show that illustrate the effectiveness of the proposed method, it is
the proposed method can provide a better solution. tested and compared to the conventional LR [5], GA
[5], EP [6], GAUC [7], ELR [8], ICGA [9] and HPSO
Index Terms—Lagrangian Relaxation, Particle Swarm [10] for 10-unit, 20-unit, and 40-unit respectively.
Optimization, Differential Evolution, Bacterial Foraging In 2001, Prof. K. M. Passino proposed an
Optimization, Unit Commitment
optimization technique known as Bacterial Foraging
Optimization Algorithm (BFOA) based on the foraging
I. INTRODUCTION
strategies of the E. Coli bacterium cells [11]. Until date
Unit commitment (UC) is used to commit the there have been a few successful applications of the
generators such that the total production cost over the said algorithm in optimal control engineering,
predicted load demand for scheduled time horizon is harmonic estimation [12], transmission loss reduction
minimized considering the spinning reserve and [13], machine learning [14] and so on. Experimentation
operational constraints of generator units [1], [2]. Unit with several benchmark functions reveal that BFOA
commitment is a high dimensional, nonlinear, non- possesses a poor convergence behavior over multi-
convex, mixed-integer combinatorial optimization modal and rough fitness landscapes as compared to
problem. Priority list method, integer programming, other naturally inspired optimization techniques like the
dynamic programming (DP), branch-and-bound Genetic Algorithm (GA), Particle Swarm Optimization
methods, mixed-integer programming, and Lagrange (PSO) and Differential Evolution (DE). Its performance
relaxation (LR) are a few methods developed up to now is also heavily affected with the growth of search space
for solving UC problem. dimensionality. In 2007, Kim et al. proposed a hybrid
In priority list method, priority of units is determined approach involving GA and BFOA for function
from full load average production cost of the unit. The optimization [15]. The proposed algorithm
method is simple but the quality of solution is low. outperformed both GA and BFOA over a few
Priority list of units is also considered in Dynamic numerical benchmarks and a practical PID tuner design
programming method. In spite of many advantages problem [16-19].
such as ability to maintain solution feasibility, for

50
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II. NOMENCLATURE objective function can be formulated mathematically as
an optimization problem as follows:
Pi t : Generation output power of unit i at hour t. T N
F ( Pi t , U it ) = ∑∑[ Fi ( Pi t ) + STi t (1 − U it −1 )]U it (1)
U it :
Status of unit i at hour t (on=1, off=0). t =1 i =1
subject to
STi t : Startup cost of generator i at hour t. (a) Power balance constraint
N
Fi ( Pi t ) : Generator fuel cost function in quadratic
Pload − ∑ Pi t U it = 0
t
(2)
form i =1
Fi ( Pi t ) = ai ( Pi t ) 2 + bi Pi t + ci ($/h) (b) Spinning reserve constraint
N
Pload + R t − ∑ Pi ,maxU it ≤ 0
t t
F ( Pi , U ) : Total production cost.
i
t
(3)
N : The number of generator units. i =1

NS : Swimming length Ns is the maximum (c) Generation limit constraints
number of steps taken by each bacterium Pi ,minU it ≤ Pi t ≤ Pi ,maxU it (4)
when it moves from low nutrient area to (d) Minimum up and down time constraints
high nutrient area. 1 if Ti ,on < Ti ,up ,
S : The number of bacteria in the population. 
U i = 0
t
if Ti ,off < Ti ,down , (5)
NC : The number of chemo-tactic steps. 
T : The number of hours. 0 or 1, otherwise,
pbest : The best previous position of k th (e) Startup cost
particles. HSTi , if Ti ,down ≤ Ti ,off ≤ Ti ,cold + Ti ,down

STi t =  (6)
gbest : The index of the best particle among all CSTi , if Ti ,off > Ti ,cold + Ti ,down

particles in the group.
F : The scale factor IV. OVERVIEW OF PARTICLE SWARM
CR : The Cross over probability OPTIMIZATION
C1 ,C 2 : Acceleration constants
The Particle Swarm Optimization was introduced by
rand, Rand : Random value in the range [0, 1] James Kennedy and Russell C. Eberhart in 1995 [20]. It
t
Pload : Load demand at hour t. is similar to other evolutionary computational
techniques those are GA and EP in that PSO initializes
t : Spinning reserve at hour t.
R a population of individuals randomly. These
Pi ,min : Minimum generation limit of generator i individuals are known as particles and have positions
and velocities. In addition, it searches for the optimum
Pi ,max : Maximum generation limit of generator i by updating generations, and population evolution is
Ti ,up based on the previous generations. PSO is motivated
: Minimum up time of generator i
from the simulation of the behavior of social systems
Ti ,down : Minimum down time of generator i such as fish schooling and birds flocking. In PSO, each
particle flies through the problem space by following
HSTi : The unit’s hot startup cost. the current optimal particles. Each individual adjusts its
CSTi : The unit’s cold startup cost. flying according to its own flying experience and its
neighbors flying experience. Based on its own thinking
Ti ,cold : The cold start hour. the particle attracts to its best position ( pbest ) the
λ ,µt t
: Lagrange multipliers particle changes its velocity based on the social-
psychological adaptation of knowledge that is the
Vλ (k ) : Velocity vector of λt in t th hour for k th
t
particle attracts its previous best position among the
bacterial population. group ( gbest ), here so the velocity of the particle is
Vµ t
(k ) : Velocity vector of µt in t th hour for k th updated to new position according to its modified
bacterial population. velocity using the information.
a) The current velocity
III. PROBLEM FORMULATION b) The distance between the current position and
From the definition of UC mentioned above, the
pbest .
objective of UC problem is to minimize the production c) The distance between the current position and
cost over the schedule time horizon (e.g., 24h). The gbest .

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Vij (iter + 1) = w ∗ Vij (iter ) solutions. Chemo-tactic process of bacterial foraging
+ C1 ∗ rand 1 ∗ ( pbest ij − x ij (iter )) (7) describes the movement of an E.coli cell through
swimming and tumbling via flagella. It can move in
+ C 2 ∗ rand 2 ∗ ( gbest i − x ij (iter ))
two different ways, it can swim for a definite period of
where time or it can tumble and it alternates between these
w = inertia weight factor two modes of operation for the entire lifetime. For
xij (iter ) = current position of i th particle in j th example if θ k denotes the k th bacterium and Ck is the
dimension at iteration (iter) size of the step taken in the random direction specified
Vij (iter ) = velocity of i th particle in j th dimension by the tumble. Then the movement of bacterium is
represented by
at iteration (iter) θk = θk + C k ∗ Delta k
Appropriate selection of inertia weight provides a
∆k (11)
balance between global and local explorations. The Delta k =
updated position of each particle is expressed as ∆k ⋅ ∆k
T

xij (iter + 1) = xij (iter ) + Vij (iter + 1) (8) where Δ is a unit length vector in the random direction
In this article we come up with a hybrid optimization
The new positions are calculated repeatedly till a pre-
technique, which synergistically couples the BFOA
specified maximum number of iterations are reached.
with the PSO, DE and Lagrange multipliers method.
The proposed algorithm performs local search through
V. OVERVIEW OF DIFFERENTIAL
the chemo-tactic movement operation of BFOA
EVOLUTION OPTIMIZATION
whereas the global search over the entire search space
Differential Evolution (DE) is a new floating point is accomplished by a PSO-DE operator. In this way it
encoded evolutionary algorithm for global optimization balances between exploration and exploitation enjoying
developed by Price and Storn in 1995 [21]. It is similar best of both the worlds.
to other evolutionary computation technique and also
starts with a population of NP. It is a D-dimensional VII. METHODOLOGY
search variable vector. A special kind of differential
Since UC was introduced, several methods have been
operator is used to create new offspring from parent
used to solve this problem. Among those methods, LR
chromosomes instead of classical crossover or
seems to be the most appropriate one [5], [8]. This
mutation. In each generation to change each population
method solves the UC problem by relaxing or
member Xi, a Donor vector vi is created. Three temporarily ignoring the coupling constraints, power
parameter vectors r1, r2, and r3 are chosen in a random balance and spinning reserve requirements, then
fashion from the current population. A scalar number F solving the problem through a dual optimization
scales the difference of any two of the three vectors and procedure. The dual procedure attempts to reach the
the scaled difference is added to the third one. Donor constrained optimum by maximizing the Lagrange
vector for j th component of k th population at iter function with respect to the Lagrange multipliers.
L( P, U , λ, µ) = F ( Pi t , U it )
generation is expressed as T
 t N

v k , j (iter + 1) = X r1, j (iter ) + ∑ t  Pload − ∑Pi t U it 
λ
(9) t= 1  i=1  (12)
+ F ∗ ( X r 2, j (iter ) − X r 3, j (iter ))
t 
T N
+ ∑µ  Pload + R − ∑Pi ,maxU it 
t t
CR is called “Crossover” constant and it appears as a t= 1  i=1 
control parameter of DE. The crossover is performed while minimize with respect to the power output and
on each of the D variables whenever a randomly picked generating unit status, that is
number between 0 and 1 is within the CR value. The generating unit status, that is
trial vector u k , j is outlined as q ∗ (λ, µ) = max q (λ, µ)
λ,µ
u k , j (iter ) = v k , j (iter ) if rand (0,1) < CR 13)
(10) where
u k , j (iter ) = X k , j (iter ) otherwise q (λ, µ) = min L( P,U , λ, µ) (14)
t t
Pi ,U i

VI. OVERVIEW OF CHEMOTACTIC PSO-DE Substitute (1) in (9), the Lagrangian function becomes
 Fi ( Pi t )
[ 
PSO and DE are excellent heuristics like other N 
T 
L = ∑∑ + STi (1 −U i )] U i

t t−1 t

evolutionary algorithms. Practical experiences suggest i= t = 
1 1 
−λ Pi tU it − µt Pi , maxU it 
t
(15)
that they reach stagnation after certain number of
generations as the population is not converged locally, ( )
T
+ ∑ λ Pload + µt ( Pload + R t )
t t t

so they will stop proceeding towards global optimal t=1

52
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DOI: 01.ijepe.01.02.10


Equations (2) and (3) are the coupling constraints If DC i ≤ 0 the unit will be committed if it does not
t
across the units. The coupling constraints in the above
equation are temporarily ignored and then the violate minimum downtime constraint, otherwise the
minimum of Lagrangian function is solved for each unit is decommitted if it does not violate the minimum
generating unit, without regard for what is happening uptime constraint. A step-by-step procedure for the
on the other generating units. proposed method is outlined as follows.
T [ F ( P t ) + ST t (1 − U t −1 )] t 
Ui 
N
 i i
min L( P, U , λ , µ ) = ∑ min ∑  t t t Section -A:
i i
 (16)
t =1  − λ P U i − µ P , maxU i
t t

t t
Pi ,U i
i =1  i i  Step1: Compute λt and µ by Chemo-tactic PSO-DE.
t
Therefore the minimum for each generating unit over
Step2: Set the unit status by calculating dual power and
all time periods is:
T [ F ( P t ) + ST t (1 − U t −1 )] t 
obtain on/off decision DC (21).
N
 i i Ui 
min q (λ , µ ) = ∑ min ∑  t t t Step3: Calculate the spinning reserve for generated ‘U’
i i
 (17)
t =1 − λ P U i − µ P , maxU i
t t
i =1  i i 
 status at each hour of the population. If excessive
subject to constraints (4) and (5). This problem is spinning reserve is observed decommit the unit with
solved through a two-state dynamic programming high full load average production cost if it does not
problem in two variables for each unit. On the other violate the minimum uptime constraint. Otherwise if
hand, in order to maximize the Lagrange function with the unit violates, the unit with next high full load
respect to the Lagrange multipliers, the adjustment of average production cost is decommitted. Now set the
Lagrange multipliers must be done carefully. Most of spinning reserve according to the new status and repeat
research works use sub-gradient method to achieve this until the spinning reserve satisfies.
task. In this paper, we use the Chemo-tactic PSO-DE Step4: If less spinning reserve than required is observed
optimization to adjust the Lagrange multipliers and commit the unit with less full load average cost if it
improve the performance of Lagrange relaxation does not violate the minimum downtime constraint.
method. Otherwise if it violates, the unit with next less full load
This is solved as a dynamic programming problem in average production cost is committed. Now set the
one variable with two possible unit states ( U i = 0 or
t spinning reserve according to the new status and repeat
until the spinning reserve satisfies.
1). Step5: Economic dispatch for the generated U-status is
U= carried out by Lagrangian multiplier method.
1

Section-B:
U=0
Step1: Randomly generate λt , µ , Vλt , and Vµ with
t t
t=0 t= t=2 t=3
1
Fig. 1 Unit on/off status
The dynamic programming part must take into account in the range for initialized bacterium. Generate unit-
all the start up costs as well as the minimum up and status U as in section-A and calculate pbest for U, λt
down time for the generating units. , µt , cost and fit (J and JF) and gbest for U, λt , µt , J
t
At U =0 state the value of function to minimize is
i and JF.
t Step2: Starting of the chemo-tactic loop (it=it+1)
zero. At U i =1 state the function to be minimized is Step3: For each bacteria (k=k+1)
JFlast = JF (k )
min[ Fi ( Pi t ) − λt Pi t ] . The minimum of this function
is found by taking the first derivative Step4: Calculate Vλt , Vµ t and check for maximum
d
[ Fi ( Pi t ) −λ Pi t ] = 0
t and minimum velocity limits.
dPi t
(18)
Vλt ( k ) = Vλt ( k )
Pi t ,opt is obtained from (19) + C1 * Rand * ( pbestλt ( k ) − λt (k )) (22)
The dual power
λt − bi + C 2 * Rand * ( gbestλt − λt ( k ))
Pi t ,opt = (19) Vµt (k ) =Vµt (k )
2c i
+ C1 * Rand * ( pbestµt ( k ) − µt ( k )) (23)
if Pi t ,opt
< Pi ,min , then Pi = Pi , min
t
+ C 2 * Rand * ( gbestµt − µt (k ))
if Pi t ,opt > Pi ,max , then Pi t = Pi ,max (20) Step5: Move to new position
λ ( k ) =λ (k ) +C λ ∗Vλ ( k )
t t t

if Pi , min ≤ Pi t ,opt
≤ Pi ,max , then Pi = Pi
t t , opt
(24)
µt (k ) = µt (k ) +C µ ∗Vµt ( k )
The dual power obtained from (20) is substituted in
on/off decision criterion DC where C λ and C µ are chemo-tactic step size
DC it = [ Fi ( Pi t ) + STi t (1 − U it −1 ) − λt Pi t − µ t Pi , max ] (21) Step6: Handle limits
λmin ≤λ ≤λmax
t
k

µmin ≤µt ≤µmax
(25)
k

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Step7:
Generate U-status as in section-A TABLE I
Step8:
Compute cost and fit – J and JF UNIT STATUS FOR 40-UNIT SYSTEM
Step9:
Swimming Peri 40-Unit Status
9a) swim-count=0 od 1-40
9b) while (swim-count<Ns) and JF (k ) > JFlast 1 1111111100000000000000000000000000000000
i. swim-count = swim-count + 1 2 1111111100000000000000000000000000000000
JFlast = JF (k ) 3 1111111100000000000100000000000000000000
ii.
4 1111111100000001011100000000000000000000
λt ( k ) = λt ( k ) + C λ ∗Vλt ( k )
iii. 5 1111111100000011111100000000000000000000
µt ( k ) = µt ( k ) + C µ ∗Vµt ( k ) 6 1111111100011111111100000000000000000000
iv. Handle limits 7 1111111100111111111100000000000000000000
Step10:
Generate unit-status U as in section-A 8 1111111111111111111100000000000000000000
Step11:
Compute cost and fit - J and JF 9 1111111111111111111111111000000000000000
Step12:
Differential Evolution 10 1111111111111111111111111111111100000000
While(rand<CR) 11 1111111111111111111111111111111111110000
12 1111111111111111111111111111111111111111
λdef t (k ) = λt (k ) + F ∗ (λt (r2 ) − λt (r3 )) 13 1111111111111111111111111111111100000000
14 1111111111111111111111111000000000000000
µ def t (k ) = µ t (k ) + F ∗ ( µ t (r2 ) − µ t (r3 )) (26)
15 1111111111111111111100000000000000000000
i. Handle limits 16 1111111111111111111100000000000000000000
Step13: Generate unit status – U def as in section-A 17 1111111111111111111100000000000000000000
18 1111111111111111111100000000000000000000
Step14: Compute cost and fit – J def , JFdef 19 1111111111111111111100000000000000000000
20 1111111111111111111111111000111111110000
Step15:
21 1111111111111111111111111000000000000000
if ( JFdef > JF ) then 22 1111111111000000111111111000000000000000
JF = JFdef , J = J def , U = U def , λ = λ def , µ = µ def 23 1111111100000000001100000000000000000000
24 1111111100000000000000000000000000000000
Step16: Update best location pbest and gbest for JF,
J, U, λ, µ
Step17: if k<S, go to step3 otherwise perform step18. TABLE II
COMPARISON OF COST
Step18: If iter < N c then go to step2 otherwise stop.
Total Cost ($)
VIII. RESULTS AND DISCUSSIONS No. of Units 10 20 40
LR [5] 565825 1130660 2258503
The 10-unit system data and load demands are taken
GA [5] 565825 1126243 2251911
from [5]. The 20 and 40 unit’s data are obtained by
duplicating the ten unit case, and the load demands are EP [6] 564551 1125494 2249093
adjusted in proportion to the system size. In the GAUC [7] 563977 1125516 2249715
simulation, the reserve is assumed to be 10% of the ELR [8] 563977 1123297 2244237
load demand. The control parameters chosen for these ICGA [9] 566404 1127244 2254123
units are CR=0.7, F=0.1 or 0.2, C1=0.5, C2=0.5, C λ and HPSO [10] 563942 --- ---
Cµ =random number between 0.1 and 1. The unit Proposed method
CPSO-DE 563977 1123297 2243363
status obtained through Chemo-tactic PSO-DE for 40
unit system is shown in Table I. From simulation it is 2450000
observed that setting up of maximum and minimum
bounds for λ and µ, such that λ max = 30 , 2400000
Total Cost

λ min = 12 , µ max = 15 and µ min = 2 , leads 2350000

the system to high-quality convergence.The proposed 2300000
method provides best production cost when compared
2250000
to literature as in Table II with a reasonable
computation time per iteration of 0.72s for 10-unit 2200000
1 131 261 391 521 651 781 911
system, 2.1s for 20-unit system and 5.7s for 40-unit
Iterations
system. The convergence characteristics for the
proposed method (40-unit system) are as shown in
figure2. Fig. 2. Cost convergence characteristics for 40-unit system

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IX. CONCLUSION [14] Kim.D.H, Abraham.A, Cho.J.H, A hybrid genetic
algorithm and bacterial foraging approach for global
Unit commitment problem is solved with a new optimization, Information Sciences, Vol. 177 (18), 3918-
methodology Chemo-tactic PSO-DE Optimization 3937, (2007).
algorithm. PSO and DE are incorporated in chemo- [15] Yao.X, Liu.Y, Lin.G, “Evolutionary programming made
tactic algorithm to update Lagrange multipliers so that faster”, IEEE Transactions on Evolutionary
they are suitable for high dimensional combinatorial Computation, vol 3, No 2, 82-102, (1999).
optimization problem. Results show that as the [16] Hao.Z, F.Guo.G.H, Huang.H, “A Particle Swarm
Optimization Algorithm with Differential Evolution”, in
dimension of the problem increases it generates good
IEEE Int.Conf. Systems, man and Cybernetics, Aug.
unit status with lower production cost. Hence it can be 2007, Vol. 2, pp. 1031-1035
concluded that the proposed method provide lower [17] Kennedy.J, Eberhart.R, “Particle swarm optimization”,
production cost than those of LR, GA, EP,GAUC and In Proceedings of IEEE International Conference on
ELR methods for 40-unit system and also the problem Neural Networks, (1995) 1942-1948.
can be extended for 60-unit, 80-unit and 100-units as in [18] Storn.R., Price.K, “Differential evolution – A Simple
this method better costs are obtained as the dimension and Efficient Heuristic for Global Optimization over
(number of generators) of the problem increases. Continuous Spaces”, Journal of Global Optimization,
11(4) 341–359, (1997).

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Solving Unit Commitment Problem Using Chemo-tactic PSO–DE Optimization Algorithm Combined with Lagrange Relaxation

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