ERK'2019, Portorož, 144-147 144
Reducing the Number of Solutions in the Unit Commitment 
Problem Using Variations with Repetition 
Izudin Softić
1
, Nedžmija Demirović
1
, Marina Pejić
1
, Midhat Umihanić
1
, Samed Bajrić
2
 
1
Faculty of Electrical Engineering,
 
University of Tuzla, Tuzla, Bosnia and Herzegovina 
2
Jozef Stefan Institute, Laboratory for Open Systems and Networks, Ljubljana, Slovenia 
E-mail: izudin.softic@untz.ba 
 
Abstract. In this paper, a new approach has been 
presented to reduce the number of solutions in the unit 
commitment (UC) problem, using variations with 
repetition, which can not be used in power system 
exploitation, due to the impossibility of meeting the 
loads and limitations associated with the minimal 
operation time after the start-up and minimal pause 
time after shutting down thermal units (minimum 
up/down). Each variation that repeats is representative 
of the corresponding drive status and the number of 
thermal units from the economic dispatching (ED)  
process will be executed for a given time interval. The 
problem is formulated as a complex mathematical 
optimization task that includes both binary (power 
status of thermal units, on/off) and real variables 
(production of active power of engaged thermal units). 
The suggested approach is tested on a system with ten 
thermal units showing very good performances. 
 
1 Introduction 
In everyday life, we often find a series of problems 
whose solution is necessary for normal function of 
power system. By developing the power of today's 
computers, part of these problems has become trivial. 
However, a large set of problems still poses an 
extremely difficult challenge. These are the problems 
that we can not solve with today's computer 
development by using complete enumeration (counting) 
of possible solutions, due to extensive requirements at 
the time of the budget. One of the typical 
representatives of this type of problem is NP-hard 
problems, which is also a UC problem. Due to the large 
number of thermal units of different characteristics, as 
well as the dynamics of the system, this is a very 
complex task of optimization, so the researchers' efforts 
are focused on seeking a suboptimal solution, with 
moderate demands on the time of budget and computer 
resources. 
 The most commonly used methods for solving UC 
problems are list priority methods [1], dynamic 
programming method [2], Lagrange relaxation problem 
[3], Benders decomposition [4]. Recent results also 
include some heuristic methods such as taboo search, 
simulated annealing, neural networks [5], genetic 
algorithms [6], and fuzzy logic [7]. 
 
2 UC problem formulation 
The UC problem aims at minimizing the total 
production cost over the scheduling horizon satisfying 
all constraints. In addition, the total production cost 
comprises fuel cost, which are related to operation of 
thermal units, start-up costs and shut down costs. In this 
sense, the general form of the UC problem can be 
described as:  
 
 
1
min ( ) ( ) ( ) ( )
TN
TC i i i i
t i i
F F t u t SC t v t

    
 
       (1) 
 
 where: 
 i : Index of thermal unit  
 t : Index of hour 
 
TC
F : Total production cost 
   ()
i
Ft: Fuel cost for thermal unit i at hour t 
    ) (t u
i
: i-th thermal unit status at hour t  
               (ON: 1 ) (  t u
i
; OFF: 0 ) (  t u
i
) 
 ()
i
SC t : Start-up cost for thermal unit i  
 ()
i
vt: i-th thermal unit start-up/shut down status at 
hour t (if the thermal unit started ( ) 1
i
vt  , otherwise 
( ) 0
i
vt  )  
     N : Number of thermal units 
     T : Scheduling horizon 
 
 The main component of the operating costs is the i-
th unit cost function F
i
(t) which can be viewed as a 
quadratic function of the power output at hour t: 
 
                 ) ( ) ( ) (
2
t P c t P b a t F
i i i i i i
                        (2) 
 
 where: 
    ) (t P
i
: Output power of i-th thermal unit at hour t 
 a
i
, b
i
 and c
i
: Cost coefficients of the i-th thermal unit 
 
 The start-up costs in the thermal units change from 
some of the highest values when the thermal units start 
from a completely cold state, to considerably lower 
values if the start-up is carried out after short cooling of 
the thermal units. In other words, the start-up costs are 
the functions of the cooling time of the boiler and 
mathematically can be presented in the form of 

145
 
: ( )
()
:
off off off
i i i i i
i
off off
i i i i
hc T X t T cs
SC t
cc X T cs

  






                   (3) 
 
 where: 
 
off
i
T : Minimum down time of i-th thermal unit  
 ()
off
i
Xt : Duration that the i-th thermal unit is 
continuously OFF 
 
i
hc : Hot start-up cost of i-th thermal unit 
 
i
cc : Cold start-up cost of i-th thermal unit 
 
i
cs : Cold start-up hours of i-th thermal unit 
 Note that if the number of hours less than cs
i
 we use 
hot start-up costs otherwise we use cold start-up costs. 
 Different constraints of the system are to be satisfied 
while solving the complex UC problem. Moreover, the 
constraints that must be fullfiled at specific time are as 
follows: 
 System power balance 
 
                    ) ( ) (
1
t P t P
N
i
D i 

                                    (4) 
 where: 
 ) (t P
D
 : Power demand at hour t 
 
 Minimum up/down time 
 
()
()
on on
ii
off off
ii
T X t
T X t







                                             (5) 
 
 where: 
      
on
i
T : Minimum up time of i-th thermal unit  
()
off
i
Xt : Duration that the i-th thermal unit is 
continuously ON  
 
 Output limit of thermal units 
 
min max
()
i i i
P P t P                                        (6) 
 
 where: 
 
max
i
P : Maximum power output for i-th thermal unit 
 
min
i
P : Minimum power output for i-th thermal unit 
 
 After the defining the UC problem together with the  
function of total production costs (1), its components (2) 
and (3), as well as different constraints (4) – (6), there is 
need to choose an appropriate method for solving the 
UC problem. 
 
3 Variations with repetition 
When the ordering of objects matters, and an object can 
be choosen more than once, we are talking about 
variations with repetition, and the number of variations 
is: 
 
kk
n
Vn  (7) 
  
 where: 
 n: Number of objects from which we can choose 
 k: Number of objects we can choose 
 For instance, the first element can be chosen in n 
ways, the second element can be chosen in n ways, the 
third element in n ways, etc., k-th element in n ways, 
because it is allowed to repeat the elements from the set. 
 Since the operating status of the i-th thermal unit at 
hour t (ON: 1; OFF: 0) is used in solving the UC 
problem, it is n = 2, while the whole number of 
repeating variations for a one-hour period is defined by 
the expression: 
 
             
12
2
000...00 , 000...01 ,..., 111...11
k
            (8) 
 
4 Solution methodology 
The daily load curve is characterized by a big disparity 
between minimal and maximum loads despite efforts to 
make the load as balanced as possible. At all times, the 
load value must be fullfiled by using a production of 
thermal units. This means that just a few number of  
variations with repetition will be used, due to the 
impossibility of loads to meet its value and constraints 
associated with the minimal operation time after the 
start-up and minimal pause time after shutting down 
thermal units. 
 In order to obtain an optimum or suboptimal 
solution, the initial solution of the problem is 
represented by a binary matrix of dimension T x N 
where all elements are assigned the value 1 (all thermal 
units are ON) as shown in Table 1. 
Table 1. Initial solution 
 1 2 3 
 
N 
1 1 1 1 1 1 
2 1 1 1 1 1 
3 1 1 1 1 1 
 
1 1 1 1 1 
T 1 1 1 1 1 
 
 Each row vector in the solution matrix is the 
representative of the corresponding drive state of the 
thermal unit for the time interval T and indicates the 
number of thermal units over which the economic 
dispatching process will be executed [8]. For the first 
hour, after the ED procedure, a low-cost solution is 
adopted and replaced in the initial solution, and verifies 
whether the solution is applicable. If the solution is 
applicable, the ED process is executed for the second 
hour and replaced in the lowest cost solution solution 
matrix, and it is again verified whether the solution is 
applicable or not. This process is repeated for all hours 

146
in the time interval T. If the solution is not applicable 
for a given hour, then it is replaced by the first higher 
occurance for that hour and so until an applicable 
solution is obtained. 
 
5 Results 
To illustrate the efficiency of the proposed method as 
well as solving technique, testing was performed on a 
test system whose data is given in Table 2 [6] and the 
24-hour interval. 
 The optimum number of thermal units involved is 
continually changing with the time-changing load. In 
practice, it is common that an one hour is the minimum 
time required to switch the selected thermal unit from 
off to on, so it can be stated that at any time step (one 
hour), the assumption of constant load could be 
supposed. This represents the conversion of a continual 
problem into a discrete problem as shown in Figure 1. 
 
 
 The exact solution of the problem can only be 
obtained by fully enumerating (counting) the possible 
working conditions of the thermal units, which is 
impossible to achieve on systems of realistic size due to 
the large number of requests at the time of the 
calculation.
 In order to reduce the number of variations with 
repetition in the UC problem with the operating status 
{0,1} and the total number (k = 10)  of the thermal units 
in the system for a time period of one hour, it is possible 
to obtain: 
 
1024 2
10 10
2
  V variations with repetition
 
     
1 2 1024
0000000000 , 0000000001 ,..., 1111111111 
 Because most of the above variations are not 
applicable to a power system exploitation due to 
impossibility of loading, Table 3 shows the number of 
variations with repetition that are applicable in the 
power system exploitation for different loads occurring 
in the system. 
 
Table 3. Number of variations with repetition that are 
applicable 
Hour Load (MW) Number  
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
700 
750 
850 
950 
1000 
1100 
1150 
1200 
1300 
1400 
1450 
1500 
1400 
1300 
1200 
1050 
1000 
1100 
1200 
1400 
1300 
1100 
900 
800 
671 
623 
498 
359 
314 
242 
222 
187 
121 
53 
34 
18 
53 
121 
187 
276 
314 
242 
187 
53 
121 
242 
422 
564 
Table 2. Parameters of thermal units 
Thermal
Units  
No. 
a
i 
b
i 
c
i 
P
i
min 
P
i
max
 T
i
on 
T
i
off
 hc
i 
cc
i 
cs
i 
Initial 
state 
1 1000 16.19 0.00048 150 455 8 8 4500 9000 5 8  + (ON) 
2 970 17.26 0.00031 150 455 8 8 5000 1000 5 8 + (ON) 
3 700 16.60 0.00200 20 130 5 5 550 1100 4 -5 - (OFF) 
4 680 16.50 0.00211 20 130 5 5 560 1120 4 -5 - (OFF) 
5 450 19.70 0.00398 25 162 6 6 900 1800 4 -6 - (OFF) 
6 370 22.26 0.00712 20 80 3 3 170 340 2 -3 - (OFF) 
7 480 27.74 0.00079 25 85 3 3 260 520 2 -3 - (OFF) 
8 660 25.92 0.00413 10 55 1 1 30 60 0 -1 - (OFF) 
9 665 27.27 0.00222 10 55 1 1 30 60 0 -1 - (OFF) 
10 670 27.79 0.00173 10 55 1 1 30 60 0 -1 - (OFF) 
 
Figure 1. Conversion of continuous into discrete problem 
 
 

147
 An example of the calculation costs using the 
described method for 24-hour interval is given in Table 
4. 
 
6 Discussion 
In addition to the fact that the problem UC is a highly 
complex mathematical optimization task that includes 
both binary and real variables, the additional complexity 
of the problem is the presence of dynamic constraints on 
thermal units. In order to solve these problems, the 
number of solutions in the UC problem has been 
reduced and from the obtained results it is possible to 
see that by applying the initial solution in which all the 
thermal units are in operation, using the variations with 
repetition it is possible to find the applicable solutions 
much faster than the initial solution is given by default. 
By limiting itself to choose the lowest cost solution in a 
given hour, a path that matches an optimal or 
suboptimal solution is chosen. 
 
7 References 
 [1]  G. B. Sheble, Solution of the unit commitment problem 
by the method of unit periods, IEEE Transactions on 
Power Systems, 5(1), 1990, 257-260. 
[2] W. L. Snyder Jr., H. D. Powell Jr., J. C. Rayburn, 
Dynamic programming approach to unit commitment, 
IEEE Transactions on Power Apparatus and Systems, 
2(2), 1987, 339-350. 
[3] W. Ongsakul, N. Petcharaks, Unit commitment by 
enhanced adaptive Lagrangian relaxation, IEEE 
Transactions on Power Systems, 19(1), 2004, 620-628. 
[4]  H. Ma, S. M. Shahidehpour, Transmission-constrained 
unit commitment based on Benders decomposition, 
International Journal of Electrical Power and Energy 
Systems, 20(4), 1998, 287-294. 
[5]  H. Sasaki, M. Watabable, J. Kubokawa, N. Yorino, R. 
Yokoyama, A solution method of unit commitment by 
artificial neural networks, IEEE Transactions on Power 
Systems, 7(1), 1992, 974-985. 
[6] A. Kazarlis, A.G. Bakirtzis and V.Petridis, A Genetic  
Algotirhm Solution to the Unit Commitment Problem, 
IEEE Transactions On Power Systems, Vol. 11, No. 1, 
pp.83-90, 1996. 
 [7] S. C. Pandian, K. Duraiswamy: Fuzzy logic 
implementation for solving the unit commitment problem, 
International Conference on Power System Technology – 
POWERCON: Singapore, November 2004. 
[8] I. Softic and S. Halilčević, The Efficiency of 
Deterministic Methods in Solving the Problem of 
Economic Dispatch, 19th Electrotechnical and Computer 
Science Conference ERK’2010, Portoroz, Slovenia, 20-
22. September 2010, str.A: 492-495. 
Table 4. Operating and startup costs, load and schedule for 24 hours time frame 
 
Hour 
Operation 
cost 
($) 
Start up 
cost 
($) 
Load 
(MW) 
Generation schedule 
1 2 3 4 5 6 7 8 9 10 
1 13683 0 700 455 245 0 0 0 0 0 0 0 0 
2 14554 0 750 455 295 0 0 0 0 0 0 0 0 
3 16302 0 850 455 395 0 0 0 0 0 0 0 0 
4 18638 1120 950 455 365 0 130 0 0 0 0 0 0 
5 19513 0 1000 455 455 0 130 0 0 0 0 0 0 
6 21860 1800 1100 455 455 0 130 60 0 0 0 0 0 
7 22879 0 1150 455 435 0 130 110 0 0 0 0 0 
8 23918 0 1200 455 455 0 130 160 0 0 0 0 0 
9 26184 1100 1300 455 455 130 130 130 0 0 0 0 0 
10 28768 340 1400 455 455 130 130 162 68 0 0 0 0 
11 30583 520 1450 455 455 130 130 162 80 38 0 0 0 
12 32542 60 1500 455 455 130 130 162 80 33 55 0 0 
13 29222 0 1400 455 455 130 130 162 0 68 0 0 0 
14 26184 0 1300 455 455 130 130 130 0 0 0 0 0 
15 23918 0 1200 455 455 0 130 160 0 0 0 0 0 
16 20896 0 1050 455 440 0 130 25 0 0 0 0 0 
17 20020 0 1000 455 390 0 130 25 0 0 0 0 0 
18 21860 0 1100 455 455 0 130 60 0 0 0 0 0 
19 23918 0 1200 455 455 0 130 160 0 0 0 0 0 
20 30485 920 1400 455 455 0 130 162 80 63 55 0 0 
21 27167 0 1300 455 455 0 130 162 73 25 0 0 0 
22 22546 0 1100 455 455 0 130 0 35 25 0 0 0 
23 17795 0 900 455 445 0 0 0 0 0 0 0 0 
24 15427 0 800 455 345 0 0 0 0 0 0 0 0 
Total 548862 5860 27100