Firefly based Unit Commitment

Firefly based Unit Commitment

Manabjyoti Daimari

Assam Engineering College Dept. of Electrical Engineering, Guwahati, India

Dr. Barnali Goswami

Assam Engineering College Dept. of Electrical Engineering, Guwahati, India

AbstractUnit commitment (UC) problem is considered one of the most vital problems for daily economic operation and planning of present power systems that optimize the operation cost with respect to the load demands. UC decision incorporates the determination of the generating units to be committed during each hour of the planning period, by considering system capacity requirements, reserve, and the constraints on the start-up and shut-down of units. Economic Load Dispatch (ELD) is a constrained non- linear optimization problem. ELD schedules the outputs of available generating units for a specific time that reduces the production cost while fulfilling equality and inequality constraints. In this paper, a firefly (FF) algorithm has been proposed for solving UC problem. The FF algorithm decides the ON-OFF status of all the available generating units while Lambda Iteration method solves the ELD problem among the committed units in each hour. The proposed algorithm has been tested with the systems having 10, 20 and 40 units.

Keywords Unit Commitment; Economic Load Dispatch; Firefly

1. INTRODUCTION

   UC is a combinatorial optimization problem in a power system, which incorporates finding a start-up and shut-down schedule of the generating units to fulfill the hourly fluctuating forecasted system demand and different requirements such that the total cost is minimized. UC can be considered as two linked optimization sub-problems namely the unit scheduling problem and the ELD problem. The UC problem is a binary-variable power system optimization problem which incorporates determining on/off status of all the available generating units whereas the ELD is a real- variable power system optimization problem which incorporates allocating the loads among the online units to balance the forecasted load demand. Various methods are proposed to determine the status of the generating units in the UC problem. The conventional methods such as dynamic programming (DP) [1], Lagrangian relaxation (LR) [2], integer programming (IP) [3], and branch and bound [4] have been applied to solve the UC problem. These days meta- heuristic methods like artificial neural network (ANN) [5], genetic algorithm (GA) [6], simulated annealing (SA) [7], particle swarm optimization (PSO) [8], and tabu search [9] are able to produce better solutions than the conventional methods like mixed integer linear programming used in the load dispatch center.

   In recent years, a new biologically-inspired meta- heuristic algorithm, known as the rey algorithm was developed by Xin-She Yang. FA is an optimization algorithm inspired by the behavior and motion of fireflies.

   In this paper, the firefly (FF) algorithm has been implemented to solve the UC problem. Main objective of this paper is to minimize the production cost of generators by optimizing the schedule of generation. A set of power systems up to 40 units system are used to test over 24 hr. time horizon.

2. PROBLEM FORMULATION

   UC can be characterized mathematically as an optimization problem as follows:

   1. Objective Function

      The objective function is specified as a sum of fuel cost and start- up cost of each generating unit over 24 hours scheduled period and mathematically is expressed as equation (1):

      T N

      minCF = [Fi(Pi(t))+SCi(t)] (1)

      t=1 i=1

      Where,

      Fi(Pi(t)) = fuel cost of unit i at hour t expressed as a quadratic function of each unit output =ai + bi*Pi(t) + ci*Pi(t)2, where ai, bi and ci represent cost coefficients of the unit.

      N = number of units;

      T = total scheduling period;

      i = index of unit (i=1.2….N);

      t = index of hour (t=1.2….T);

      Pi(t) = power generation of unit i at hour t;

      CF = aggregate cost;

      SCi(t) = startup cost of unit i;

      The startup cost depends on the duration during which the generating unit has been decommitted. A cold start-up cost is applied, if the unit has been off for a long period. A hot start-up cost is applied, if the unit has been off for a short period.

      i

      i

      As per the two-step function, the time dependent start-up cost is simplified using H off defined as equation (2):

      off

      off

      SCi(t) = h-costi: MDTi Xi off (t) Hi (2)

      i

      i

      = c- costi: Xi off (t) > H off

      off

      off

      Where, Hi = MDTi + c-s-houri;

      MDTi = minimum down time of ith unit;

      h-costi = hot start cost of ith unit;

      c- costi = cold start cost of ith unit;

      c-s-houri = cold start hour of ith unit;

      For each generating unit, shut down cost is usually a constant value. In the standard systems the typical value of the shut down cost is zero.

   2. Constraints

   In minimizing the objective function, following constraints must be satisfied.

   1. Power balance:

      The generated power from all the online units must fulfill the forecasted load demand, which is defined as equation (3):

      N

      PD (t) = Pi (t) (3)

      i=1

   2. Spinning Reserve requirements:

      The spinning reserve is the additional real power generation accessible from all the synchronized unit to provide the load in the event of any fault or sudden tripping or maintaining any generating units:. The mathematical equation is expressed by equation (4):

      N

      Ui (t). Pmaxi PD (t) + RE (t) (4)

      i=1

      Where, Ui (t) = ON/OFF status of unit i at hour t; PD (t) = load demand in the hour t;

      RE (t) =spinning reserve requirement in the hour t;

   3. Real power generation limits:

      The generation of the accessible unit must lie between its minimum and maximum limit. The formulation can be expressed by equations (5) and (6):

      Pmini Pi (t) Pmaxi (without ramp-rate constraint) (5)

      ModPmini(t) Pi(t) ModPmaxi(t) (with ramp-rate constraint) (6)

      Where,

      Pmaxi=maximum real power generation limit of unit i; Pmini= minimum real power generation limit of unit i; ModPmini(t) =modified minimum generation limit of unit i at t;

      ModPmaxi(t) =modified maximum generation limit of unit i

      at t;

   4. Unit minimum up/down time:

      Once unit is committed/decommited, there is a pre-defined minimum time after it can be decommitted/committed. The formulation of these constraints can be seen in equations (7) and (8):

      i

      i

      MUTi X ON (7)

      i

      i

      MDTi X OFF (8)

   5. Unit initial status:

   The initial status at the beginning of the planning period must be taken into account.

3. FIREFLY ALGORITHM

   Firey algorithm uses the following three idealized rules: (1)All the fireflies have a distinct characteristic of being attracted to other fireflies whatever may be the others sex i.e. they are unisexual; (2) the level of the attractiveness of a rey is related to its brightness or light intensity, therefore the less brighter one will be attracted to the brighter one for any two ashing reies. Both attractiveness and brightness will be more if the distance between two reies decreases. If no one is brighter one than a specific rey, it will move randomly; (3) the brightness of a firefly is determined by the value of the objective function to be optimized. For a maximization problem, the intensity of light of each rey is proportional to the value of the objective funcion whereas it is converse in case of minimization problem.

   Firey algorithm involves three important parameters which are given as follows.

   1. Attractiveness and light intensity:

      There are two important points associated with the firefly algorithm: the variation of the light intensity and the formulation of the attractiveness. As the intensity of light I(r) varies with distance r monotonically and exponentially, that is expressed as equation (9):

      I(r) = I0 e-r2 (9)

      Where I0 is the initial light intensity and is the light absorption coefficient. Since attractiveness of a firefly is proportional to the light intensity seen by adjacent fireflies, now the attractiveness of a firefly can be expressed by equation (10):

      (r) = 0 e-r2 (10)

   2. Distance between reies:

      The distance between any two fireflies u and v at xu and xv respectively, can be characterized as Cartesian distance by equation (11):

      ruv = d n=1 (xu,n – xv,n)2 (11)

      Where d is the number of dimensions and xu,n is the nth component of the spatial coordinate of xu the uth firefly.

   3. Movement of rey:

   The movement of a firefly u is attracted by another brighter firefly v and is determined by equation (12):

   xu/ = xu + (r) (xu-xv) + (rand-0.5) (12)

   Where the second term is due to the attraction while the third term indicates randomization with the randomization parameter and rand is a random number which is uniformly distributed in [0, 1].

   Where,

   MUTi = minimum up time of ith unit;

   MDTi = minimum down time of ith unit;

   i

   i

   X ON(t) =duration for which unit i is continuously on;

   i

   i

   X OFF(t) =duration for which unit i is continuously off;

4. PROPOSED ALGORITHM

   The main steps of the proposed algorithm are as follows:

   Initialize randomly individuals of population and set the initial values of FF control parameters

   Initialize randomly individuals of population and set the initial values of FF control parameters

   1. Initialize randomly the individuals of the population (N by T matrix) and set the initial values of FF control parameters.

   2. These schedules (individuals) are checked for solution feasibility (generation >= load+reserve) in which infeasible strings are prohibited and a new random individuals are created.

   3. If the above solution is feasible, then checked for the satisfaction of minimum up time/down time constraints.

   4. Evaluate the level of attractiveness of each firefly using the equation (10).

   5. Modify the firefly position by using the equation (12).

   6. Calculate the fitness function [F= (FCT + SCT)]. No

   7. Selection of more brighter/attractive firefly and minimum

      cost.

   8. Reinsert best commitment/generation schedule for the next generation.

   9. If the maximum number of iteration is reached, the running process is stopped. Otherwise jump to step (3). The flowchart of the proposed algorithm is presented in

   /

   /

   Fig. 1.

   Start

   Solution feasibility?

   Satisfy MUT and MDT

   Satisfy MUT and MDT

   Evaluate the level of attractiveness of each firefly by using (r) = 0 e-r2

   Evaluate the level of attractiveness of each firefly by using (r) = 0 e-r2

   Yes

   Modify the firefly position by using xu =

   xu + (r) (xu-xv) + (rand-0.5)

   Modify the firefly position by using xu =

   xu + (r) (xu-xv) + (rand-0.5)

   Calculate the fitness function

   F= (FCT + SCT)

   Calculate the fitness function

   F= (FCT + SCT)

   Select brightest firefly and minimum cost

   Select brightest firefly and minimum cost

   Reinsert best commitment & generation schedule

   Reinsert best commitment & generation schedule

   No Max Iteration?

   Yes

   Stop

   Fig.1. Flowchart of the proposed algorithm

   1

   Hour	Unit1	Unit2	Unit3	Unit4	Unit5	Unit6	Unit7	Unit8	Unit9	Unit10
   1	1	1	0	0	0	0	0	0	0	0
   2	1	1	0	0	0	0	0	0	0	0
   3	1	1	1	0	0	0	0	0	0	0
   4	1	1	0	0	1	0	0	0	0	0
   5	1	1	1	0	1	0	0	0	0	0
   6	1	1	1	1	1	0	0	0	0	0
   7	1	1	1	1	1	0	0	0	0	0
   8	1	1	1	1	1	0	0	0	0	0
   9	1	1	1	1	1	1	1	0	0	0
   10	1	1	1	1	1	1	1	1	0	0
   11	1	1	1	1	1	1	1	1	1	0
   12	1	1	1	1	1	1	1	1	1	1
   13	1	1	1	1	1	1	1	1	0	0
   14	1	1	1	1	1	1	1	0	0	0
   15	1	1	1	1	1	0	0	0	0	0
   16	1	1	1	1	1	0	0	0	0	0
   17	1	1	1	1	1	0	0	0	0	0
   18	1	1	1	1	1	0	0	0	0	0
   19	1	1	1	1	1	0	0	0	0	0
   20	1	1	1	1	1	1	1	0	0
   21	1	1	1	1	1	1	1	0	0	0
   22	1	1	0	0	1	1	1	0	0	0
   23	1	1	0	0	1	0	0	0	0	0
   24	1	1	0	0	0	0	0	0	0	0

   Hour	Unit1	Unit2	Unit3	Unit4	Unit5	Unit6	Unit7	Unit8	Unit9	Unit10
   1	1	1	0	0	0	0	0	0	0	0
   2	1	1	0	0	0	0	0	0	0	0
   3	1	1	1	0	0	0	0	0	0	0
   4	1	1	0	0	1	0	0	0	0	0
   5	1	1	1	0	1	0	0	0	0	0
   6	1	1	1	1	1	0	0	0	0	0
   7	1	1	1	1	1	0	0	0	0	0
   8	1	1	1	1	1	0	0	0	0	0
   9	1	1	1	1	1	1	1	0	0	0
   10	1	1	1	1	1	1	1	1	0	0
   11	1	1	1	1	1	1	1	1	1	0
   12	1	1	1	1	1	1	1	1	1	1
   13	1	1	1	1	1	1	1	1	0	0
   14	1	1	1	1	1	1	1	0	0	0
   15	1	1	1	1	1	0	0	0	0	0
   16	1	1	1	1	1	0	0	0	0	0
   17	1	1	1	1	1	0	0	0	0	0
   18	1	1	1	1	1	0	0	0	0	0
   19	1	1	1	1	1	0	0	0	0	0
   20	1	1	1	1	1	1	1	1	0	0
   21	1	1	1	1	1	1	1	0	0	0
   22	1	1	0	0	1	1	1	0	0	0
   23	1	1	0	0	1	0	0	0	0	0
   24	1	1	0	0	0	0	0	0	0	0

5. RESULTS AND DISCUSSIONS TABLE 2

   The proposed method has been tested on systems with 10, 20 and 40 generating units. The unit data and load demand data for 24 hours for the systems with 10 units has been shown in the Tables A.1 and A.2 of the appendix respectively. The data for other bigger systems has been acquired by copying the data of 10 unit system and modifying the load demand in extent to the system size. The generation-load curve is shown in Fig.2. The best production cost of the proposed method is compared with ICGA [10], SFLA [11], EPL [12] and LR [13] and shown in

   Table 1. The analysis of this table demonstrates that the proposed method gives global ideal solution which compares to lower production cost than that of different techniques. In this paper population size, absorption coefficient, randomness parameter, attraction coefficient base value and maximum generations for 10, 20 and 40 units has been considered as 30, 1,0.2,0.9 and 500 respectively. The final commitment schedule and generation schedule for 10 unit system are presented in the Tables 2 and 3 respectively.

   Hour	Unit1	Unit2	Unit3	Unit4	Unit5	Unit6	Unit7	Unit8	Unit9	Unit10
   1	455	245	0	0	0	0	0	0	0	0
   2	455	295	0	0	0	0	0	0	0	0
   3	455	265	130	0	0	0	0	0	0	0
   4	455	455	0	0	40	0	0	0	0	0
   5	455	395	130	0	20	0	0	0	0	0
   6	455	360	130	130	25	0	0	0	0	0
   7	455	410	130	130	25	0	0	0	0	0
   8	455	455	130	130	30	0	0	0	0	0
   9	455	455	130	130	80	30	20	0	0	0
   10	455	455	130	130	162	38	20	10	0	0
   11	455	455	130	130	162	78	20	10	10	0
   12	455	455	130	130	162	85	20	43	10	10
   13	455	455	130	130	162	38	20	10	0	0
   14	455	455	130	130	85	25	20	0	0	0
   15	455	455	130	130	30	0	0	0	0	0
   16	455	315	130	130	20	0	0	0	0	0
   17	455	265	130	130	20	0	0	0	0	0
   18	455	360	130	130	25	0	0	0	0	0
   19	455	455	130	130	30	0	0	0	0	0
   20	455	455	130	130	162	33	25	10	0	0
   21	455	455	130	130	80	25	25	0	0	0
   22	455	455	0	0	140	25	25	0	0	0
   23	455	420	0	0	25	0	0	0	0	0
   24	455	345	0	0	0	0	0	0	0	0

   Hour	Unit1	Unit2	Unit3	Unit4	Unit5	Unit6	Unit7	Unit8	Unit9	Unit10
   1	455	245	0	0	0	0	0	0	0	0
   2	455	295	0	0	0	0	0	0	0	0
   3	455	265	130	0	0	0	0	0	0	0
   4	455	455	0	0	40	0	0	0	0	0
   5	455	395	130	0	20	0	0	0	0	0
   6	455	360	130	130	25	0	0	0	0	0
   7	455	410	130	130	25	0	0	0	0	0
   8	455	455	130	130	30	0	0	0	0	0
   9	455	455	130	130	80	30	20	0	0	0
   10	455	455	130	130	162	38	20	10	0	0
   11	455	455	130	130	162	78	20	10	10	0
   12	455	455	130	130	162	85	20	43	10	10
   13	455	455	130	130	162	38	20	10	0	0
   14	455	455	130	130	85	25	20	0	0	0
   15	455	455	130	130	30	0	0	0	0	0
   16	455	315	130	130	20	0	0	0	0	0
   17	455	265	130	130	20	0	0	0	0	0
   18	455	360	130	130	25	0	0	0	0	0
   19	455	455	130	130	30	0	0	0	0	0
   20	455	455	130	130	162	33	25	10	0	0
   21	455	455	130	130	80	25	25	0	0	0
   22	455	455	0	0	140	25	25	0	0	0
   23	455	420	0	0	25	0	0	0	0	0
   24	455	345	0	0	0	0	0	0	0	0

   TABLE 3

   No. of unit	Total Operating Cost ($)
   	ICGA [10]	SFLA [11]	EPL [12]	LR [13]	Proposed Algorithm
   10	566404	564769	563977	565825	562490
   20	1127244	1123261	1124369	1130660	1126900
   40	2254123	2246005	2246508	2258503	2246000

   No. of unit	Total Operating Cost ($)
   	ICGA [10]	SFLA [11]	EPL [12]	LR [13]	Proposed Algorithm
   10	566404	564769	563977	565825	562490
   20	1127244	1123261	1124369	1130660	1126900
   40	2254123	2246005	2246508	2258503	2246000

   Fig. 2.Generation- load Curve TABLE 1

6. CONCLUSION

This paper has presented a firefly algorithm for determination of optimal solution of UC with respect to load demand. The feasibility of the proposed method has been implemented with the system of 10, 20 and 40 generating units in respect to load demand. Finally, the obtained result of the proposed method is compared with that of other different methods. From the results, it is observed that the proposed method provides the quality solution which is low cost.

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APPENDIX

TABLE A.1. Unit data for 10 unit system

TABLE A.2. Load Deman Data for 10 unit system (Reserve is taken as 10% of load demand)