Pso Based Short-term Hydrothermal Scheduling with Prohibited Discharge Zones

— This paper presents a new approach to determine the optimal hourly schedule of power generation in a hydrothermal power system using PSO technique.. The simulation results reveal that the proposed PSO approach appears to be the powerful in terms of convergence speed, computational time and minimum fuel cost.


INTRODUCTION
The optimal scheduling of generation in a hydrothermal system involves the allocation of generation among the hydroelectric and thermal plants so as to minimize the total operation costs of thermal plants while satisfying the various constraints on the hydraulic and power system network.In Short-term scheduling it is normally assumed that the largest dam levels at the end of the scheduling period have been set a medium term scheduling process that takes into account longer term river inflow modeling and load predictions.The short term scheduler than allocates this water (Power) to the various time intervals in an effort to minimize thermal generation costs while attempting to satisfy the various unit and reservoir constraints.
The main constraints include the time coupling effect of the hydro sub problem, where the water flow in an earlier time intervals affects the discharge capability at a later period of time, the time varying system long demand, the cascade nature of the hydraulic network, the varying hourly reservoir inflows, the physical limitations on the reservoir storage and turbine flow rate and loading limits of both thermal and hydro plants.Further constraints could be depending on the particular requirements of a given power system, such as the need to satisfy activities including, flood control, irrigation, fishing, water supply etc., The hydrothermal scheduling problem has been the subject of intensive investigation for several decades now.
Most of the methods that have been used to solve the hydrothermal co-ordination problem make a number of simplifying assumptions in order to make the optimization problem more tractable.
The performances of different stochastic techniques have been studied in the literature [6][7][8][9][10][11][12][13][14].Though stochastic techniques have been proved to be very efficient and having faster performances than the conventional methods, there are some limitations in the goodness of the solutions to the problem that are obtained in [13].From the literature it is found that particle swarm optimization technique has the fastest convergence rate to the global solution amongst all algorithms and has highest potential of finding more nearly global solutions to hydrothermal co-ordination problems [13].Early works on PSO have shown the rich promise of emergence of a relatively simple optimization technique this is easier to understand compared to other evolutionary computation techniques presently available eg. Genetic algorithm and evolutionary programming.
Another advantage of PSO can be the possibility of tuning smaller number of free, tunable parameters to arrive at the desired goal.The PSO technique has been applied to various fields of power system optimization.Yu et al applied PSO technique to solve short-term hydrothermal scheduling [16] with an equivalent thermal unit having smooth cost functions connected to hydel systems.Here the constraints were handled by penalty function method [16].But the performance of PSO to Short-term hydrothermal scheduling for interconnected individual thermal units with non-smooth cost function has not been tested yet.
In this paper PSO method is proposed for short-term optimal scheduling of generation in a hydrothermal system which involves the allocation of generation among the multireservoirs cascaded hydro plants and thermal plants with prohibited discharge zones and valve point loading effects so as to minimize the fuel cost of equivalent thermal plant while satisfying the various constraints on the hydraulic and power system network.
To validate the PSO based hydrothermal scheduling algorithm, the developed algorithm has been illustrated for a test system [11].The same problem has been solved by GA and the results are compared.The performance of the proposed method is found to be quite encouraging as compared with other methods.

C
Composite Cost function www.ijacsa.thesai.org Output power of i th thermal unit at time "m" P GHjM Output power of j th hydro unit at time "m" P GTi min ,P  Due to Zero incremental cost of hydro generating units, the prime objective of the short-term hydrothermal scheduling problem becomes to minimize the fuel cost of thermal plants, while making use of the availability of hydropower as much as possible, such that the load demands P D supplied from hydro plants and a thermal plant in the intervals of the generation scheduling horizon can be met and simultaneously, all the equality and inequality operation constraints are satisfied.
The objective function and associated constraints of the Hydrothermal scheduling problem are formulated as follows.

A. Objective Function
The total fuel cost for running the thermal system to meet the load demand in scheduling horizon is given by C. The objective function is expressed mathematically, as When considering valve-point effects, the fuel cost function of each thermal generating unit is expressed as the sum of a quadratic and a sinusoidal function.The total fuel cost in-terms of real power output can be expressed as : subject to a number of unit and power system network constraints.

B. constraints
This non-linear constrained hydrothermal scheduling optimization problem is subjected to a variety of constraints depending upon practical implications like the varying system load demand, the time coupling effect of hydro subsystem, the cascading nature of the hydraulic network, the time varying hourly reservoir inflows, thermal plant and hydro plant operating limits, system losses, reservoir storage limits, water www.ijacsa.thesai.orgdischarge rate limits, hydraulic continuity constraints and initial and final reservoir storage limits.These constraints are discussed below.

1) Power balance constraints(Demand Constraints)
This constraint is based on the principle of equilibrium between the total active power generation from the hydro and thermal plants and the total system demand plus the system losses in each time interval of scheduling "m" 2) Thermal Generator Constraints The operating limit of equivalent thermal generator has a lower and upper bound so that it lies in between these bounds.

3) Hydro Generator Constraints
The operating limit of hydro plant must lie in between its upper and lower bounds.

HYDRAULIC NETWORK CONSTRAINTS
The hydraulic operational constraints comprise the water balance (Continuity) equations for each hydro unit (System) as well as the bounds on reservoir storage and release targets.
These bounds are determined by the physical reservoir and plant limitations as well as the multipurpose requirements of the hydro system.These constraints include :

1) Reservoir Capacity Constraints
The operating volume of reservoir storage limit must lie in between the minimum and maximum capacity limits.
V Hj min ≤V Hjm ≤V Hj max , jN H , mM (6)

2) The Water Discharge Constraints
The variable net head operation is considered and the physical limitation of water discharge of turbine, Q Hjm , Must lie in between maximum and minimum operating limits, as given by The desired volume of water to be discharged by each reservoir over the scheduling period,

4) Hydraulic Continuity Equation Constraint
The storage reservoir volume limits are expressed with given initial and final volumes as Where  lj is the water delay time between reservoir l and its upstream u at interval" m".R u is the set of upstream units directly above the hydro plant "j".

5) Power Generation Characteristics
The Power generated from a hydro plant is related to the reservoir characteristics as well as the water discharge rate.A number of models have been used to represent this relationship.In general, the hydro generator power output is a function of the net hydraulic head, H, reservoir volume, V H , and the rate of water discharge, Q H , P GHjm = f(Q Hjm , V Hjm ) andV Hjm =f(H jm ) (10) The model can also be written in-terms of reservoir volume instead of the reservoir net head, and a frequently used functional is Net head variation can only be ignored for relatively large reservoirs, in which case power generation is solely dependent on the water discharge.In setting the generation levels of the thermal plants, a quadratic cost function is used to model the fuel input power output characteristic of thermal units.

IV. PARTICLE SWARM OPTIMIZATION
Particle swarm optimization is one of the most recent developments in the category of combinatorial metaheuristic optimizations.This method has been developed under the scope of artificial life where PSO is inspired by the natural phenomenon of fish schooling or bird flocking.PSO is basically based on the fact that in quest of reaching the optimum solution in a multi-dimensional space, a population of particles is created whose present coordinate determines the cost function to be minimized.After each iteration the new velocity and hence the new position of each particle is updated on the basis of a summated influence of each particle"s present velocity, distance of the particle from its own best performance, achieve so far during the search process and the distance of the particle from the leading particle, i.e. the particle which at present is globally the best particle producing till now the best performance i.e. minimum of the cost function achieved so far.
Let x and v denote a particle position and its corresponding velocity in a search space, respectively.Therefore, the i th particle is represented as x i = (x i1 , x i2 , . ..,x id ) in the "d" dimensional space.The best previous position of the i th particles recorded and represented as pbest i = (pbest i1 , pbest i2 , . .., pbest id ).The index of the best particle among all the particles in the group is represented by the gbest d .The rate of www.ijacsa.thesai.org the velocity for the particle i is represented as v i =(v i1 , v i2 , . .., v id ).
The modified velocity and position of each particle can be calculated using the current velocity and the distance from pbest id to gbest d as shown in the following formulas: where, N P is the number of particles in a group, Ng the number of members in a particle, k the pointer of iterations, w the inertia weight factor, C 1 , C 2 the acceleration constant, rand()the uniform random value in the range [0,1], v i k the velocity ofa particle i at iteration k, v d min ≤ v id k ≤ v d max and x i k is the current position of a particle i at iteration k.In the above procedures, the parameter v max determined the resolution, with which regions are to be searched between the present position and the target position.
If v max is too high, articles might fly past good solutions.If v max is too small, particles may not explore sufficiently beyond local solutions.The constants C 1 and C 2 represent the weighting of the stochastic acceleration terms that pull each particle toward the pbest and gbest positions.Low values allow particle to roam far from the target regions before being tugged back.On the other hand, high values result in abrupt movement toward or past, target regions.Hence, the acceleration constants C 1 and C 2 were often set to be 2.0 according to past experiences.Suitable selection of inertia weight "w" provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution.
As originally developed," w "often decreases linearly from about 0.3to -0.2 during a run.In general, the inertia weight w is set according to the following equation: where iter max is the maximum number of iterations and "iter" is the current number of iterations.

V. PSO BASED HYDROTHERMAL SCHEDULING
Taking the number of particles to be N, the no. of Scheduling intervals as m and the number of hydro unit, as N H , each initial trial vector Q (j, m, p) denoting the particles of population to be evolved for P = 1, 2, ….N is selected.The discharge of j th hydro plant at m th interval is randomly generated as Q GHjm  (Q GHj min , Q GHj max ) Let P K = [P GT1 , P GT2 , …..P GTi , ….. PG TNT , Q GH , Q GH1 , Q GH2 , …Q GHj …, Q GHNH ] T be a trail matrix designating the K th individual of population to be evolved and The elements P GTim and Q GHjm are the power output of the i th thermal unit and the discharge rate of the j th hydro plant at time interval m.The range of elements P GTim and Q GHjm should satisfy the thermal generating capacity and the water discharge rate constraints in equations ( 3) and ( 7) respectively.
Assuming the spillage in Eq (9) to be zero for simplicity the hydraulic continuity constraints are To meet exactly the restrictions on the initial and final reservoir storage in eq.( 9), the water discharge rate of j th hydro plant in the dependent interval "d" is then calculated by The dependent water discharge rate must satisfy the constraints is Eq (7).After knowing the water discharges, the reservoir volumes of different intervals are determined.Then, the hydro generations are calculated from Eq (11).Knowing the calculated hydro generations, P GHjm and the given load demand P Djm for m =1, 2 …..m, thermal generations P GTi can be calculated as N H P GTim = P Dm + P Lossm -P GHjm (17) j=1 Also to meet exactly the power balance constraints in Eq (3), the thermal power generation P GTdm of the dependent thermal generating unit can then be calculated using the following equation.
The dependent thermal generation must satisfy the constraints in Eq. ( 4).All the generation levels, discharges, reservoir water volumes and initial and final reservoir storage volumes must be checked against their limiting values as per eq"s.(4)- (11).

Stopping Rule :
The iterative procedure of generating new solutions with minimum function value is terminated when a predefined maximum number of iterations (generations) is reacted.www.ijacsa.thesai.org

VI. PSO ALGORITHM
The computational process of PSO technique can be described in the following steps.
Step 1 Input parameters of the system and specify the upper and lower boundaries of each variable.

Step 2
Initialize randomly the particles of the population according to the limit of each unit including individual dimensions, searching points and velocities.There initial particles must be feasible candidate solutions that satisfy the practical operating constraints.

Step 3
Let, Qp = [q 11 , q 12 , ……, q 1m , q 21 , q 22 , …. q 2m ,…q n1 , q n2 , …., q nm ], be the trait vector denoting the particles of population to be evolved.The elements of q jm are the discharges of turbines of reservoirs at various intervals subjected to their capacity constraints in (7).q id , be the dependent discharge of i th hydro plant at d th interval is randomly selected from among the committed "m" intervals.Then, knowing the hydro discharges, storage volumes of reservoirs V jm are calculated by ( 9).Then P GHjm is calculated from ( 11) for all the intervals.

Step 4
Compare each particle (4 x 24) evaluation value with its P best the best evaluations value among P best is denoted as g best .

Step 6
Each particle is evaluated according to its updated position, only when satisfied by all constraints.If the evaluation value of each particle is better than the previous P best .The current value is set to be P best .
If the best P best is better than g best , the value is set to be g best .

Step 7
If the stopping criterion is reacted, then go to Step-8, otherwise go to Step-2.

Step 8
The individual that generates the latest g best is the solution of the problem and then print the result and stop.

A. Test System
To verify the applicability and to evaluate the performance of the proposed PSO algorithm, a test system has been adapted from [22], [23].It consists of a multi chain cascade of four hydro units, and a number of thermal units represented by an equivalent thermal plant.The schedule horizon is one day with 24 intervals of 1 hour each.
The cost of thermal generation can be obtained in two ways: a) By using a standard economic dispatch technique to find the optimal operation cost of the on-line thermal generators.b) By assuming the thermal generation is represented by an equivalent single plant, where characteristic can be determined as described in [1].
The hydraulic Sub-system is characterized by the following: c) A multi chain cascade flow network, with all of the plants on one stream; d) Reservoir transport delay between successive reservoirs; e) Variable head hydro plants; f) Variable natural inflow rates into each reservoir; g) Variable load demand over scheduling period.
The data of the test system considered here are the same as in [10] and the additional data with valve point loading effect are also same as in Reference [11].
The hydro Sub-system configuration is shown in fig 1 .The hydraulic test network models most of the complexities encountered in practical hydro networks.The load demand, hydro units power generation Coefficients, river inflows, reservoir limits are given in reference [11].
The fuel cost function of the equivalent thermal plant unit with valve point loading is C i (P GTi ) = 5000 + 19.2 P GTi + 0.002 P G2Ti + 700 Sin (0.085 (P GTi min -P GTi ) And the inequality constraint limit of this unit is The Spillage rate for the hydraulic system is not taken into account for simplicity and further the electric loss from the hydro plant to the load is taken to be negligibly small.
To demonstrate the effectiveness of the proposed PSO method, the system is considered with prohibited discharge zones and with valve point loading effects.

B. Simulation Results
In short term hydrothermal scheduling problem, the two important parameters, that can be allowed to vary, are the satisfaction of the final reservoir levels and the cost of thermal generation.The present work has been implemented in command line of Matlab-7.0for the solution of hydrothermal scheduling.The program was run on a 2.70 GHz, Pentium-® Dual core, with 1GB RAM PC.After a number of trails of run with different values of PSO parameters tuning , such as inertia weight, number of particles, maximum allowable velocity, the details of key parameters selected are: w max =0.9,w min =0.4,N=20,c 1 =c 2 =2.0,iter max =100.
The optimal hydro generations , optimal hydro discharges, hydro reservoir levels with minimum cost obtained by the proposed PSO methods are reported in tables 6-8 respectively.VIII.CONCLUSION In this paper an approach of particle swarm optimization has been proposed and demonstrated to solve shortterm hydrothermal scheduling problem.In the algorithm, the thermal generator units are represented by and equivalent unit.The generator load power balance equations and total water discharge equation have been subsumed into system model .constraintson the operational limits of the thermal and hydro units on the reservoir volume limits are also included in the algorithm.the numerical results show that the proposed approach is better than generic algorithm in terms of having better solution quality and good convergence characteristics.The PSO approach can easily be extended to other complex optimization problems faced by the utilities.

Final
storage volume of j th Reservoir III.MATHEMATICAL FORMULATION Hydrothermal Scheduling involves the optimization of a problem with a non-linear objective function, with a mixture of linear, non-linear and dynamic network flow constraints.The problem difficulty is compounded by a number of practical considerations and unless several simplifying assumption are made, this problem is difficult to solve for practical power systems as shown in fig 1.

Fig. 1
Fig.1 Practical Hydrothermal Power system Network

TABLE : 1
HYDRO THERMAL POWER GENERATIONS AND OPERATING COST OF EQUIVALENT THERMAL UNIT.
Fig.5 Convergence characteristic of PSO Algorithm for the test case.

TABLE : 4
SUMMARY OF TEST RESULTS