1 60 CHAPTER 3 SECURITY CONSTRAINED OPTIMAL SHORT-TERM HYDROTHERMAL SCHEDULING 3.1 INTRODUCTION Optimal short-term hydrothermal scheduling of power systems aims at determining optimal hydro and thermal generations in order to meet the load demands over a scheduled horizon of one day or a week while satisfying the various constraints on the hydraulic and power system network (Wood and Wollenberg 1996). The objective is to minimize total operation costs of thermal plants. The problem is a complex mathematical optimization problem with a highly nonlinear and computational expensive environment. In literature the hydrothermal scheduling with various constraints have been solved effectively by using GA based algorithms. However, none of these works reported in the literature considered the security constraints in the hydrothermal scheduling formulation, which are important from the point of view of practical application. In the present work, security constrained hydrothermal scheduling is solved using decomposition approach and GA based OPF. Also, GA based algorithm is implemented to solve the security constrained hydrothermal scheduling problem. The hydro subproblem is solved using proposed GA and thermal subproblem is solved using lambda iteration technique. GA based OPF is implemented only for the constraints violated intervals.
2 STATEMENT OF OPTIMAL HYDROTHERMAL SCHEDULING The generation mix of hydro and thermal plants for optimal scheduling is considered. For a number of time intervals N, the set of hydro plants is NH and the set of thermal units are NT. A typical short-term hydrothermal generation scheduling problem is formulated as (Mohan et al 1993): NT N Min. TPC F mj(pt mj) m1 j1 (3.1) subject to: the characteristic equations of the hydro plants where Y, i j 1 Y ij AL ij - D ij NH µ ik k 1 ik = 1 if reservoir i is down stream to reservoir k = 0 otherwise D ik ; i 1,2,...NH; j 1,2,...N (3.2) the active power generation of hydro plants PH ij (H oi/g)[1 C i(yij Y i, j 1)/2]D ij; i 1,2,...NH; j 1,2,...N the limits on water storage level in reservoirs (3.3) Y i (min) Yij Yi (max) ;i 1,2,...NH; j 1,2,...N 1 (3.4) with Y i1 and Y i, N+1 fixed for i = 1,2, NH
3 62 the limits on water discharge D i (min) Dij Di (max) ;i 1,2,...NH; j 1,2,...N (3.5) the limits on active power generation of hydro units PH i (min) PHij PHi (max) ;i 1,2,...NH; j 1,2,...N (3.6) The optimal power flow problem for j th interval with transmission security constraint is formulated as: NT Min. TFCj (ampt m m 1 s 2 mj b m PT mj subject to: the power balance constraints cm) aspt 2 sj bspt sj cs ; j 1,2,...N (3.7) NT PT mj PDth, j PL j ; j 1,2,...N (3.8) m1 where PD th, j PD j - NH i 1 PH i, j generation limit constraints PT PT PT ;m 1,2,...NT; j 1,2,...N (3.9) m, (min) mj m, (max) the slack bus constraint PT PT PT ; j 1,2,...N (3.10) s, (min) sj s, (max)
4 63 line flow constraint k, (min) k k, (max) ; k = 1, 2,.. NL (3.11) the power flow equations F i (X, U, C) = 0; i = 1, 2,.. NB (3.12) where the state vector X comprises of the bus voltage phase angles and magnitudes. The control vector U comprises of all the controllable system variables like real power generations. The parameter vector C includes all the uncontrollable system parameters such as line parameters, loads, etc. 3.3 SECURITY CONSTRAINED OPTIMAL SHORT-TERM HYDROTHERMAL SCHEDULING USING DECOMPOSITION APPROACH The optimal hydrothermal scheduling problem is solved in two phases. In the first phase the initial feasible water storage trajectory is obtained using Discharge Proportional to Demand Method (DPDM). For this water discharge a hydrothermal scheduling is obtained. In the next phase, the schedule is improved using local variation approach and lambda iteration technique to improve the hydrothermal scheduling. Line flow constraints are checked for its limits and GA based OPF is applied only to the intervals at which line flow constraints violate the limits. The algorithmic steps for the implementation of the proposed algorithm is as follows: Algorithm for Initial Water Storage Trajectory Step 1: Choose an initial water storage trajectory for each hydro plant using discharge proportional to demand method.
5 64 Determine the water discharge D ij and the hydro plant output Ph ij for i = 1,2 NH and j = 1,2 N. Step 2: Step 3: Step 4: For each time interval, j = 1,2 N compute the difference between system demand and total hydro power generation, PD th, j. PD th, j is considered as demand for the thermal units. The thermal subproblem is solved using lambda iteration technique. The fuel cost is calculated for each interval including transmission losses Discharge Proportional to Demand Method Step 1: Step 2: Step 3: Obtain total demand (PD tot ) by summing up the demands of the time intervals, j = 1,2 N. Obtain the total discharge (D i, tot ) by summing up the discharge of the time intervals, j = 1,2 N for each hydro plant i = 1,2 NH. Calculate the initial feasible water discharge for each hydro plant using the formula: D ij (PDj / PDtot) Di, tot ; i 1,2,...NH; j 1,2,...N (3.13) where, PD tot is the total system demand D i, tot is the total discharge of all hydro plants i =1,2 NH
6 Improving the Water Storage Trajectory Step 1: Set the hydro plant index, i = 1. Step 2: Set the time interval index, j = 1. Step 3: Step 4: Step 5: Step 6: Perturb the storage level of the hydro plant at the end of j th interval by +Y. Compute the discharge D ij and hydro generations, PH ij for the ith plant in the j th interval. Corresponding to the hydro generation PH ij, compute the optimal thermal generation schedule, P mj for the thermal plants m = 1,2 NT using lambda iteration technique. Repeat steps 4 and 5 for (j + 1) th interval. Step 7: Compute the total cost for j th and (j + 1) th intervals. Check for cost reduction by comparing pre-perturbed trajectory. If cost is less, proceed to step 8. Otherwise repeat steps 4 to 7 with a perturbation of -Y. If the cost is less, go to step 8. Otherwise, retain the pre-perturbed storage level and go to step 8. Step 8: Step 9: Step10: Increment the interval index, j = j + 1. If j < N go to step 3. Otherwise go to step 9. Increment the hydro plant index, i= i + 1. If i < NH go to step 2. Otherwise go to step10. Repeat the above procedure until the maximum generation count is reached.
7 Genetic Algorithm Based OPF The optimal solution for the GA based OPF includes power balance equality constraints, limits on the active power generations and limits on line phase angle as inequality constraints. Units under decommitted hours, power generation varies significantly at the generator buses. Subsequently, variation in the line flows may lead to overloading of the lines. The approach of genetic algorithm limits the flow in the overloaded lines by adjusting the real power generations of the committed units. Fast-decoupled load flow method is used to calculate the line losses and the line flows. The various steps of the algorithm for solving the OPF problem with line flow constraints for each interval are same as discussed in section NUMERICAL EXAMPLES AND RESULTS The proposed algorithm has been tested on two sample systems, one with 9 buses, 11 transmission lines, 4 thermal plants and 3 hydro plants and an adapted Indian utility system comprising 66 buses, 93 transmission lines, 12 thermal plants and 11 hydro plants. 66-bus and 9-bus system data are provided in the Appendices 2 and 3 respectively. To prove the effectiveness of the discharge proportional to demand method, it is compared with the Average Inflow Method (AIFM). In the AIFM the discharge during each interval is made equal to the average inflow, which is obtained by summing the inflows to the reservoir during all the intervals and dividing it by the number of intervals.
8 67 Figures 3.1and 3.2 shows the cost convergence characteristics of 9- bus and 66-bus systems respectively AIFM DPDM Total Fuel Cost (Rs) Hydro Thermal Iterations Figure 3.1 Cost convergence of 9-bus system DPDM AIFM Total Fuel Cost (RS) Hydro Therm al Iterations Figure 3.2 Cost convergence of 66-bus system
9 68 It is seen from Figures 3.1 and 3.2 that DPDM takes less number of iterations to reach the optimum solution and also it is seen that convergence is faster in the first few iterations and slow during the subsequent iterations and also the convergence curve is almost flat in the later part. AIFM takes more number of iterations to reach the near optimum solution. A number of trial studies were made on both the systems to choose the best initial incremental step size for Y and its subsequent reduction during trajectory perturbation from the convergence point of view. It is observed that the initial value of Y equal to 30 % of the initial discharge is the best choice, also its value should be reduced by 50% in the second and third iterations and thereafter maintained constant. Table 3.1 provides the cost comparison of both the methods for 9- bus and 66-bus systems. In DPDM both the initial and final cost obtained is less as compared to AIFM. For a 9-bus system, using DPDM the cost saving is 3.8 % as compared to AIFM and in the 66-bus system the cost saving using DPDM is 0.24 % as compared to the AIFM. Table 3.1 Cost comparisons of AIFM and DPDM System Solution DPDM (Rs) AIFM (Rs) 9-bus 66-bus Initial Final Initial Final Figures 3.3 and 3.4 gives the discharge trajectory of plant 1 in the 9-bus system for AIFM and DPDM. In AIFM the initial Discharge trajectory is 200cms, so it takes more number of iterations to reach the
10 69 optimum discharge. In DPDM the initial discharge trajectory is very close to the final discharge and it takes less number of iterations to reach the optimum discharge initial final Dishcarge (cms) Time Interval (hours) Figure 3.3 Discharge trajectory for plant 1 in 9-bus system AIFM initial final Discharge (cms) Time Interval (hours) Figure 3.4 Discharge trajectory for plant 1 in 9-bus System DPDM
11 70 Table 3.2 gives the comparison of optimal discharge obtained for the 9-bus system using AIFM and DPDM for 24 intervals. Table 3.2 Comparison of optimal discharge obtained for 9-bus system Interval (Hours) Optimal discharges of hydro plants in CMS (AIFM) Plant 1 Plant 2 Plant 3 Optimal discharges of hydro plants in CMS (DPDM) Total Total Plant 1 Plant 2 Plant 3 discharge discharge
12 71 Figure 3.5 shows the daily load curve of the 66-bus utility system. A set of 12 limiting lines is chosen for observing line flow constraint violations. Demand (MW) Time Interval (hours) Figure 3.5 Load curve of 66-bus system Table 3.3 gives the line phase angles of the violated line number 7 of a 66-bus system. AIFM gives the line violations at intervals 1, 2, 3, 23 and 24. The GA based OPF removes the line flow violations by adjusting the real power generations and consequently the cost is increased from Rs to Rs DPDM gives the line violations at intervals 1, 2, 23 and 24. The GA based OPF removes the line flow violations by adjusting the real power generations and the cost is increased from Rs to Rs
13 72 Table 3.3 Comparison of line phase angle (degrees) for AIFM and DPDM Violated line no. Rating (degrees) Interval no. Before OPF (degrees) AIFM After OPF (degrees) Before OPF (degrees) DPDM After OPF (degrees) SECURITY CONSTRAINED OPTIMAL SHORT-TERM HYDROTHERMAL SCHEDULING USING GA GA is applied to solve the security constrained optimal hydrothermal scheduling problem. The hydro subproblem is solved using GA and thermal subproblem is solved using lambda iteration technique without line losses. Both the hydro and thermal subproblems are solved alternatively. The total fuel cost over the time period is calculated including line losses for the best hydrothermal schedule obtained using proposed GA. Line flow constraints are checked for its limits at each interval. GA based OPF is implemented only for the constraint violated intervals. Fast-decoupled load flow method is used to calculate the line flows and losses. Computation of line flows and losses in each generation of genetic algorithm increases the computational time and increases the complexity of the problem. This proposed GA reduces the complexity, computation time and also gives near global optimum solution.
14 73 The hydro subproblem is solved using GA by creating the initial populations randomly. The strings generated in the population represent the discharge of each interval for all the plants. For the discharge, equivalent hydropower generations are calculated. The sum of hydropower generations of all the plants for each interval gives the demand for thermal subproblem. The thermal subproblem is solved using lambda iteration technique without considering losses. The cost obtained from economic dispatch and the penalty functions for the constraint violations are considered as objective function. Line losses and line flow constraints are computed only for the best solution obtained using GA. The various steps of the algorithm for solving the proposed GA for hydrothermal scheduling are given below Initialization of Population For the application of GA to the hydro scheduling problem a simple binary alphabet was chosen to encode a solution. Let the number of hydro units be NH, the string (in binary codes) be S and the number of time intervals be N, then each parent population is represented as follows: Step 1: Step 2: A number of initial binary-coded solutions (genotypes) are generated at random to form the initial parent of population size N p. Each string (S) represents the discharge for that particular interval of that unit (Figure 3.6). Binary strings are decoded to real values D ij (discharges) for the i th reservoir, i = 1,2 NH during the j th discrete time interval, j = 1,2 N.
15 74 S 1 S 2 S 3 S N-1 S N U U U NH Figure 3.6 Binary representation of hydro discharges Step 3: Step 4: Step 5: Step 6: Each discharge D ij is checked for minimum and maximum limits. If discharge D ij is less than the minimum discharge level it is made equal to minimum discharge and if the discharge D ij is greater than the maximum discharge level it is made equal to maximum discharge. Corresponding generation schedule of the hydro plants, Ph ij ; i = 1,2 NH is calculated. In each time interval, j = 1,2 N compute the balance demand to be met from thermal units PD th, j by taking the difference between the system demand and the total hydropower generation. The scheduling of thermal units were done for the demand of PD th, j. This economic dispatch problem is solved using lambda iteration technique. The fuel cost is calculated for each interval excluding transmission losses.
16 75 Step 7: The fitness function for each parent population F pi is computed as: NH N pi, lim pi, lim pi Tpi 1 i 2 j p i 1 j 1 F FC k DH k PT ; pi 1, 2,..., N (3.14) where K 1, and K 2 are penalty factors for the constraint violations, FC Tpi is the total fuel cost for pi-th parent and the constraint violations are given by DH pi, lim i N j 1 if ((Y if D ij N j 1 N j 1 - ((Y D i, (min) D ij ((Y - Y ij i, (min) i, (max) ((Y - Y i, (min) ) i, (min) i, (max) - Y N j 1 - Y ) N i, (max) j 1 ) AL ij) i, (max) AL N j 1 ) ij N j 1 D N ), j 1 AL ij, AL ij ij ) ) (3.15) NT NT PDth, j PTm, (max), if PDth, j PTm, (max) pi, lim m 1 m 1 PTj (3.16) NT NT PTm, (min) - PDth, j, if PDth, j PTm, (min) m 1 m 1 Equation (3.14) represents the fitness function of parent population N p, equation (3.15) represents the constraint violation of total discharge for NH hydro plants and equation (3.16) represents the constraint violation of total thermal generation of NT thermal plants.
17 Reproduction The reproduction operator is a prime selection operator. Two genotypes are selected using Roulette wheel parent selection algorithm that selects a genotype with a probability proportional to genotypes relative fitness within the population. Then, a new offspring genotype is produced by means of the two basic genetic operators namely crossover and mutation Crossover To get the new patterns of genetic strings during the evolution process, two levels of crossover operation, i.e. string level crossover and population level crossover are introduced. Both type of crossover is done with fixed probability of String level crossover A good scheduling is expected by exchanging the strings of the units within the genotype. Since the partial string of genotype has no fitness function value, the selection processes are performed randomly with certain probability Population level crossover This operator is applied with certain probability. When applied, the parent genotypes are combined to form two new genotypes that inherent solution characteristics from both parents. In the opposite case the offspring are identical replications of their parents. Crossover is done between the parent genotypes obtained from roulette wheel parent selection. The crossover scheme used is single-point crossover.
18 Mutation Mutation introduces new genetic material into the gene at some low rate. With a small probability, randomly chosen bits of the offspring genotypes change from 0 to 1 and vice versa Selection The entire population, including parent and offspring are arranged in descending order. The best N p solutions, which survive are transcribed along with their elements to form the basis of the next generation. The above procedure is repeated until the given maximum generation count is reached Genetic Algorithm Based OPF The optimal solution for the GA based OPF includes power balance equality constraints, limits on the active power generations and limits on line phase angle as inequality constraints. Units under decommitted hours, power generation varies significantly at the generator buses. Subsequently, variation in the line flows may lead to overloading of the lines. The approach of genetic algorithm limits the flow in the overloaded lines by adjusting the real power generations of the committed units. Fast-decoupled load flow method is used to calculate the line losses and the line flows. The various steps of the algorithm for solving the OPF problem with line flow constraints for each intervals are as discussed in section
19 NUMERICAL EXAMPLES AND RESULTS The proposed algorithm has been tested on two sample systems, the first one consisting of 9 buses, 11 transmission lines, 4 thermal plants and 3 hydro plants and the second with an adapted Indian utility system comprising 66 buses, 93 transmission lines, 12 thermal plants and 11 hydro plants. 66-bus and 9-bus system data are provided in the Appendices 2 and 3 respectively. The proposed GA is used to solve the hydrothermal scheduling problem. In order to avoid the misleading results due to stochastic nature of the GA, 20 trial runs were made with each run starting with different random populations. The population size was 50 genotypes in all the runs. The hydro thermal scheduling convergence characteristic of fitness function for the best five individuals of a 9-bus and 66-bus systems using proposed GA is presented in Figure 3.7 and Figure 3.8 respectively. The fitness function convergence characteristic is drawn by taking the parent with minimum fitness value at the end of iterations. It is seen from Figure 3.7 and Figure 3.8 that fitness function converges smoothly to the optimum value without any abrupt oscillations. This shows the convergence reliability of the proposed algorithm. Figure 3.9 shows the optimal discharge trajectories of the hydro plant 1 of 9-bus system. Figure 3.10 shows the daily load curve of the 9-bus system. It is seen from the Figure 3.9 that the hydro discharge trajectory obtained by the proposed GA closely matches with the daily load curve. Figure 3.11 shows the daily load curve of the 66-bus utility system. A set of 12 limiting lines is chosen for observing line flow constraint violations.
20 objective function ind 1 ind 2 ind 3 ind 4 ind No. of iterations Figure 3.7 Convergence characteristics of 9-bus system objective function ind 1 ind 2 ind 3 ind 4 ind No. of Iterations Figure 3.8 Convergence characteristics of 66-bus system
21 Discharge (cms) Time interval (hours) Figure 3.9 Discharge of hydro plant 1 of 9-bus system Load (MW) T im e In terval (h ou rs) Figure 3.10 Load curve of a 9-bus system
22 81 Demand (MW) Time Interval (hours) Figure 3.11 Load curve of 66-bus system Table 3.4 gives the line phase angles of the violated line number 7 of a 66-bus system. Hydrothermal scheduling obtained by the proposed GA gives the line flow violations at intervals 1, 2 and 24. The GA based OPF removes the line flow violations by adjusting the real power generations and consequently the cost is increased from Rs to Rs Table 3.4 Line flows in limiting line no. 7 of a 66-bus system Violated line no. Rating (degrees) Interval no. Without line flow constraints (degrees) With line flow constraints (degrees)
23 SUMMARY This chapter presents a security constrained hydrothermal scheduling using decomposition approach. Initial water storage trajectory is obtained using DPDM. Using local variation method the initial storage trajectory is improved. The thermal problem is solved using lambda iteration technique. To avoid the complexity and to reduce the computational time GA based OPF is applied only to the best solution obtained from decomposition approach. Investigations reveal that the proposed method is efficient, simple and reliable. GA based security constrained hydrothermal scheduling is proposed in this chapter. The hydro subproblem is solved using GA and thermal subproblem is solved using lambda iteration technique. To avoid the complexity and to reduce the computational time, GA based OPF is applied only to the best solution obtained from GA based hydrothermal scheduling. Investigations on both systems reveal that the proposed method is relatively simple and reliable.
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