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.
3.2 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
mj mj
m 1 j 1
Min. TPC F (PT )
(3.1)subject to:
the characteristic equations of the hydro plants
NH 1 k
ik ik ij
ij ij
1 j
i, Y AL -D µ D ;i 1,2,...NH;j 1,2,...N Y
(3.2) where ik = 1 if reservoir i is down stream to reservoir k
= 0 otherwise
the active power generation of hydro plants
PHij(H /G)[1 C (Yoi i ijY ,i j1)/2]D ; i ij 1,2,...NH; j 1,2,...N
(3.3) the limits on water storage level in reservoirs
1 1,2,...N j
1,2,...NH;
i
; Y Y
Yi(min) ij i(max) (3.4)
with Yi1 and Yi, N+1 fixed for i = 1,2,…NH
the limits on water discharge
1,2,...N j
1,2,...NH;
i
; D D
Di(min) ij i(max) (3.5)
the limits on active power generation of hydro units 1,2,...N j
1,2,...NH;
i
; PH PH
PHi(min) ij i(max) (3.6)
The optimal power flow problem for jth interval with transmission security constraint is formulated as:
1,2,...N j
; c PT b PT a ) c PT b PT (a TFC
Min.
NT
s m
1 m
sj s 2 s
s sj mj m
2 m m mj
j
(3.7) subject to:
the power balance constraints
1,2,...N j
PL PD
PT th,j j; NT
1 m
mj
(3.8)where
NH 1 i
j i, j
j
th, PD - PH
PD
generation limit constraints
1,2,...N j
1,2,...NT;
m
; PT
PT
PTm,(min) mj m,(max) (3.9)
the slack bus constraint
1,2,....N j
; PT
PT
PTs,(min) sj s,(max) (3.10)
line flow constraint
k, (min) k k, (max); k = 1, 2, . . NL (3.11)
the power flow equations
Fi(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:
3.3.1 Algorithm for Initial Water Storage Trajectory
Step 1: Choose an initial water storage trajectory for each hydro plant using discharge proportional to demand method.
Determine the water discharge Dij and the hydro plant output Phij for i = 1,2…NH and j = 1,2…N.
Step 2: For each time interval, j = 1,2…N compute the difference between system demand and total hydro power generation, PDth, j.
Step 3: PDth, j is considered as demand for the thermal units.
Step 4: The thermal subproblem is solved using lambda iteration technique. The fuel cost is calculated for each interval including transmission losses.
3.3.2 Discharge Proportional to Demand Method
Step 1: Obtain total demand (PDtot) by summing up the demands of the time intervals, j = 1,2…N.
Step 2: Obtain the total discharge (Di, tot) by summing up the discharge of the time intervals, j = 1,2…N for each hydro plant i = 1,2…NH.
Step 3: Calculate the initial feasible water discharge for each hydro plant using the formula:
1,2,...N j
1,2,...NH;
i
; D ) PD / (PD
Dij j tot i, tot
(3.13) where, PDtot is the total system demand
Di, tot is the total discharge of all hydro plants i =1,2…NH
3.3.3 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: Perturb the storage level of the hydro plant at the end of jth interval by +Y.
Step 4: Compute the discharge Dij and hydro generations, PHij for the ith plant in the jth interval.
Step 5: Corresponding to the hydro generation PHij, compute the optimal thermal generation schedule, Pmj for the thermal plants m = 1,2…NT using lambda iteration technique.
Step 6: Repeat steps 4 and 5 for (j + 1)th interval.
Step 7: Compute the total cost for jth 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: Increment the interval index, j = j + 1. If j < N go to step 3. Otherwise go to step 9.
Step 9: Increment the hydro plant index, i= i + 1. If i < NH go to step 2. Otherwise go to step10.
Step10: Repeat the above procedure until the maximum generation count is reached.
3.3.4 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 2.11.2.
3.4 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.
Figures 3.1and 3.2 shows the cost convergence characteristics of 9- bus and 66-bus systems respectively.
5000000 5200000 5400000 5600000 5800000 6000000 6200000 6400000
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99
Hydro Thermal Iterations
Total Fuel Cost (Rs)
AIFM DPDM
Figure 3.1 Cost convergence of 9-bus system
4100000 4140000 4180000 4220000 4260000 4300000
1 6 11 16 21 26 31 36 41 46 51
Hydro Therm al Iterations
Total Fuel Cost (RS)
DPDM AIFM
Figure 3.2 Cost convergence of 66-bus system
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 Initial 5325855 6245051
Final 5162751 5369144
66-bus Initial 4210379 4278965
Final 4138107 4148423
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
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.
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time Interval (hours)
Dishcarge (cms)
initial final
Figure 3.3 Discharge trajectory for plant –1 in 9-bus system – AIFM
0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time Interval (hours)
Discharge (cms)
initial final
Figure 3.4 Discharge trajectory for plant –1 in 9-bus System – DPDM
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)
Optimal discharges of hydro plants in CMS (DPDM)
Plant 1 Plant 2 Plant 3 Total
discharge Plant 1 Plant 2 Plant 3 Total discharge 1 166.75 41.33 104.78 312.86 130.80 118.23 77.29 326.32 2 103.20 115.19 106.61 325.00 129.59 80.19 100.59 310.37 3 105.47 126.51 111.45 343.43 97.26 114.10 127.46 338.82 4 120.68 141.84 128.92 391.44 111.61 137.88 140.34 389.83 5 299.71 196.31 174.52 670.54 285.63 199.74 211.59 696.96 6 346.47 257.44 299.03 902.94 332.39 258.39 343.48 934.26 7 232.03 318.71 330.42 881.16 259.10 309.22 332.52 900.84 8 239.45 317.68 254.19 811.32 256.41 328.09 254.25 838.75 9 217.21 254.16 248.41 719.78 230.20 274.74 245.65 750.59 10 199.51 218.79 213.67 631.97 215.03 235.39 214.83 665.25 11 188.17 189.63 198.80 576.6 205.92 201.81 201.89 609.62 12 163.55 165.61 177.59 506.75 189.76 167.29 186.12 543.17 13 183.49 177.97 157.83 519.29 185.22 187.65 188.89 561.76 14 183.95 150.95 187.25 522.15 193.54 193.85 175.97 563.36 15 162.43 186.00 199.64 548.07 193.57 183.45 197.82 574.84 16 187.20 206.56 191.73 585.49 193.10 203.57 215.56 612.23 17 212.65 247.28 237.25 697.18 223.68 256.13 251.73 731.54 18 266.47 268.99 268.52 803.98 288.62 276.61 266.61 831.84 19 256.64 286.61 235.48 778.73 278.49 277.45 246.09 802.03 20 196.14 184.55 197.10 577.79 197.28 197.37 203.02 597.67 21 200.20 202.14 200.19 602.53 156.73 155.82 159.96 472.51 22 195.99 191.84 195.99 583.82 156.81 156.84 156.88 470.53 23 192.00 180.09 196.08 568.17 153.56 147.41 153.81 454.78 24 180.63 173.82 184.54 538.99 135.72 138.79 147.65 422.16
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.
0 200 400 600 800 1000 1200 1400 1600 1800
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time Interval (hours)
Demand (MW)
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. 4148421.38 to Rs. 4194384.76. 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. 4138117.65 to Rs. 4167539.94.
Table 3.3 Comparison of line phase angle (degrees) for AIFM and DPDM
Violated line no.
Rating (degrees)
Interval no.
AIFM DPDM
Before OPF (degrees)
After OPF (degrees)
Before OPF (degrees)
After OPF (degrees)
7 2.44
1 2.7465 2.4152 2.6875 2.4236 2 2.6597 2.4118 2.5394 2.4351
3 2.4735 2.1903 - -
23 3.2721 2.4340 2.9723 2.4329 24 3.2756 2.4300 2.9961 2.4264
3.5 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.
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.
3.5.1 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: A number of initial binary-coded solutions (genotypes) are generated at random to form the initial parent of population size Np. Each string (S) represents the discharge for that particular interval of that unit (Figure 3.6).
Step 2: Binary strings are decoded to real values Dij (discharges) for the ith reservoir, i = 1,2…NH during the jth discrete time interval, j = 1,2…N.
S1 S2 S3 SN-1 SN U1 11010101 10011000 11001101 00111101 11011101 U2 11101011 11001101 01101000 01101001 01001001
. .
. .
. .
UNH 10110101 01001010 01101110 01011001 01011100
Figure 3.6 Binary representation of hydro discharges
Step 3: Each discharge Dij is checked for minimum and maximum limits. If discharge Dij is less than the minimum discharge level it is made equal to minimum discharge and if the discharge Dij is greater than the maximum discharge level it is made equal to maximum discharge.
Step 4: Corresponding generation schedule of the hydro plants, Phij; i = 1,2…NH is calculated.
Step 5: In each time interval, j = 1,2…N compute the balance demand to be met from thermal units PDth, j by taking the difference between the system demand and the total hydropower generation.
Step 6: The scheduling of thermal units were done for the demand of PDth, j. This economic dispatch problem is solved using lambda iteration technique. The fuel cost is calculated for each interval excluding transmission losses.
Step 7: The fitness function for each parent population Fpi is computed as:
NH pi, lim N 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 K1, and K2 are penalty factors for the constraint violations, FCTpi is the total fuel cost for pi-th parent and the constraint violations are given by
) AL )
Y - ((Y
D if
, ) D
AL )
Y - ((Y
) AL )
Y - ((Y
D if
), AL )
Y - ((Y
D
DH
N 1 j
N 1 j
ij (max)
i, (min) i, ij
N 1 j
ij N
1 j
ij (max)
i, (min) i,
N 1 j
N 1 j
ij (max)
i, (min) i, ij
N 1 j
N 1 j
ij (max)
i, (min) i, ij
lim pi, i
-
(3.15)
PT PD
if , PD - PT
PT PD
if , PT
PD
PT NT
1 m
(min) m, j
th, j
th, NT
1 m
(min) m,
NT 1 m
(max) m, j
th, NT
1 m
(max) m, j
th, lim
pi,
j (3.16)
Equation (3.14) represents the fitness function of parent population Np, 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.
3.5.2 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.
3.5.3 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 0.7.
3.5.3.1 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.
3.5.3.2 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.
3.5.4 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.
3.5.5 Selection
The entire population, including parent and offspring are arranged in descending order. The best Np 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.
3.5.6 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 2.11.2.
3.6 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.
0 10000000 20000000 30000000 40000000 50000000 60000000 70000000 80000000 90000000
1 26 51 76 101 126 151 176
No. of iterations
objective function
ind 1 ind 2 ind 3 ind 4 ind 5
Figure 3.7 Convergence characteristics of 9-bus system
0 5000000 10000000 15000000 20000000 25000000
1 26 51 76 101 126 151 176
N o. of Iterations
objective function
ind 1 ind 2 ind 3 ind 4 ind 5
Figure 3.8 Convergence characteristics of 66-bus system
0 100 200 300 400
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time interval (hours)
Discharge (cms)
Figure 3.9 Discharge of hydro plant 1 of 9-bus system
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0
0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 T im e In terval (h ou r s )
Load (MW)
Figure 3.10 Load curve of a 9-bus system
0 200 400 600 800 1000 1200 1400 1600 1800
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time Interval (hours)
Demand (MW)
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. 4310906.89 to Rs. 4341229.57.
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)
7 2.44
1 2.8077 2.4282
2 2.5776 2.4086
24 2.6252 2.4292
3.7 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.
