CHAPTER 7 SHORT TERM HYDROTHERMAL SCHEDULING WITH PROHIBITED OPERATING ZONES

Transcription

1 135 CHAPTER 7 SHORT TERM HYDROTHERMAL SCHEDULING WITH PROHIBITED OPERATING ZONES 7.1 INTRODUCTION A modern power system consists of a large number of thermal and hydel plants connected at various load centers through a transmission network. An important objective in the operation of such a power system is to generate and transmit power to meet the system load demand at minimum fuel cost by an optimal mix of various types of plants. The study of the problem of optimum scheduling of power generation at various plants in a power system is of paramount importance, particularly where the hydel sources are scarce and high cost of thermal generation has to be relied upon to meet the power demand. The hydel resources being extremely limited, the worth of water is greatly increased. If optimum use is made of their limited resource in conjunction with the thermal sources, huge saving in fuel and the associated cost can be made (Ohishi 1991). All hydro-systems are basically different from each other in their characteristics. The reason for this difference are plenty- the chief points being their natural difference in their water areas, difference between release elements, control constraints, non-uniform water flow etc. Sudden alteration in the volume of water flow due to natural constraints, occurrence of flood, draught and other natural calamities also affect the hydro scheduling. Navigational requirement of agricultural water may also govern the hydro

2 136 scheduling. Sometimes, water release may be dictated by treaties between the states and due to the fishing requirements. In certain sectors, however, the hydel source is sufficiently large, particularly in rainy seasons the inflows into the hydel reservoirs exhibits an annual cycle. Furthermore, there may be a seasonal variation in power demand on the system and this too exhibits an annual cycle. The optimization interval of one year duration is thus a natural choice for long range optimal generation scheduling studies. The solution to the scheduling problem in this case consists of determination of water quantities to be drawn from the reservoirs for hydro generation in each sub-interval and the corresponding thermal generations to meet the load demand over each sub-interval utilizing the entire quantity of water available for power generation during the total interval. The long range scheduling (generally persisting from months to year) involves mainly the scheduling of water release. Long range scheduling also involves meteorological and statistical analysis (Huang Chen and Chang 1996). The benefit of this scheduling is to save the cost of generation, in addition to meeting the agricultural and irrigational requirements. Long range scheduling involves optimization of the operating policy in the context of major unknowns such as load, hydroelectric inflows, unit availability etc. The short range problem usually has an optimization interval of a day or a week. This period is normally divided into sub-intervals for scheduled purposes. Here, the load, water inflows and unit availabilities are assumed to be known. A set of starting conditions (i.e. reservoirs levels) being given, the optimal hourly schedule can be prepared that minimizes a desired objective while meeting system constraints successfully. Cost optimization of hydro stations can be achieved by assuming the water heads constants and converting the incremental water (i.e. fuel) rate characteristics into incremental fuel cost curves by multiplying it with cost of water per cubic meter and applying the conventional technique of minimizing the cost function (Baskar 2002).

3 HYDROTHERMAL SCHEDULING IN POWER SYSTEM Optimal scheduling of power plant generation is the determination of the generation for every generating unit such that the total system generation cost is minimum while satisfying the system constraints. The objective of the hydrothermal scheduling problem is to determine the water releases from each reservoir of the hydro system at each stage such that the operation cost is minimized along the planning period. The operation cost includes fuel costs for the thermal units, import costs from neighboring systems and penalties for load shedding. The basic question in hydrothermal co-ordination is to find a trade-off between a relative gain associated with immediate hydro generation and the expectation of future benefits coming from storage (Huang Chen and Chang 1996). Classification of Hydrothermal Scheduling Problem 1. Long range problem 2. Short range problem Long Range Problem Long range problem includes the yearly cyclic nature of reservoir water inflows and seasonal load demand and correspondingly a scheduling period of one year is used. The solution of the long range problem considers the dynamics of head variations through the water flow continuity equation. The co-ordination of the operation of hydroelectric plants involves, of course, the scheduling of water releases. The long-range hydro-scheduling problem involves the long-range forecasting of water availability and the scheduling of reservoir water releases for an interval of time that depends on the reservoir

4 138 capacities. Typical long-range scheduling goes anywhere from 1 week to 1 year or several years. For hydro schemes with a capacity of impounding water over several seasons, the long-range problem involves meteorological and statistical analysis. The purpose of the long-term scheduling is to provide a good feasible solution that is close to the long-term cost minimization of the whole system. The problem is usually very difficult to solve due to its size, the time span (up to several years) and the randomness of the water inflows over the long term. Long-range scheduling involves optimizing a policy in the context of unknowns such as load, hydraulic inflows and unit availabilities (steam and hydro). These unknowns are treated statistically and long-range scheduling involves optimization of statistical variables. Short Range Problem The load demand on the power system exhibits cyclic variation over a day or a week and the scheduling interval is either a day or a week. As the scheduling interval of short range problem is small, the solution of the shortrange problem can assume the head to be fairly constant. The amount of water to be utilized for the short-range scheduling problem is known from the solution of the long-range scheduling problem. Short-range hydro-scheduling (1 day to 1 week) involves the hour-by-hour scheduling of all generation on a system to achieve minimum production cost for the given time period. two groups The short term hydrothermal scheduling problem is classified into 1. Fixed head hydro thermal scheduling 2. Variable head hydro thermal scheduling

5 PROBLEM FORMULATION The hydrothermal scheduling problem is a power system optimization problem with an objective function, which is a concatenation of linear, non-linear and dynamic network flow constraint (Baskar et al 2002). Since the hydro generating units have zero incremental cost, the hydrothermal scheduling problem is aspired to optimize the system thermal cost, while trying to maximize the hydro electric power generation. The objective function and associated constraints of the hydrothermal scheduling problem are formulated as follows. Objective Function The objective function of the hydrothermal scheduling problem is the minimization of the thermal power generation cost (Baskar et al 2002). T n F CTk = FC i (P sti (t)) t=1 i=1 (7.1) Constraints (i) Power balance equation n m D t = P sti (t) + P hyj (t) P L i=1 j=1 (7.2) The hydro generation P hyj (t) is a function of water discharge rate and storage volume. (ii) Thermal generation capacity P stimin P sti (t) P stimax (7.3)

6 140 (iii) Hydro generation capacity P hyjmin P hyj (t) P hyjmax (7.4) (iv) Hydraulic Continuity V j (t+1) = V j (t) + q j (t m) + s j (t m) q j (t) s j (t) + r j (t) (7.5) Where m is the water delay time between reservoir j and its upstream 1 at interval t. (v) Initial and final reservoir storage V j (0) = V 0 ; V j (T) = V T (7.6) (vi) Reservoir storage V jmin V j (t) V jmax (7.7) (vii) Water discharge rate q jmin q j (t) q jmax (7.8) (viii) where Total water discharge T q jtot = q i (t) t=1 (7.9) D t : System load demand at interval t FC i (P sti (t)): Fuel cost function of the i th thermal unit F CTk : Objective function value of k th individual of a population

7 141 n : Number of thermal generating units m : Number of hydro generating units P sti (t) : Thermal generation of unit i at interval t P hyj (t) : Hydro generation of plant j at interval t P L : Total Transmission loss P stimin, P stimax : P hyjmin, P hyjmax : Minimum and maximum generation capacity limits of thermal unit Minimum and maximum generation capacity limits of hydro unit q j (t) : Water discharge rate of plant j at interval t r j (t) : Inflow rate into the storage reservoir of plant j at interval t s j (t) : Spillage of reservoir j at interval t T : Number of hours in the study period V j (t) : Reservoir storage volume of plant j at interval t V 0, V T : Initial and final reservoir storage Prohibited Operating Zones A thermal or hydro generating unit may have a prohibited operating zone, depending on the physical limitations of the power plant components like generator shaft etc. At certain power output, vibrations occur in the components and when the frequency of vibration equals the natural frequency, resonance occurs and it may damage the components. Those

8 142 power outputs at which this phenomenon occurs are called Prohibited Operating Zones. Hence the thermal and hydro units should not be operated in the specific prohibited operating zone. For a POZ, the unit can operate only above or below the zone. In this work, POZ has been taken as an additional constraint for the test cases. 7.4 NUMERICAL EXAMPLES AND SIMULATION RESULTS Test case 1: One hydro and one thermal system In order to prove the validity and usefulness of the proposed algorithm, result of short term hydrothermal scheduling problem was considered. The test system is from (Yang et al 1996). It consists of a hydro plant and an equivalent thermal plant. The load demands over six 12-hour intervals are shown in Table 7.1. plant is as follows: The fuel cost function in dollars per hour of the equivalent thermal F(P st ) = P st P st ; 150 P st 1500 denoted as, The hydro power generation relationship to water discharge is q = 4.97 P hy ; 0 P hy 1000 q = 0.05 (P hy 1000) (P hy 1000) ;1000 P hy 1100

9 143 Table 7.1 Load demand data for test system-1 Interval Number Day Interval Demand (MW) 1 1 st day 0 hour 12.0 hour st day 12.0 hour 24.0 hour nd day 0 hour 12.0 hour nd day 12.0 hour 24.0 hour rd day 0 hour 12.0 hour rd day 12.0 hour 24.0 hour 1300 The data for the hydro plant is given in Table 7.2. Table 7.2 Hydro plant data for test system-1 V min V max q min q max V 0 V 6 R (m 3 ) (m 3 ) (m 3 /hr) (m 3 /hr) (m 3 ) (m 3 ) (m 3 /hr) The following control parameters have been chosen after running a number of simulations: The parameters used in BFA are as follows: Number of Bacteria - 40 Number of iteration in Chemotactic Loop - 6 Swimming Length - 4 Number of Reproduction - 2

10 144 Probability of elimination and dispersal d attract, attract, h repellent, repellent , 3e -6, 0.25, 15e -5. The test case was solved by proposed bacteria foraging algorithm and the solution obtained was compared with the other methods like gradient search, simulated annealing, genetic algorithm, EP method and hybrid EP method. All the programs were developed using MATLAB and the test case was simulated for 10 independent trials. For the purpose of comparison of performance of different algorithms, the results of gradient search, simulated annealing and GA were directly taken from (Yang et al 1996) and results of EP and hybrid EP were taken from (Baskar et al 2002). Table 7.3 gives the comparison of results obtained for the test case using different optimization algorithms. Table 7.3 Comparison of results from different algorithms-test system-1 Sl. No Optimization Method Generation Cost ($) CPU Time (Sec) 1 Gradient Search (Yang et al 1996) Simulated Annealing (SA) (Yang et al 1996) Genetic Algorithms (GA) (Yang et al 1996) Evolutionary Programming (EP) (Yang et al 1996) Hybrid EP method (HEP) (Yang et al 1996) Proposed bacteria foraging algorithm

11 145 Table 7.4 gives the best optimal power output of thermal generator, hydro generator, storage volume and discharge for different time intervals obtained from proposed algorithm. Table 7.4 Optimal hydrothermal scheduling by proposed algorithmtest system-1 Time Interval P hy (MW) P st (MW) Reservoir Storage Volume (V), m 3 Water Discharge Rate(q),m 3 /hr Generation Cost ($) Test case 2: Three thermal and one hydro unit System A hydro thermal system consisting of 3 thermal and 1 hydro generating units is considered. The fuel cost function in dollars per hour of the equivalent thermal plants is as follows: f (P S1 ) = 0.01 P S P S ; 50MW P S 200MW f (P S2 ) = 0.02P S P S ; 40MW P S 170MW f (P S3 ) = 0.01 P S P S ; 30MW P S 215MW

12 146 denoted as, The hydro power generation relationship to water discharge is q = 4.97 P H ; 0 P H 1000MW q = 0.05 (P H 1000) (P H 1000) ; 1000MW P H 1100MW The load demand data for the test case 2 has been shown in Table 7.5 and the hydro system data has been shown in Table 7.6. The prohibited operating zones are shown in Table 7.7. Table 7.5 Load demand data for test system-2 Interval No. Day Interval Demand (MW) 1. 1 st day 0 hour- 12 hour st day 12 hour-24 hour nd day 0 hour- 12 hour nd day 12 hour-24 hour rd day 0 hour- 12 hour rd day 12 hour-24 hour 900 Table 7.6 Hydro plant data for test system-2 V min V max q min q max V 0 V 6 r (m 3 ) (m 3 ) (m 3 /hr) (m 3 /hr) (m 3 ) (m 3 ) (m 3 /hr)

13 147 Table 7.7 Prohibited operating zones of the test system-2 ZONE I ZONE II ZONE III ZONE IV The Tables 7.8 and 7.9 show the results obtained in the initial run and the final run by the proposed algorithm with prohibited operating zones as additional constraint for the chosen test problem.the steam powers, hydro powers, rate of discharge, volume of the reservoir and total cost are displayed. Table 7.8 Results obtained by the proposed algorithm in initial run for the test system-2 Interval Steam Power1 P s1 (MW) Steam Power2 P s2 (MW) Steam Power3 P s3 (MW) Hydro Power P h (MW) Volume of Rate of Reservoir,v Discharge,q (m 3 ) (m 3 /hr) Total Fuel cost ($) ,00, ,01, , , , , ,

14 148 Table 7.9 Results obtained by the proposed algorithm in final run for the test system-2 Steam Steam Interval Power1 Power2 P s1 P s2 P s3 (MW) (MW) (MW) Steam Hydro Volume of Rate of Power3 Power P h Reservoir,v Discharge,q (MW) (m 3 ) (m 3 /hr) , Total Fuel cost ($) , , , , , ,565 The convergence characteristics of the proposed algorithm for the test case 2 is shown in Figure 7.1 and the reliability characteristics of the proposed algorithm is shown in Figure x10 4 Convergence characteristics of the proposed algorithm 6.8 Total fuel cost ($) Number of runs Figure 7.1 Convergence characteristics of the proposed algorithm

15 149 11x10 4 Reliability characteristics of the proposed algorithm 10 Total fuel cost ($) Number of runs Figure 7.2 Reliability characteristics of the proposed algorithm From the comparison of results, it is very clear that the proposed method is able to give global optimal solution and computation time of the proposed algorithm is very low compared with EP method and hybrid EP method. Thus the potential of finding optimal solution with affordable time to the short term hydrothermal scheduling problem by proposed bacteria foraging algorithm is proved. 7.5 CONCLUSION In this chapter, new integrated bacteria foraging algorithm is proposed to solve the short term hydrothermal scheduling problem with prohibited operating zones. When the problem is highly nonlinear, this algorithm outperforms other algorithms in terms of the quality of the solution and computation expenses. In the proposed algorithm, bacteria foraging algorithm is used to narrow down the solution space and the local searches are applied to locate the global optimal solution. The test system 1 which has one

16 150 hydro and one thermal unit has been solved by the proposed method and the results are compared with other methods as shown in Table 7.3. The proposed method resulted in a total fuel cost of $ which is less than other methods addressed in the literature and all the hydro constraints are satisfied exactly. The test system 2 which has prohibited operating zones has been solved by the proposed algorithm which resulted in a total fuel cost of $75565 and all the hydro and thermal constraints are satisfied. For the example problem 2, the convergence of bacteria foraging algorithm is verified and the reliability of the proposed method is tested for 10 independent runs. From the results it is proved that the proposed algorithm can be applied to short term hydrothermal scheduling problem and can obtain the global optimal solution with lesser computation time.