Since the allowed power shortage in each week is 50 MW, and in weeks 2-9 the power shortage is 39.2 MW, the first iteration solution of the master problem ...
Implementation of the benders decomposition in hydro generating units maintenance scheduling I. Kuzle and H. Pandzic Faculty of Electrical Engineering and Computing Unska 3 Zagreb, 10000 Croatia
M. Brezovec Hrvatska elektroprivreda HPP Generation area “North” Medimurska 26c Varazdin, 42000 Croatia
Introduction Before the liberalization of the power sector, the generating units maintenance had one single constraint – the maintenance of associated transmission lines. Since the transmission lines and generating utility were both controlled by national electric utility, there were no confronted entities with different goals. The great advantage of centrally planned scheduling for each electric facility was optimizing the system reliability and reducing the maintenance cost of the entire system. After restructuring of electric utilities, the maintenance of generating utility can not be planned nor done independently. Owners of generating facilities will want to deliver electricity when its price is high, and will want to perform maintenance when the price is low1. Maximization of the profit is the prime goal of all generating companies in restructured power systems. If excluded that shareholders and economists will want to keep running their generators only with inevitable maintenance in order to maximize profit, each generating company will want to perform maintenance when the market clearing price is low or when it is unable to deliver electricity. But regularly this will not be approved by the ISO, the entity which ensures the instantaneous balance of the power system. If ISO would approve all desired maintenance schedules of all generating companies, the power grid reliability standards could not be met. Another object ISO has to meet is to keep the transmission lines under the congestion level. All this can be introduced as the constraint to the objective, i.e. maximizing the profit by performing the maintenance according to desired schedule. The system reliability requirement from the ISO is not the only constraint the generating companies have to obey. There is also a problem with the number of specialized maintenance crews. These crews can not perform the specific maintenance in more generating companies simultaneously, so the availability of maintenance crew is also a constraint to generating companies.The availability of spare parts also has to be taken into account when deciding to undertake a maintenance procedure. Another common constraint is the sequence of maintaining of the parts of generating facility, i.e. a specific part of the facility can be maintained only after the maintenance of another part is over. In addition, all signed treaties for delivering electricity must be taken into account or rigorous penalties will have to be paid.
1. Maintenance Management Maintenance control handles the balance between demands for improving generating facility effectiveness and efficiency and economic imperative for reducing expenses and maximizing generator operating hours for increased electric energy delivery. Maintenance management consists of the three basic functions: planning, scheduling and control2. In the planning stage, all maintenance demands are reviewed until all economic and engineering aspects are satisfied. It is important to determine the critical operations that have to be done without any delay. Afterwards, this proposed maintenance plan is sent to ISO for authorization. ISO usually has some objections and notes. Creating a new schedule with many constraints imposed by the ISO is a major challenge, and proper implementation of mathematic programming is recommended. Planning could be either deterministic or stochastic. While deterministic approach assumes that all details are known in advance, the stochastic is more realistic because it assumes that future events are unknown and can only be
predicted with a certain accuracy2. Therefore, the generating unit maintenance has the property of stochastic planning based on heuristics. Due to this stochastic nature of the maintenance system, an unexpected disruption in the generating system is not negligible. This is known as fictitious cost approach and is used to penalize alterations of the ideal power plant maintenance schedule3. Another, much more often applied, approach is the maintenance window approach, in which the ideal schedule is represented by time intervals in which the maintenance should be performed3. This means that costs do not depend on time, as long as the maintenance actions are done within these time intervals. Mathematical model presented in this paper is based on the maintenance window approach. For integral and thorough generation maintenance scheduling problem, all three hierarchical levels must be included: the generating plant, the utility and the interconnected system4. The generating plant is the lowest hierarchical level and the maintenance scheduling from its point of view is attaining the adequate maintenance interval in order to keep the equipment availability above the desired level. Crew availability, technological aspects and ageing are the most obvious constraints in this level. At the utility level, which is the concern of the ISO, additional constraints include area load supply, system reliability and interchange contracts. Finally, the highest hierarchical level, interconnected system, includes, besides the generating unit and the utility constraints, interchange capability constraints, interconnection system reliability, the practice of mutual assistance between states, and others. These hierarchical levels are visually shown in fig. 1.
Fig. 1. The hierarchical levels of the interconnected power system4
2. The Maintenance Parameters 2.1 The Timing Generating units are usually scheduled to be maintained once a year. Naturally, it is not always possible to perform maintenance exactly 12 months after the previous maintenance, but deviation in both directions causes negative effects5. If a unit is maintained before the period of 12 months has expired, the maintenance will be more costly. Although the unit could have worked without maintenance for full 12 months, some of that time was thrown away (fig. 2). Taking into consideration only this aspect of maintenance, one could deduce that savings on maintenance will be greater the more the maintenance is prolonged, which is not the case.
Fig. 2. The amount of wasted money as a function of the time since the previous maintenance5
On the other hand, performing maintenance after expiration of 12 months dramatically increases expected maintenance costs. Progressive deterioration takes place because appropriate maintenance was not done in time and the maintenance cost rapidly increases, as shown in fig. 3.
Fig. 3. The required investment money as a function of the time since the previous maintenance5
Overall maintenance costs comprise both previous aspects of maintenance and the optimum maintenance period is obtained by the summation of curves in fig. 2 and 3. The minimum of this new curve (fig. 4) represents optimal time period for performing maintenance.
Fig. 4. The sum of the amount of wasted money and the required investment money as a function of the time since the previous maintenance5
2.2 The Constraints If all maintaining units were independent, the maintenance criteria would be rather easy to establish. Maintenance would be performed in optimum time periods. However, generating units are interconnected and their maintenance periods have to be coordinated by the ISO. In order to design the generating unit maintenance schedule which can be implemented in practice, it is essential to properly implement numerous constraints into the mathematical model. There are two types of constraints. The coupling constraints are mutually independent over a time period6. These include regular overhaul of generating units, which is essential in order to keep the forced outage rates at low percentages, and the availability of the
maintenance crew. Seasonal limitations and the availability of the resources are also coupling constraints. The decoupling constraints include network restrictions, such as peak load, lines capacity, and generators capacity.
3. Mathematical Formulation of the Benders Decomposition The prior objective is to minimize maintenance costs for the following year. Two types of constraints are significant. The first group is the unit maintenance constraints which include maintenance window, crew constraints, maintenance completion deadline and resource constraints7. The other group is the system constraints imposed by the ISO. These include load and reliability constraints. Since the restructured power system is composed of many independent entities, the Benders decomposition algorithm is suitable for solving the maintenance problem. The idea of this algorithm is to decompose the original problem into a master problem and a sub-problem. The master problem contains only few constraints or even no constraints at all. After finding the solution, it is tested by the sub-problem in order to verify if the proposed optimal solution satisfies the remaining constraints, included in the sub-problem. The least satisfied constraint, called the deepest cut, is than added to the master problem, which is being solved again with more constraints. These iterations are continued and new constraints are being added to the master problem until all the constraints of the sub-problem are satisfied. Generally, the generating unit is either down for maintenance or running, and therefore it is a binary variable with values 0 or 1. Accordingly, the master problem is a mixed integer programming problem. The sub-problem is a linear programming problem. The formulation of the maintenance problem is as follows6: Min∑∑ {Cit (1 − xit ) + cit git } (1) t
i
S.T. maintenance constraints:
system constraints:
xit = 1 for t ≤ ei or t ≥ li + di xit = 0 for si ≤ t ≤ si + di
(i) (ii)
xit = {0,1} for ei ≤ t ≤ li
(iii)
seasonal limitations crew availability
(iv) (v)
Sf + g + r = d g≤ g⋅x
f ≤ f
∑r
it
≤ε
i
where is: Cit
generation maintenance cost for unit i at time t
cit
generation cost of unit i at time t
xit
unit maintenance status
g it
generation of unit i at time t
ei
earliest possible period to begin generating unit i maintenance
li
latest possible period to begin generating unit i maintenance
di
duration of generating unit i maintenance
S f g r
node-branch incidence matrix active power flow vector active power generation vector at time t lack of active energy vector at time t
(vi) (vii) (viii) (ix)
d g
demand vector in every bus at time t maximum generation capacity vector
f
maximum line flow capacity vector acceptable amount of missing active energy
ε
The unknown variables xit in (1) are integer variables, while Cit, cit, and git are continuous variables. The goal is to minimize the objective function (1), which represents maintenance and operational costs of the generating units. Constraints (i)-(iii) represent the maintenance window interval. If a unit i is off-line for maintenance in time interval t, xit is 0, otherwise it is 1. Accordingly, xit is always 1 except in the maintenance window. Seasonal limitations (iv) can be incorporated in constraints (i)-(iii). Limitation (v) on crew availability prevents simultaneous maintenance of too many generating units if there is lack of maintenance personnel. Constraint (vi) is a peak load balance equation, while constraints (vii) and (viii) represent generation and transmission capacity. Finally, constraint (viii) grants that the amount of electrical energy not served is under the set limit. 3.1 The Problem Solving Procedure When using Benders decomposition, first it is necessary to divide an original problem into a master problem and a sub-problem. The master problem is a mixed integer programming problem and it generates preliminary solution to the problem. The master problem contains only few constraints so it is considered as a relaxation of the original problem. This trial solution is the lowest possible cost that the original maintenance scheduling problem can achieve. Therefore, this trial solution is the lower bound of the optimal value of the original minimizing maintenance costs problem. After calculating the optimal xit values, these are used to solve a set of sub-problems. After solving each sub-problem, a set of dual multipliers is generated and they are added to the master problem in the next iteration forming one or more additional constraints, known as the Benders cuts. After finding a feasible solution close enough to the lower bound, i.e. the optimal maintenance schedule satisfying all the constraints, the iteration process stops. The flow diagram is shown in fig. 5.
Fig. 5. The flow diagram of the Benders decomposition6
4. The Application of the Benders Decomposition The following example is a demonstration of the Benders decomposition on the part of the Croatian power system, shown in fig. 6.
Fig. 6. The analyzed part of the Croatian power grid. Varazdin, Cakovec and Dubrava are hydro plants, and numbers beside each arrow show the peak active power demand for each bus
The regarded part of the system contains three hydro power plants in a cascade on the river Drava (Varazdin, Cakovec, and Dubrava), each containing two generators:. All generating units data are given in the table I. The transmission capacity of all 110 kV lines in the system is 115 MW. Three busses (Nedeljanec, Koprivnica and Krizevci) are connected to the rest of the power system. Overall demand in this subsystem is 166 MW, and hydro power plant capacity 253.6 MW. The whole calculation is based on the weekly time periods. The allowed active power shortage in each week during the maintenance is 50 MW, which will be replaced by nearby thermal power plants. TABLE I GENERATOR UNITS DATA
Max Maint. Maint. Unit Capacity Cost duration [pu] [weeks] [MW] A 47 100 5 Varazdin B 47 100 4 A 39.9 100 4 Cakovec B 39.9 100 4 A 39.9 100 5 Dubrava B 39.9 100 5 Hydro Plant
The optimal maintenance periods for generators are given by the penalty factors for each week. These penalty factors are based on the information on the previous water flow of the river Drava. The smallest weekly avarage water flow (140 m3/s in week 6) is given the penalty factor 1.00, and we presumed that the penalty factors for all the other weeks have a linear increase. The invariable cost of maintenance for each generator is 100 pu and variable cost depends on the penalty factors. These factors are based on water flow because if it is too big, part of the flowing water is wasted. In other words, variable expenses represent an opportunity cost. Ideally, variable expenses would be 0 and each unit maintenance cost would be 100 pu, which makes total of 600 pu. Since all three hydro power plants are on the same river and have small reservoirs, all three have the same avarage weekly water flow and the penalty factors (table II). It is important to emphasize that each generator maintenance weeks have to be consecutive, e.g. the generator has to be maintained for 4 or 5 weeks in a row.
TABLE II PENALIZING FACTORS FOR EACH GENERATOR BY WEEKS
Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Avarage water flow [m3/s] 176 164 157 151 147 140 160 154 169 171 179 208 212 239 244 253 287 312
Penalty factor 1.26 1.17 1.12 1.08 1.05 1.00 1.14 1.10 1.21 1.21 1.28 1.49 1.51 1.71 1.74 1.80 2.05 2.23
4.1 The Calculus The initial solution of the maintenance master problem is given in the table III. TABLE III THE INITIAL GENERATING UNITS MAINTENANCE SCHEDULE
Hydro Plant
Unit
Varazdin Cakovec Dubrava
A B A B A B
Maintenance Weeks 6,7,8,9,10 2,3,4,5 6,7,8,9 2,3,4,5 6,7,8,9,10 2,3,4,5
The maintenance schedule in table III is the cheapest possible and if maintenance would be executed according to it, it would cost 672.65 pu, where 600 pu are invariable expenses, and 72.65 are variable expenses. However, this schedule has to meet all constraints in the optimization sub-problem and limit shortage of active power in each week to 50 MW. The results of the sub-problem are given in the table IV. TABLE IV OVERALL ACTIVE POWER SHORTAGE IN FIRST ITERATION
Week 2 3 4 5 6 7 8 9
Active power shortage [MW] 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2
Since the allowed power shortage in each week is 50 MW, and in weeks 2-9 the power shortage is 39.2 MW, the first iteration solution of the master problem satisfies all the given constraints in the subproblem. Therefore, already in the first iteration the optimal maintenance plan is achieved.
5. Conclusion Successful management of the maintenance scheduling is crucial in a restructured power system and the decomposition approach will have a significant role in planning maintenance scheduling in the future. Using Benders cut, an optimal solution was successfully obtained already in the first iteration. This simple example is used to demonstrate the simplicity of the Benders decomposition. The more complex and constrained examples would require more iterations and the use of the Benders cut, which is not discussed in this paper. Our future work will include more complex subsystems with more restrictions and the transmission facilites maintenance.
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The Authors Igor Kuzle was born in Tuzla, Bosnia and Herzegovina, in 1967. He went to primary and secondary school in Pozega, Croatia. He received his B.Sc.EE, M.Sc.EE and PhD (in the field underfrequency load shedding) degree from the Faculty of Electrical Engineering in Zagreb, in 1991, 1997 and 2002 respectively. He was awarded with faculty annual award "Josip Loncar". Since graduation, he has been working at the Faculty of Electrical Engineering, Department of Power Systems. He is presently an Assistant Professor and IEEE Croatia Section Vice-Chair. He published more than 150 scientific papers and practical project studies. He has been a licensed engineer of electrical engineering since 1999. His scientific interests are problems in measuring, automation and electric power systems control, mathematical modeling and simulation of dynamics phenomena in electric power systems (especially frequency variations). Hrvoje Pandzic was born in Zagreb, Croatia, in 1984. He received his B.Sc.EE in 2007. on the Faculty of Electrical Engineering in Zagreb. He is currently a postgraduate student, working at the Faculty of Electrical Engineering, Department of Power Systems. He worked on many faculty projects, including the development of an e-learning system which is being tested by the lecturers of the Faculty of Electrical Engineering and the Zagreb School of Economics and Management. He is the IEEE Student Branch Zagreb Vice-Chair. He was awarded annual reward from the “Hrvoje Pozar” foundation of Croatian Energy Society. Miljenko Brezovec was born in Varazdin, Croatia, in 1974. He received his B.Sc.EE and M.Sc.EE at the Faculty of Electrical Engineering and Computing in Zagreb, in 1997 and 2002 respectively. He got faculty annual award "Josip Loncar" and his B.Sc. thesis was awarded from Foundation "Hrvoje Pozar" of the Croatian Energy Society. Since graduation, he has been working in Croatian electrical utility "Hrvatska elektroprivreda d.d." (HEP), in HPP Generation Area "North", pursuing operation and maintenance, especially remote control and optimization. His main fields of interest include also modeling and simulation of hydroelectric power units.
