164 Chapter 8. Maximising expected energy production 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 0 1 0 2 0 3 0 4 0 5 0 200 400 600 Cap Cap 200 400 600 Weeks Units 0 10 20 30 40 50 Weeks Cap Cap 200 400 600 Scenario G Scenario H Capacity 600 400 200 600 400 200 (MW)
(a) Two maintenance schedules for the 21-unit test system corresponding to different GMS criteria
0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 Weeks Manp o w er
Manpower available Manpower required: G Manpower required: H
(b) The manpower required over the duration of the scheduling window for the two maintenance schedules in Figure 8.5(a) 4 750 5 000 5 250 5 500 0 10 20 30 40 50 Weeks Capacit y (MW)
Max capacity Available capacity: G Available capacity: H Demand
(c) The available system capacity over the duration of the scheduling window for the two maintenance schedules in Figure 8.5(a)
Figure 8.5: Comparison between the maintenance schedules of Scenarios G and H for the 21-unit test system.
8.1. Piecewise linear approximation results 165
As previously mentioned, in order to minimise the probability of unit failure, maintenance schedules typically exhibit maintenance commencement dates that are early during the schedul-ing window. This is the case in Scenario G where maintenance on PGUs with large capacities is performed early in the scheduling window. This objective function also seeks to schedule PGUs with large failure rates as early as possible so as to decrease the probability of PGU failure. The objective in Scenario H, however, aims to schedule PGUs with large capacities (PGUs that can produce more energy) close to the peaks of the expected energy curves of these PGUs, which are typically close to the end of the scheduling window. It may be seen in Figure 8.5(a) that PGUs with large capacities are typically scheduled later, but still within their respective PGU maintenance windows. It may also be seen in Figure 8.5(a) that, compared to Scenario G, some PGUs with higher failure rates are scheduled for planned maintenance later during the scheduling window in Scenario H. PGU 19 is, for example, scheduled for maintenance during the last week of the scheduling window due to having the highest failure rate of 0.3733 — this is 22 weeks later than in Scenario G. A similar observation may be made for PGU 12, which is scheduled for maintenance 14 weeks later in Scenario H than in Scenario G due to its large failure rate of 0.1527.
The different effects of the two scheduling objectives may also be observed in Figure 8.5(b). In Scenario G, the maximum amount of manpower is often required during the early stages of the scheduling window. It is also observed that towards the end of both halves of the scheduling window (e.g. weeks 1–27 and weeks 28–52) no manpower is required as no maintenance is scheduled during these times. In Scenario H, on the other hand, it is observed that towards the end of both halves of the scheduling window (e.g. weeks 1–27 and weeks 28–52) the manpower required is at the maximum available number as most of the PGUs exhibit peaks of their expected energy production curves towards the end of the scheduling window.
Similar observations may also be made in respect of Figure 8.5(c). In both Scenarios G and H, the available system capacity drops down to 0.696% above the demand. For Scenario G, this drop is observed during the early stages of the scheduling window whereas for Scenario H the drop is observed during the middle stages of the scheduling window, towards the end of the first half of the scheduling window, due to the maintenance window constraints. The maximum capacity is mainly available just before the end of the first and second halves of the scheduling window in Scenario G. In Scenario H, on the other hand, the maximum available capacity is available during the beginning of the first and second halves of the scheduling window.
Sensitivity analysis
A piecewise linear approximation approach towards solving the nonlinear GMS model of§4.3 for large power systems or very unconstrained systems (in terms of maintenance window constraints) is not expected to be feasible. The feasibility of an exact solution approach by CPLEX is also influenced by the nature of the objective function (e.g. linear or nonlinear). It was demonstrated above that employing such a piecewise linear approximation model solution approach in the context of the 21-unit test system is feasible.
In order to analyse the effects of alterations in the system specifications on the feasibility of the piecewise linear approximation model solution approach, six cases are again analysed in this section in terms of the computation times required by CPLEX to solve the nonlinear model of §4.3. These cases are the same as those considered in §7.1 and involve combinations of increasing the peak demand of the system by a certain margin and relaxing the maintenance window constraints to have an earliest starting time of 1 and latest starting time of 53 less the duration of maintenance of each PGU. The first case is the original 21-unit test system which
166 Chapter 8. Maximising expected energy production
Table 8.3: Various statistics pertaining to optimal solutions to a piecewise linearisation of the nonlinear model of§4.3 in a sensitivity analysis in respect of demand and PGU maintenance windows for the 21-unit test system. The last column contains optimality gap values with respect to provable upper bounds on the maximum objective function value. An asterisk denotes that a time-out budget of 4 hours of computation time was reached by CPLEX.
Demand Maintenance Objective function Time Gap
Cases (%) window value (MW·week) (s) (%)
1 100 Original 219 717 36 0 2 103 Original 217 862 13 0 3 106.5 Original 215 048 27 0 4 100 Relaxed 260 095∗ 14 400 0.83 5 103 Relaxed 255 114 5 963 0 6 106.5 Relaxed 243 566∗ 14 400 6.14
is considered as a reference case for the other five cases. The second case involves a 3% increase in the peak demand, but adheres to the original test system’s maintenance window constraints. The third case involves a 6.5% increase in the peak demand, but also adheres to the original maintenance window constraints. In the fourth case, the peak demand is kept as specified for the original 21-unit test system, but the maintenance window constraints are relaxed as described above. The fifth case involves a 3% increase in the peak demand and relaxed maintenance window constraints. Finally, the sixth case involves an increase in the peak demand of 6.5% and relaxed maintenance window constraints. Various statistics pertaining to optimal solutions to a piecewise linearisation of the nonlinear model in§4.3 are shown for these six cases in Table 8.3. Case 1 requires 36 seconds of computing time by CPLEX to obtain an optimal solution to the piecewise linear approximation model. In Case 2, the demand is increased by 3%, which requires 13 seconds of computing time and yields a 0.844% worsening of the objective function value. It is observed that there is a decrease in computing time and that an optimal solution is still obtainable within a reasonable timeframe. Furthermore, when increasing the demand by 6.5%, in Case 3, a computation time of 27 seconds is required to obtain the optimal solution, which results in a 2.125% worsening of the objective function value compared to that in Case 1. It is observed that increasing the demand has a small impact on the computation time required to solve the nonlinear model of§4.3 (approximately) for the 21-unit test system. This computation time decreases by only 9 seconds over the course of a 6.5% increase in demand. The reason for the decrease in computation time as the demand is increased may be attributed to the smaller solution space through which the algorithm has to search in order to obtain an optimal solution. When the maintenance window constraints are relaxed, however, the number of feasible solutions to the nonlinear model of §4.3 increases drastically, and so does the required computation time. Case 4, in which the demand is kept as specified for the original 21-unit test system and the maintenance window constraints are relaxed, results in a large increase in computation time. In fact, no optimal solution can be obtained within the time-out budget of 14 400 seconds of processing time. An optimality gap of 0.83% is obtained between the best objective function value (of 260 095) and the smallest provable upper bound on the objective function value within the allowed processing time. A decrease in computation time is observed in Case 5, where the demand is increased by 3% and the maintenance window constraints are relaxed. The first optimal solution is obtained within 5 963 seconds of computation time and yields an objective function value that is 16.11% better than that of Case 1. Finally, in Case 6, it is found that no optimal solution can be obtained within the time-out budget of 14 400 seconds of processing time. A gap of 6.14% is obtained between the best objective function value (of 243 566) and
8.1. Piecewise linear approximation results 167
the smallest provable upper bound on the objective function within four hours of computation time. It is therefore observed that relaxing the maintenance window constraints has a significant impact on the computation time required to solve the nonlinear model of §4.3 (approximately) for the 21-unit test system.
8.1.2 The IEEE-RTS system
This section contains a presentation of the piecewise linear approximation solution obtained by CPLEX for the nonlinear model of §4.3 as applied to the IEEE-RTS [188]. The solution is contrasted with a solution obtained in the literature upon adoption of another reliability-related scheduling criterion (minimisation of the SSR). The solution is also compared with the solution reported in §7.1.2 when adopting the minimisation of the probability of unit failure objective function. The feasibility of a piecewise linear approximation solution approach for the nonlinear model of§4.3 within the context of the IEEE-RTS is finally also assessed in the form of a sensitivity analysis involving various relaxations of the demand and maintenance scheduling window constraints under an allowable computation time budget of 24 400s (8 hours).
Numerical results
A piecewise linear approximation solution to the nonlinear GMS model of§4.3 is obtained for the IEEE-RTS by CPLEX within 93 538 seconds (25.983 hours). The optimal decision variable values of this solution are given in integer decision vector form by xxx = [25, 12, 21, 30, 15, 51, 18, 33, 32, 48, 35, 14, 22, 42, 30, 38, 30, 31, 43, 23, 44, 17, 38, 36, 36, 49, 51, 51, 43, 13, 46, 38], which corresponds to an objective function value of 130 630 MW·week (21 945 840 MWh). A graphical representation of the maintenance schedule is presented in Figure 8.6 with the colour scale in Figure 8.6(a) indicating the rated capacity (in MW) of each PGU and the colour scale in Figure 8.6(b) indi-cating the failure rate of each PGU. The manpower required over the duration of the scheduling window in order to implement the optimal solution in Figure 8.6 is shown in Figure 8.7(a). The available system capacity over the duration of the scheduling window associated with the optimal solution in Figure 8.6 is shown in Figure 8.7(b).
Figures 8.6 and 8.7 may be used to analyse the optimal solution obtained to the nonlinear GMS model of §4.3 obtained by the piecewise linear approximation solution approach for the IEEE-RTS. The influence that the rated capacity of a PGU has on its scheduled starting time is clear in Figure 8.6(b). PGUs with large rated capacities (such as PGUs 23 and 32) are scheduled to start as close as possible to the peaks of their expected energy curves. This is also the case for PGU 22, but as this PGU has a maintenance window that ends at planning period 27, it therefore must be scheduled for maintenance starting no later than planning period 27. From these examples it is concluded that good solutions to the nonlinear GMS model of §4.3 will typically aim to schedule maintenance on PGUs with large rated capacities as close as possible to the dates at which they are expected to produce the most energy (e.g. the peaks of their expected energy production curves). In terms of the failure rate of each PGU in the IEEE-RTS, it is observed that PGUs with large failure rates are either scheduled for maintenance early or late, as may be seen in Figure 8.6(b). The reason for this might be that PGUs with large failure rates have nearly flat expected energy curves. Therefore, the effect on the expected energy produced of moving a PGU with a large failure rate from one maintenance commencement date to another is small.
The manpower requirement for the maintenance schedule presented in Figure 8.6 is shown in Figure 8.7(a). It is observed that during the first 12 weeks of the maintenance schedule no
168 Chapter 8. Maximising expected energy production Units Weeks 1 2 3 4 5 6 78 9 10 11 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 30 32 31 0 10 20 30 40 50 Capacity 400 300 200 100 (MW)
(a) An optimal maintenance schedule obtained by piecewise linear approximation of the nonlinear GMS model objective, with the colour scale indicating the capacities of the PGUs in the IEEE-RTS
Units Weeks 1 2 3 4 5 6 78 9 10 11 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 30 32 31 0 10 20 30 40 50 Failure rate 0.3 0.2 0.1
(b) An optimal maintenance schedule obtained by piecewise linear approximation of the nonlinear GMS model objective, with the colour scale indicating the failure rates of the PGUs in the IEEE-RTS
Figure 8.6: An optimal solution to a piecewise linearisation of the nonlinear model of §4.3 for the IEEE-RTS.
8.1. Piecewise linear approximation results 169 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 25 26 0 10 20 30 40 50 Weeks Manp o w er Manpower available Manpower required
(a) The manpower required over the scheduling window corresponding to the optimal maintenance schedule in Figure 8.6 2 250 2 500 2 750 3 000 3 250 0 10 20 30 40 50 Weeks Capacit y (MW) Max capacity Demand Available capacity
(b) The system capacity over the scheduling window corresponding to the optimal maintenance schedule in Figure 8.6
Figure 8.7: Evaluation of the manpower required and the system capacity available over the duration of the scheduling window for the IEEE-RTS.
manpower is required as no maintenance is scheduled during this time period. Thereafter, the manpower required increases sharply and is at a maximum for a number of time periods towards the end of the first half of the scheduling window. It is then observed that for 3 weeks after planning period 27, no manpower is again required, whereafter the manpower once again increases and is at a maximum during a number of planning periods for the remainder of the scheduling window. The reason for the maximum manpower required during the latter stages of the scheduling window is that most PGUs exhibit peaks in their expected energy curves towards the end of the scheduling window. Maintenance is therefore scheduled as close as possible to these peaks.
170 Chapter 8. Maximising expected energy production
In Figure 8.7(b), the available capacity is observed to be at a maximum at the beginning of the scheduling window, but diminishes towards planning period 27, which is typically when the second set of maintenance window constraints starts. After planning period 27, the available capacity is again at a maximum, whereafter it diminishes towards the end of the scheduling window. Towards the end of the scheduling window, it is also observed that the difference between the energy demand and available system capacity is very small.
Comparison with results from the literature
The effects of adopting a scheduling criterion which seeks to maximise the expected energy production, as in the newly proposed objective function of §4.3.2, may be better analysed by comparing the numerical results reported above with results found in the literature when adopt-ing other reliability-related scheduladopt-ing criteria, such as minimisation of the SSR. The numerical results reported above are therefore compared in this section with the results obtained by Schl¨unz and van Vuuren [188], who adopted the SSR scheduling criterion in (2.8). The results obtained by Schl¨unz and Van Vuuren [188] (referred to here as Scenario I) are compared with the results reported above (referred to here as Scenario J). Both GMS objectives adopted in Scenarios I and J reside within the class of the reliability scheduling criteria and the results of Schl¨unz and Van Vuuren [188] for the IEEE-RTS (with the minimisation of SSR as objective) represent the best results available in the literature for this particular scheduling criterion and problem instance combination. A graphical representation of the two maintenance schedules are shown in Figure 8.9 and their corresponding effects on the manpower required and the available system capacity for the IEEE-RTS are shown in Figures 8.9(a) and 8.9(b), respectively. In Figure 8.8, the maintenance schedules of the two scenarios are compared, with the colour scale indicating the rated capacity of the PGUs in the system.
A comparison between the results for the two scenarios in terms of both scheduling objectives
0 10 20 30 40 50 Un its 100 200 300 400 Cap Cap 100 200 0 10 20 30 40 Weeks Units 400 300 200 100 400 300 200 100 Capacity Scenario I Scenario J 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 30 32 31 (MW)
Figure 8.8: Comparison between the maintenance schedules of Scenarios I and J for the IEEE-RTS. Stellenbosch University https://scholar.sun.ac.za
