4.6 Case studies
4.6.1 Case 1: Test case with a 43 MW generator and 10 producing
producing wells
In the first case, we consider a small-scale combined-cycle NGPP running a gen- erator with maximum 43 MW output capacity, e.g. the ALSTOM GTX100 gas turbine. The NGPP is supplied with gas from a shale-gas field with 10 producing wells. We assume that the EUC/GENCO persistently seeks full load of its genera- tor in order to maximize the efficiency, causing a constant demand dNGPP
k as shown
with black dots in Fig. 4.9, and we use a 8 weeks time horizon for the example. Fig. 4.9 compares the solution of the receding horizon scheme with the solution obtained by applying the open-loop solution to the shale-gas field. In the open- loop solution, we assume that the operator for the first four weeks applies the initial solution from the LR scheme computed on June 15, and then re-optimize the scheme four weeks later, i.e. on July 13, in order to generate the optimal well
June 15. July 1. July 15. August 1. 1.4 1.6 1.8 2 2.2 2.4 2.6 x 105 Time [date] G as rat e [m 3/d ] Firm demand dNGPP k
Receding horizon - firm supply to NGPP Open-loop - firm supply to NGPP
Fig. 4.9: Comparison of the receding horizon scheme’s and the open-loop scheme’s ability to track the demand rate dNGPP
k .
schedule for the next four weeks. We assume that the first unplanned well shut-in occurs 5 days after initialization of the optimal well schedule. In Fig. 4.9, it can be seen that the open-loop solution gives satisfactory tracking of dNGPP
k for the first
few days when there are no disturbances, while the tracking performance of the open-loop scheme is substantially deteriorated in the event of an unplanned shut- in. The large oscillations seen in the supply are due to the peak rates occurring after shut-ins of shale-gas wells (Knudsen and Foss, 2013). The deviations between dNGPP and that actual gas supply to the NGPP is in this case unacceptably high.
In contrast, the solution of the receding horizon scheme meets in this case the de- mand dNGPP
k exactly over the entire time horizon. This is achieved by the feedback
introduced through receding horizon optimization, in which the scheme in each iteration recomputes tubinghead pressures pt,jk and well schedules yjkd to compen-
sate for unplanned shut-ins and thereby meet the demand. The receding horizon scheme is clearly superior to the open-loop scheme for handling unplanned events and disturbances, and actually a necessity in order to tightly meet the demand dNGPP
k as time progress.
Solving the Lagrangian subproblems (4.19) for the 10-well case required on average 0.5 seconds, ranging from 0.03 to 4.9 seconds. The convex hull reformulation (4.32)–(4.34) of the GDP formulation (4.10d) for the routing decisions for each well, renders as such a tight MILP formulation, requiring limited computation time for solving each of the Lagrangian subproblems. To demonstrate the performance of the primal recovery fixing heuristic in Section 4.4.1, we show in Fig. 4.10 the progress in improvement of the lower bound ZLBand the % of the binary variables fixed for
the eighth RH iteration in case 1. Note that the % number of binary variables fixed includes the additional binaries yb
jkimposed for the piecewise linear approximations
4.6. Case studies
RH iteration tk−1as starting point for the Lagrangian scheme, the primal recovery
heuristic initially fixes a high number of variables since the provided starting point is almost primal feasible. The value of the Lagrangian tends to increase slightly in the next iterations as cuts need to be generated and added to the bundle B in order to refine the polyhedral approximation ˜ZLR in (4.24) and hence produce
good multiplier updates (λn+1, πn+1). During these iterations the number of fixed
binary variables is seen to drop, as the solution of the Lagrangian is further from being primal feasible. After this initial drop, the number of fixed binary variables is seen to remain at a level between 85-95%, and gradually increase with the number of LR iterations n, during which the lower bound ZLB is seen to be consistently
improved.
The same progress of the primal recovery heuristic as shown in Fig. 4.10 is also observed for the majority of the remaining receding horizon iterations. Out of all the Lagrangian relaxation iterations applied in case 1, the initial fixing procedure in the primal recovery heuristic fixed averagely 86% of the binary variables. The fixing of binaries yb
jk associated with the piecewise linearization is, however, somewhat
troublesome; in 41% of the total number of LR iterations in case 1, the initial binary fixing caused the primal MILP to be integer infeasible. However, after the relaxation of the SOS associated binaries yjkb as described in Section 4.4.1, leaving
averagely 67% of the binaries fixed, then only 0.2% of the LR iterations were not able to return a primal feasible solution. Consequently, with an 1.1% average duality gap at the termination of each RH iteration, the proposed primal recovery heuristic and Lagrangian scheme is able to efficiently produce solutions of high quality. 0 5 10 15 20 25 30 35 40 60 65 70 75 80 85 90 95 100 [% ]
Iteration n in Lagrangian scheme [-]
4.45 4.5 4.55 4.6 x 105 O b je ct iv e val u e [$]
[%] binary variables fixed
[%] bin. var. fixed after relaxing SOS binaries yb jk
ZLBfrom fixing heuristic
Fig. 4.10: The percentage of variables fixed and the improvement of the lower bound for the Lagrangian scheme in RH iteration number 8, case 1. The final duality gap in this RH iteration is 1.0%.