5.4
Example: Shale-well scheduling
The computational results in Section 5.3 show that the proposed OFP heuristic compares favorably in terms of balancing short computation time and solution quality for convex MINLP. These results raise two questions; the first is how the heuristic applies to large-scale real-world problems, for instance as part of indus- trial decision-support tools (DSTs). In this type of application, where optimization problems are solved repeatedly with a certain time limit, low computation times and sufficient solution qualities are often more important than global optima. The second question is to which extent the results of the proposed algorithm rely on convexity of the NLP relaxation (5.2), and as such if the algorithm can be ex- tended favorably to nonconvex MINLPs. A detailed study and assessment of these questions are beyond the scope of this paper. However, to briefly explore how the proposed OFP performs on these difficult types of problems, we apply the algo- rithm on an industrial type shale-well scheduling problem taken from Knudsen et al. (2014a). Transmission/ distribution line Compressor Wellhead choke: y1 jk Separator Well
Low-pressure line plow line: yjk2 = 1
High-pressure line phighline: yjk2 = 0
Fig. 5.5: Illustration of shale-well scheduling problem.
Consider the shale-well and compressor system illustrated in Fig. 5.5. The sys- tem consists of|J | geographically distributed wells, each connected with two flow- lines to a gathering system: one low-pressure line connected upstream of a shared compression unit, and one high-pressure line bypassing the compressor. The com- pression is either performed by a midstream company, requiring a fraction CC of
the gas sales price G, or the compression is performed by the well operator, causing an equivalent compression cost. Routing gas flow to the low-pressure line increases well deliverability by allowing the well to be operated at a lower wellhead pressure, while routing the gas flow past the compressor gives the operator the full sales price. The compressor requires a minimum inflow rate qlow
tot to avoid surge, and has
a maximum load capacity quptot.
Each well j∈ J is described by a dynamic, nonlinear well and reservoir model given as a set of constraints Fj(mjk+1, qjk+1, pt,jk, pwf,jk), where mjk is a pseudo-
pressure, pt,jk is tubinghead pressure, pwf,jkis bottomhole pressure and qjk is gas
of a parabolic partial differential equation, while k∈ K is the discrete time index. Each well has to be operated above the respective line pressure, plow
line and p high line,
depending on the line scheduling modeled by the binary y2
jk; if yjk2 = 1 the gas
flow is routed to the low-pressure line, and if y2
jk= 0, the gas flow is bypassed the
compressor, cf. Fig. 5.5. Moreover, to avoid so-called liquid loading, the gas rate qjk has to be above a lower bound qgc (Turner et al., 1969), or the well must be
shut-in (Knudsen and Foss, 2013). We model this on/off state of the wells with a binary y1jk. See Knudsen et al. (2014a) for further details on the model formula-
tion. Summarized, the shale-well scheduling problem is modeled by the condensed nonconvex MINLP max GX k∈K X j∈J (1− CCy2 jk)qjk∆k, (5.22a) s.t. X j∈J qjky2jk≤ q up tot, ∀k ∈ K, (5.22b) X j∈J qjky2jk≥ qtotlow, ∀k ∈ K, (5.22c) Fj(mjk+1, qjk+1, pt,jk, pwf,jk) = 0, ∀j ∈ J , k ∈ Km, (5.22d) mj0= minitj , ∀j ∈ J , (5.22e) pt,jk≥ plowlineyjk2 + (1− yjk2 )p high line, ∀j ∈ J , k ∈ K, (5.22f) qjk≥ yjk2 yjk1 qlowgc + (1− y2jk)y1jkqgchigh, ∀j ∈ J , k ∈ K, (5.22g) y1jk, yjk2 ∈ {0, 1} , ∀j ∈ J , k ∈ K, qjk, pt,jk, pwf,jk∈ R, mjk∈ RI, ∀j ∈ J , k ∈ K.
5.4.1
Results
To asses the performance of the proposed OFP in Algorithm 3 on the shale-well scheduling problem, we construct an example of (5.22) consisting of|J | = 6 wells, and a two-month planning horizon K with a fixed ∆k = 2-day time step. The resulting nonconvex MINLP consists of 3276 constraints, 1620 continuous variables and 360 binary variables. We include a set of test problems with totally 15 instances of the problem (5.22), in which each problem has a different initial condition minit
j .
The computational environment and parameter values for the OFP were set to those given in Section 5.3, except for the user-defined weight u2 for the original
objective function which is set to u2:= 1. That is, we assume the users desires an
equal emphasis on solution quality and quickly obtaining a feasible solution. We impose a maximum CPU time of 7200 seconds.
There are a few instances from the set of test problems used for the com- putational study in Section 5.3 which have similar dimensions to the shale-well scheduling problem. This includes the problems: RSyn0810M04H, RSyn0815M04H, RSyn0830M03H, RSyn0840M03H and Syn40M04H. The dimensions of these prob- lems can be found in Appendix 5.A.
5.4. Example: Shale-well scheduling
Table 5.3 compares average results from the 15 instances of the shale-well scheduling problem when applying the proposed OFP in Algorithm 3, the FP heuristics of Bonami and Gon¸calves (2010), and the BB method in BONMIN. It can be seen that the OFP turns out to be superior to the FP both with respect to solution time and solution quality when applied to (5.22). The average solution time of the OFP is smaller by a factor of 2.64 compared to the FP, and the average optimality gap of the OFP is approximately 12 times smaller than that of the FP. This is in contrast to the results of the OFP from Section 5.3, where the OFP was observed to improve the solution quality compared to the FP for a majority of the test problems, while requiring on average more time to compute a feasible solution. This discrepancy in performance is likely to be a result of an intrinsic property of the problem formulation (5.22), in which the feasible space makes it possible for the OFP to quickly compute feasible solutions. The original FP does also suffer from excessive stalling on these instances. To escape integer infeasible points causing stalling of the algorithm, the structure of the optimization problem must be changed, either by the feasible region or by the objective function (Boland et al., 2012). The changing weight αiin the OFP objective may as such help reduc-
ing stalling, by serving as an additional escape mechanism to the variable flipping. The FP variant applied for comparison in this paper resembles the FP-3 variant for nonconvex MINLPs described in D’Ambrosio et al. (2012), which is reported to be very fast but often hits the time limit due to a high number of iterations. As this may indicate frequent stalling, a further study on nonconvex MINLPs could assess if the proposed OFP circumvents some of these issues and hence is a viable FP-approach also for nonconvex MINLPs.
Table 5.3: Average objective values and CPU times for the OFP, FP and the first solution found by the branch-and-bound method on 15 test instances of the shale-well scheduling problem (5.22)
OFP FP BB GM obj [106 $]a 1.12 0.70 1.14
GM gap [%] 3.21 39.32 2.11 GM time [s] 6.64 17.5 252.4
a: geometric mean.
The performance of the OFP with regards to objective value is in the first row of Table 5.3 seen to be nearly as good as that of the first solution found by the branch-and-bound method. While the average optimality gap of BB is smaller by a factor of 0.65 compared with the OFP, the average solution time of BB is larger by a factor of 38. The significant contrasts in performance of the various methods applied to (5.22) is clearly visible in the performance profiles for the relative optimality gap and solution time shown in Fig. 5.6 and 5.7, respectively. The large relative distance between the individual performance profiles for the different methods indicates clearly which solver is superior for each of the two selected performance metrics.
0 0.5 1 1.5 0
0.5 1
not more than 10x worse relative optimality gap than the best
fraction of problems solved
OFP FP BB
Fig. 5.6: Performance profile of the optimality gap (5.20) with the respect to the best known solution, comparing the results of OFP, FP and the first solution found by the branch-and-bound method on 15 test instances of the shale well scheduling problem (5.22).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0 0.5 1
not more than 10x slower than the best
fraction of problems solved
OFP FP BB
Fig. 5.7: Performance profile of the CPU time required to find a feasible solution, comparing the results of OFP, FP and the first solution found by the branch-and-bound method on 15 test instances of the shale well scheduling problem (5.22).