Computational optimization of gas compressor stations: MINLP models versus continuous reformulations
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Abstract
When considering costoptimal operation of gas transport networks, compressor stations play the most important role. Proper modeling of these stations leads to nonconvex mixedinteger nonlinear optimization problems. In this article, we give an isothermal and stationary description of compressor stations, state MINLP and GDP models for operating a single station, and discuss several continuous reformulations of the problem. The applicability and relevance of different model formulations, especially of those without discrete variables, is demonstrated by a computational study on both academic examples and realworld instances. In addition, we provide preliminary computational results for an entire network.
Keywords
Discretecontinuous nonlinear optimization Gas networks Gas compressor stations Mixedinteger optimization Continuous reformulationsMathematics Subject Classification
9008 90C11 90C30 90C33 90C901 Introduction
Natural gas is one of the most important energy sources. In 2013, it accounted for 25 % of the fossil energy used in Europe (Eurostat 2013). It is used in industrial processes, for heating, and, more recently, for natural gas vehicles. Especially in Germany, its low price leads to its role as a “bridging energy” during the transition to a future energy mix based primarily on regenerative energy. In Europe, natural gas is transported through pipeline networks with a total length of 100,000 km. Gas transport in pipeline networks is pressuredriven, i.e., the gas flows from higher to lower pressure. Thus, pipelinebased gas transport requires compressor stations. The power required to compress the gas is delivered by drives that use either electrical power or gas from the network itself. The energy consumption of compressors is responsible for a large fraction of the variable operating costs of a gas network.

Can the station be operated in a way that satisfies the given boundary conditions? In other words: are those boundary conditions feasible?

If the boundary conditions are feasible: What is a minimum cost operation that satisfies the boundary condition?
One main branch of research investigates the problems of minimum cost operation and feasibility testing for the stationary as well as the transient case. These problems require models of complete gas networks comprising various types of elements like pipes, compressors, (control) valves, etc. The combination of nonlinear gas physics with the switching of controllable network elements typically leads to mixedinteger nonlinear optimization models (MINLPs); see, e.g., CobosZaleta et al. (2002) and Domschke et al. (2011). One standard approach for tackling these MINLPs is the application of (piecewise) linearizations of the nonlinearities in order to reduce the problems to mixedinteger linear models (MILPs) (Geißler 2011; Geißler et al. 2013; Martin et al. 2006, 2007; Martin and Möller 2005). Other investigations focus on the nonlinear aspects under fixed discrete controls (Ehrhardt and Steinbach 2004, 2005; Schmidt et al. 2015a, c; Steinbach 2007), or attempt to approximate discrete aspects by continuous reformulations (Schmidt 2013; Schmidt et al. 2013, 2015b). For more references and reviews in the areas of cost minimization and feasibility testing we refer to Koch et al. (2015) and, especially, the chapter Schewe et al. (2015) therein.
The second major research branch that we want to highlight considers selected types of network elements and studies them in more detail. For pipes, theoretical studies include the controllability and stabilization of the governing system of partial differential equations, the Euler equations (cf. Banda and Herty 2008; Banda et al. 2006; Brouwer et al. 2011; Gugat et al. 2015). The bestinvestigated problem for compressor stations is the one considered in this paper, which is of nonconvex MINLP type: minimum cost operation under given boundary conditions (Carter 1996; Carter et al. 1994; Jeníček and Králik 1993; Osiadacz 1980; Wright et al. 1998). See also Králik (1993) for a simulation based model of compressor stations and Odom and Muster (2009) for a recent survey on modeling centrifugal gas compressors. As already mentioned, the compressor stations are the dominant variable cost factor in gas network operation.
In this article we study discretecontinuous models for the problem of minimum cost compressor station operation. We give an almost complete isothermal description of all relevant devices and their interplay. This description is comparable in accuracy with isothermal simulation models. However, we neglect some minor model aspects that would only complicate the presentation without influencing the solutions significantly. We explicitly mention these simplifications later in the description of the considered problem. The specific structure of the resulting MINLP allows for a large variety of continuous reformulations that will be discussed in detail and that are applied to the problem under consideration. The approach of continuous reformulation is in line with recent publications that develop techniques based on mathematical programs with equilibrium constraints (MPECs) for reformulating other discrete aspects of network optimization models (Pfetsch et al. 2015; Schmidt 2013; Schmidt et al. 2013, 2015b). Thus, by combining the MPEC techniques from the cited publications with the reformulation schemes discussed in this paper, it is possible to state purely continuous NLP type models of the genuinely discretecontinuous problems of minimum cost operation or feasibility testing. The outcome of this achievement is twofold. First, it allows us to state highly detailed models of gas networks. This in particular is not viable with approaches based on linearizations since the resulting MILP models tend to be very hard for stateoftheart MILP solvers. Second, it allows us to solve the resulting models with local NLP solvers, which are typically faster than global MI(N)LP solvers. Obviously, this comes at the price of finding only locally optimal solutions. For further applications of continuous reformulations of discretecontinuous optimization problems, especially in the field of process engineering, see Baumrucker et al. (2008), Kraemer et al. (2007), Kraemer and Marquardt (2010), and Stein et al. (2004). We remark that, despite the fact that continuous reformulations are being used in practice, an extensive numerical study like ours is yet missing in the literature.
The paper is organized as follows. The problem of optimizing a gas compressor station under steadystate boundary conditions is presented in Sect. 2. Afterwards, Sect. 3 introduces a mixedinteger nonlinear formulation of the problem and briefly discusses an equivalent general disjunctive programming formulation. Then, in Sect. 4, the concept of pseudo NCP (nonlinear complementarity problem) functions is introduced and a collection of continuous reformulation techniques is discussed that can be applied to the MINLP model of Sect. 3. These reformulations are applied to artificial and realworld compressor stations in Sect. 5 and the solutions are compared. In addition, this section also provides preliminary results for an entire largescale network that is fully reformulated using continuous variables and smooth constraints. Finally, Sect. 6 gives some remarks on further work and open questions.
2 Problem description
A compressor station hosts a fixed number of compressor machines called units. It can operate in finitely many discrete states that arise from the three operation modes (closed, bypass mode, and active) and a certain number of configurations in the active mode. Every configuration consists of a serial combination of parallel arrangements of compressor units, see Fig. 2. Every compressor unit has an associated drive that provides the power for compressing the gas. For ease of exposition we assume that every drive powers just a single compressor unit. We also neglect the frictional pressure loss caused by station piping, measurement devices, etc., which is usually modeled by fictitious elements called resistors.
In this section we present the required models of all types of compressor machines and drives and then describe their interplay. Full details can be found in Schmidt et al. (2015c) where models of all network elements have been developed. For later reference, the models are presented in constraint form, with constraint functions being denoted by \(c\) and superindexed with an abbreviated name indicating the semantics of the constraint.
2.1 Physical quantities
Principal physical quantities and constants
Symbol  Explanation  Unit 

p  Gas pressure  Pa 
T  Gas temperature  K 
\(\rho \)  Gas density  \(\hbox {kg} \hbox { m}^{3}\) 
z  Compressibility factor  1 
q  Mass flow rate  \(\hbox {kg s}^{1}\) 
Q  Volumetric flow rate \((Q = q/\rho )\)  \(\hbox {m}^3 \hbox { s}^{1}\) 
m  Average molar mass of gas mixture  \(\hbox {kg mol}^{1}\) 
\(p_\mathrm{c}\)  Pseudocritical pressure of gas mixture  Pa 
\(T_\mathrm{c}\)  Pseudocritical temperature of gas mixture  K 
R  Universal gas constant  \(\hbox {J mol}^{1} \hbox { K}^{1}\) 
\(R_\mathrm{s}\)  Specific gas constant \((R_\mathrm{s} = R/m)\)  \(\hbox {J kg}^{1} \hbox { K}^{1}\) 
2.2 Boundary conditions
Compressor quantities
Symbol  Explanation  Unit 

\(H_{\text {ad}}\)  Specific change in adiabatic enthalpy  \(\hbox {kg J}^{1}\) 
\(\eta _{\text {ad}}\)  Adiabatic efficiency  1 
\(P\)  Compressor input power  \(\hbox {W}\) 
\(q^{\text{ fc }}\)  Fuel consumption  \(\hbox {kg s}^{1}\) 
\(b\)  Specific energy consumption  \(\hbox {W}\) 
\(H_{\text {u}}\)  Lower calorific value  \(\hbox {J mol}^{1}\) 
\(n\)  Compressor speed  \(\hbox {s}^{1}\) 
\(\kappa \)  Isentropic exponent  1 
\(M\)  Shaft torque  \(\hbox {N m}\) 
\(V_{\text{ op }}\)  Operating volume  \(\hbox {m}^{3}\) 
2.3 Compressor machines
We distinguish turbo compressors and piston compressors and start with the common parts of their models. For an overview of relevant compressor quantities see Table 2.
2.4 Turbo compressors
2.5 Piston compressors
2.6 Drives
2.7 Configurations
2.8 Objective function
There are various reasonable objective functions in our context, which can mainly be categorized as feasibility or optimization goals. If one merely wishes to know whether given boundary conditions are feasible, it suffices to use a zero objective, \(f\equiv 0\). If the boundary conditions are infeasible, it may be useful to add slack variables to a certain set of constraints and minimize the total infeasibility, measured by a suitable norm of the vector of slack variables. The reader interested in problemspecific slack variable formulations for gas network planning is referred to Schmidt et al. (2015a).
If one is convinced of having feasible boundary conditions, it is straightforward to minimize operating costs, power, or fuel consumption. Specific objective functions will be formulated after stating the optimization models in the following section.
3 Mixedinteger and general disjunctive programming models
One generic way to model the problem described in the previous section is presented in Sect. 3.1: a mixedinteger nonlinear program (MINLP) that incorporates binary variables for the discrete states of the compressor station. An equivalent general disjunctive programming (GDP) formulation of the problem is given in Sect. 3.2. Continuous reformulations of the MINLP and GDP models are discussed later in Sect. 4.
One further possibility of tackling the problem is to enumerate all states of the compressor station and solve all the resulting (continuous) problems by a local or global method. However, this is only suitable when single compressor stations are considered, whereas continuous reformulations can also be used for models of the entire transport network; cf. Sect. 5.5. Another way is to apply suitable heuristics, see, e.g., Schmidt et al. (2015b).
We denote individual continuous variables by \(x\) and discrete ones by \(s\). Variable vectors are written with bold letters, like \(\mathbf {x}\) or \(\mathbf {s}\). Subindices refer to corresponding elements of the compressor station or to sets of elements. In the constraint notation of Sect. 2, we now also use subindices with the same meaning as for variables. Additional subindices \(\mathscr {E}\) or \(\mathscr {I}\) distinguish equality and inequality constraints.
3.1 MINLP formulation
For a suitable formulation of the discrete decisions and their implications on other parts of the model, we review the concept of indicator constraints.
3.1.1 Indicator constraints
Finally we need two more notions: an indicator expression y is a term whose value is \(y \ge 1\) if the associated state is enabled and zero otherwise, like \(s\) in (7). Conversely, a negation expression is a term whose value is \(y \ge 1\) if the associated state is disabled and zero otherwise, like \(1  s\) in (5), (6).
3.1.2 Discrete states

the station is active in configuration \(i\in \mathscr {C}\) if and only if \(s_i= 1\);

the station is in bypass mode if and only if \(s_\text {bp}= 1\);

the station is closed if and only if \(s_\text {cl}= 1\).
3.1.3 Configurations
3.1.4 Compressor units and drives
3.1.5 Model summary
3.2 General disjunctive programming formulation
4 Continuous reformulations
As discussed in Sect. 1, it is reasonable to study continuous reformulations of the MINLP and GDP models of Sects. 3.1 and 3.2 in order to tackle optimization problems for compressor stations or entire gas networks with continuous (local) optimization methods, which tend to be faster than global methods for mixedinteger nonlinear and nonconvex problems.
This section discusses five model reformulation schemes from the literature that can be applied to any model with binary variables that exhibit the structure of a logical disjunction, such as the problem under consideration.
First we introduce the concepts of NCP and pseudo NCP functions in Sect. 4.1. Then we present all reformulation schemes and discuss their feasible sets and regularity properties with a view towards solving them by local algorithms.
4.1 Pseudo NCP functions
4.2 Reformulation schemes
It is wellknown that local solvers tend to be very sensitive to the specific formulation of a nonlinear model. This is the reason why it is often useful in practice to have different equivalent formulations to evaluate which formulation is best suited for the used solver. In this section we discuss five schemes that allow to reformulate the MINLP and GDP models of Sects. 3.1 and 3.2 with continuous variables and additional smooth constraints. Since the geometry of the feasible regions of the resulting continuous models and their regularity properties also have a strong influence on the solution process, we analyze these aspects for every reformulation.
 1.
Every feasible solution of the reformulation has to be uniquely translatable into a feasible solution of the original MINLP (or GDP). This means that there exists a mapping (a lefttotal rightunique relation) from the feasible set of the reformulation to the feasible set of the original model.
 2.
For every binary variable \(s\in \{0,1\}\), the reformulation has variables from which indicator and negation expressions can be constructed (cf. Sect. 3.1.1).
 3.
The reformulation has variables from which the SOS1 constraint can be constructed.
As already stated in Sect. 1, all reformulation schemes below can be found in the existing literature—in particular, see Baumrucker et al. (2008), Kraemer and Marquardt (2010), and Stein et al. (2004)—except that we use pseudo NCP functions rather than NCP functions.
4.2.1 Exact bivariate reformulation
4.2.2 Approximate bivariate reformulation
4.2.3 Exact univariate reformulation
4.2.4 Alternative exact univariate reformulation
4.2.5 Approximate univariate reformulation
Summary of all reformulation schemes. Index 1 denotes the first version and index 2 denotes the second version of the reformulation
Section  Variables  \(\mathscr {E}_{\text {1}}\)  \(\mathscr {E}_{\text {2}}\)  \(\mathscr {I}_{\text {1}}\)  \(\mathscr {I}_{\text {2}}\)  Exact/approx. 

2m  \(2m + 1\)  2m  0  1  Exact  
2m  \(m + 1\)  –  m  –  Approx.  
m  \(m + 1\)  m  0  1  Exact  
m  \(m + 1\)  m  0  1  Exact  
m  m  \(m1\)  1  2  Approx. 
5 Computational study
In the preceding sections we have discussed the problem of compressor station optimization and we have presented several model formulations. With these formulations at hand, the question arises whether all models are comparably well suited for numerical computations, or whether there are benefits or disadvantages for any of them. In this section we present an extensive computational study and compare the results of local and global solvers applied to different model formulations. We will see that the continuous reformulations work quite well in comparison to MINLP approaches.
As mentioned in the introduction, one main contribution of this paper is that—in combination with techniques presented in Pfetsch et al. (2015), Schmidt (2013), Schmidt et al. (2013) and Schmidt et al. (2015b)—our approach allows the purely continuous reformulation of genuinely discretecontinuous models of entire gas networks. In order to demonstrate that this is also viable in practice, in the sense that one can compute locally optimal values of the corresponding MINLPs, we present some promising first results.
Section 5.1 introduces two compressor stations with boundary conditions as test instances and describes the hardware and software used in the study. Section 5.2 then discusses performance profiles for measuring performance and robustness of different model formulations and solvers. Next, Sects. 5.3 and 5.4 present the numerical results for single compressor stations while Sect. 5.5 presents a preliminary computational study on entire gas networks. Finally, Sect. 5.6 gives a summary of the results.
5.1 Test instances and computational setup
We consider minimum cost problems using the objective (8) with cost coefficients Open image in new window and Open image in new window , and feasibility testing using the objective \(f\equiv 0\). These objectives are combined with all presented models for two different compressor stations. The first station, called GasLib582 station in the following, is compressorStation_5 from the network GasLib582 (Humpola et al. 2015). It contains one turbo compressor and one piston compressor and can be operated in three configurations. This station is comparatively small and serves as a proof of concept for the applicability of the continuous reformulations. Moreover, the data of this test set are publicly available, so that other researchers can compare their models or algorithms on the same data. The second station, called HG station in the following, is a realworld compressor station of our former industry partner Open Grid Europe^{1} (OGE). It is one of OGE’s largest compressor stations, containing five turbo compressors that can be operated in 14 configurations in our model. The results on this station illustrate the applicability of the presented models on realworld data.
All models are implemented using the modeling language GAMS (McCarl 2009) and the C software framework LaMaTTO (LaMaTTO++ 2015). As global solvers for the MINLP model and its continuous (NLP type) reformulations we use BARON 12.3.3 (Tawarmalani and Sahinidis 2002, 2004, 2005) and SCIP 3.0 (SCIP 2015; Vigerske 2012). Additionally, we use the convex MINLP solver KNITRO 8.1.1 (Byrd et al. 2006) as a heuristic for the nonconvex MINLPs and as NLP solver for the continuous reformulations. As local solvers for the continuous reformulations we use the interiorpoint code Ipopt 3.11 (Wächter and Biegler 2006) and the reducedgradient code CONOPT4 (Drud 1994, 1995, 1996) as well as the three MINLP solvers. The solvers are run with default settings throughout, even for solution tolerances, as it is virtually impossible to find settings that make the results comparable in a strict mathematical sense.
All computations are executed on a sixcore AMD Opteron Processor 2435 with 2600 MHz and 64 GB RAM. The operating system is Debian 7.5.
5.2 Measuring performance and robustness
5.3 The GasLib582 test set
The set \(\mathscr {F}_1\) contains 48 instances of \(\mathscr {T}_1\), hence 36 of the 84 instances are infeasible. The computing times are up to 62 s in cases where solutions are found and up to 59 s in cases where a solver detects infeasibility. Some of the instances are solved in fractions of a second. This can happen, for instance, in the preprocessing of BARON due to bound strengthening. As one would expect, the largest differences of computing times are observed for the global solvers.
GasLib582 test set: numbers of false infeasibility reports and solver failures. Top: solver does not find a feasible solution with any model, bottom: no solver finds a feasible solution with any variant of given model
BARON  SCIP  KNITRO  Ipopt  CONOPT4  

0  7  7  7  8 
MINLP  EBR  ABR  EUR  AEUR  AUR 

7  1  0  0  2  5 
For every problem \(p \in \mathscr {F}_1\), we compute the minimum and maximum objective values \(f^*_{p,\min }\) and \(f^*_{p,\max }\) as well as the maximal absolute gap \(g^*_p = f^*_{p,\max }  f^*_{p,\min }\). The minimal values \(f^*_{p,\min }\) range from 0 to 0.008735 (operating cost in €/s), with maximal gaps ranging from 0 to 0.008765. The average maximal gap over \(p \in \mathscr {F}_1\) is approximately 0.0012, but most of the individual gaps are actually zero. We assume that differing objective values are mainly caused by different numerical properties of the solvers. We also compared the discrete states of the compressor stations in the optimal solutions for every instance. Different active states are found for 9 of the 48 feasible instances, 3 of which have boundary values of the form \(Q_{0}= 0\) and \(p_u= p_v\), where the bypass mode and the closed mode are both feasible and globally optimal with zero cost. The remaining 6 instances have different active configurations. Thus, we have the surprising observation that on the current test set the local solvers always yield optimal values close to the global minima. A possible reason could be that many instances admit just one feasible discrete configuration. Unfortunately we cannot find out whether this is true since it would require the huge effort of testing feasibility for all discrete configurations.
Next, we turn to the issue of infeasibility detection. A substantial fraction of the boundary values of our test set are infeasible: 36 out of 84. In theory, if a global solver detects infeasibility of an instance, this is considered as an infeasibility proof. However, due to numerical inaccuracy, this is not always true in practice. To give a more detailed overview, we also list the numbers of feasible instances that are not solved (i.e., infeasibility is reported or the solver simply fails) in Table 4. Among the solvers, BARON clearly shows the most reliable results: for every feasible instance there is at least one model formulation for which BARON produces a solution. All other solvers report false infeasible results or fail in 7 or 8 cases. Surprisingly, this is also true for the global solver SCIP. Regarding the different continuous reformulations, the approximate bivariate and exact univariate reformulations are solved for every feasible instance (at least by one solver). The worst result is obtained for the MINLP model (7 failures). However, this result has to be carefully interpreted because the MINLP is handled by only 3 solvers whereas all 5 solvers can handle the continuous reformulations. Within the set of reformulations, the approximate univariate reformulation has by far the largest number of false infeasibility reports and solver failures.
GasLib582 test set: fastest solvers for every combination of MINLP model or continuous reformulation with indicator constraint type and pseudo NCP function (BP: bilinear product)
Model  big\(M\)/\(\phi _\text {FB}\)  big\(M\)/\(\phi _\text {prod}\)  BP/\(\phi _\text {FB}\)  BP/\(\phi _\text {prod}\) 

MINLP  BARON  BARON  
Exact bivar. reform.  BARON  BARON  BARON  BARON 
Approx. bivar. reform.  KNITRO  BARON  BARON  BARON 
Exact univar. reform.  BARON  BARON  BARON  BARON 
Alt. exact univar. reform.  Ipopt  BARON  BARON  CONOPT4 
Approx. univar. reform.  BARON  BARON  BARON  CONOPT4 
First, it can be seen that the global solver BARON is the fastest solver for 18 of the 22 model formulations. The local solvers Ipopt or CONOPT4 are faster in only three cases. CONOPT4 is the overall fastest solver for the alternative exact and approximate univariate reformulations. Second, it is apparent that the bilinear product used as pseudo NCP function yields clearly faster runs than the Fischer–Burmeister function: all the bold model/solver combinations use the bilinear product. Third, a best choice of the type of indicator constraints (big\(M\) vs. bilinear product) is not apparent by this criterion. The full data set actually shows that the choice does not have a significant impact here.
GasLib582 test set: most robust solvers for every combination of MINLP model or continuous reformulation with indicator constraint type and pseudo NCP function (BP: bilinear product)
Model  big\(M\)/\(\phi _\text {FB}\)  big\(M\)/\(\phi _\text {prod}\)  BP/\(\phi _\text {FB}\)  BP/\(\phi _\text {prod}\) 

MINLP  BARON  SCIP  
Exact bivar. reform.  BARON  BARON  BARON  BARON 
Approx. bivar. reform.  BARON  BARON  BARON  BARON 
Exact univar. reform.  BARON  BARON  BARON  BARON 
Alt. exact univar. reform.  BARON  BARON  BARON  BARON 
Approx. univar. reform.  BARON  BARON  BARON  BARON 
Here BARON appears even more often than in the case of computing times: It is always the most robust solver, except for the single case where SCIP is applied to the MINLP model with bilinear products as indicator constraints. In contrast to the case of computing times, the choice of pseudo NCP functions does not seem to have a significant impact here.
Overall, the results look more “homogeneous” than in Fig. 5. All combinations produce comparatively good values \(\rho _s^*\) from 85 to 98 % of solved instances. The two bivariate reformulations produce the best values (significantly above 90 %). Note further that these model formulations use big\(M\) indicator constraints. The good robustness is probably caused by the convexity conserving property of the big\(M\) formulation, which contrasts with the inherently nonconvex bilinear products.
5.4 The HG test set
The set \(\mathscr {F}_2\) contains 41 instances of \(\mathscr {T}_2\), hence 43 of the 84 instances are infeasible. The computing times are up to 67 s in cases where solutions are found and up to 45 s in cases where a solver detects infeasibility.
The minimal objective values (in €/s) now range from 0 to 0.069505 and the maximal gaps from 0 to 0.022102. The average maximal gap over all \(p \in \mathscr {F}_2\) is significantly smaller than for \(\mathscr {F}_1\) at approximately 0.0006778 and most of the individual gaps are again zero. Except for one case, the active states in the optimal solutions for a given set of boundary values are identical for all \(s \in \mathscr {S}\) or consist of different sets of identical compressor units yielding the same objective value.^{2} The exceptional case is exactly the one leading to the maximal gap. Excluding this case reduces the maximal gap over all instances by one order of magnitude.
HG test set: numbers of false infeasibility reports and solver failures. Top: solver does not find a feasible solution with any model, bottom: no solver finds a feasible solution with any variant of given model
BARON  SCIP  KNITRO  Ipopt  CONOPT4  

1  3  2  8  5 
MINLP  EBR  ABR  EUR  AEUR  AUR 

3  2  2  2  2  4 
HG test set: fastest solvers for every combination of MINLP model or continuous reformulation with indicator constraint type and pseudo NCP function (BP: bilinear product)
Model  big\(M\)/\(\phi _\text {FB}\)  big\(M\)/\(\phi _\text {prod}\)  BP/\(\phi _\text {FB}\)  BP/\(\phi _\text {prod}\) 

MINLP  KNITRO  BARON  
Exact bivar. reform.  BARON  CONOPT4  CONOPT4  BARON 
Approx. bivar. reform.  BARON  CONOPT4  Ipopt  Ipopt 
Exact univar. reform.  KNITRO  BARON  Ipopt  Ipopt 
Alt. exact univar. reform.  Ipopt  SCIP  Ipopt  CONOPT4 
Approx. univar. reform.  BARON  SCIP  Ipopt  CONOPT4 
Figure 7 shows the performance profiles for every bold combination of Table 8. The approximate bivariate reformulation (ABR) solved with CONOPT4 is the overall fastest combination, performing best on more than 30 % of all instances. The approximate univariate (AUR), exact bivariate (EBR), and alternative exact univariate (AEUR) reformulations combined with CONOPT4 come next, each performing best on roughly 20 % of all instances. Since the approximate univariate formulation shows better results than EBR and AEUR for small values \(\tau > 0\), we may summarize that the approximate reformulations (combined with \(\phi = \phi _\text {prod}\)) tend to be the fastest combinations. A possible reason might be that for local solvers the enlarged feasible set is preferable to the feasible sets of lower dimension of the exact reformulations. In addition, the overall preferable approximate bivariate scheme is also the “most regular” formulation with respect to the LICQ (cf. Sect. 4.2). Finally we note that in terms of solution times both the exact univariate reformulation and the MINLP model cannot compete with the other reformulations.
HG test set: most robust solvers for every combination of MINLP model or continuous reformulation with indicator constraint type and pseudo NCP function (BP: bilinear product)
Model  big\(M\)/\(\phi _\text {FB}\)  big\(M\)/\(\phi _\text {prod}\)  BP/\(\phi _\text {FB}\)  BP/\(\phi _\text {prod}\) 

MINLP  KNITRO  SCIP  
Exact bivar. reform.  BARON  BARON  BARON  BARON 
Approx. bivar. reform.  BARON  BARON  BARON  BARON 
Exact univar. reform.  BARON  SCIP  BARON  BARON 
Alt. exact univar. reform.  BARON  SCIP  BARON  BARON 
Approx. univar. reform.  BARON  SCIP  KNITRO  BARON 
5.5 Continuous reformulations of entire networks
Let us now extend our approach of continuous reformulations to MINLP models of entire gas networks. To this end, we combine the reformulations presented in this paper with suitable MPEC based reformulations of all controllable network devices (except compressor stations) (Pfetsch et al. 2015; Schmidt 2013; Schmidt et al. 2013, 2015b). In our computational study, we combine the MPEC based model of the network with all combinations of pseudo NCP functions, indicator constraints, continuous reformulations, and local solvers discussed in the previous sections. Each of these combinations is applied to 50 randomly chosen instances of the GasLib582 network.^{3} All instances are solved with the goal of feasibility testing, i.e., with objective function \(f \equiv 0\); cf. Sect. 5.2. The time limit of all computations is set to 2 min, and we use the same hardware and software as in the preceding sections.
Number of solved instances (k) for every combination of pseudo NCP function (pNCP), indicator constraint type (\(c^{\text {ind}}\)), continuous reformulation, and local solver. All combinations for which no feasible solution could be found are not listed
pNCP  \(c^{\text {ind}}\)  Reformulation  Solver  k 

\(\phi _\text {FB}\)  big\(M\)  ABR  Ipopt  9 
\(\phi _\text {FB}\)  big\(M\)  ABR  CONOPT4  5 
\(\phi _\text {FB}\)  big\(M\)  EBR  Ipopt  9 
\(\phi _\mathbf{FB}\)  big\(M\)  EBR  CONOPT4  14 
\(\phi _\text {FB}\)  bilin. prod.  AAUR  Ipopt  2 
\(\phi _\text {FB}\)  bilin. prod.  AEUR  Ipopt  4 
\(\phi _\text {FB}\)  bilin. prod.  EUR  Ipopt  3 
\(\phi _\text {FB}\)  bilin. prod.  ABR  Ipopt  1 
\(\phi _\text {FB}\)  bilin. prod.  ABR  CONOPT4  2 
\(\phi _\text {FB}\)  bilin. prod.  EBR  Ipopt  1 
\(\phi _\text {FB}\)  bilin. prod.  EBR  CONOPT4  1 
\(\phi _\text {prod}\)  big\(M\)  AAUR  Ipopt  1 
\(\phi _\text {prod}\)  big\(M\)  AAUR  KNITRO  1 
\(\phi _\text {prod}\)  big\(M\)  AEUR  Ipopt  1 
\(\phi _\text {prod}\)  big\(M\)  EUR  Ipopt  1 
\(\phi _\text {prod}\)  big\(M\)  ABR  Ipopt  7 
\(\phi _\text {prod}\)  big\(M\)  ABR  CONOPT4  3 
\(\phi _\text {prod}\)  big\(M\)  EBR  Ipopt  10 
\(\phi _\text {prod}\)  bilin. prod.  AAUR  Ipopt  4 
\(\phi _\text {prod}\)  bilin. prod.  AEUR  Ipopt  8 
\(\phi _\text {prod}\)  bilin. prod.  EUR  Ipopt  5 
\(\phi _\text {prod}\)  bilin. prod.  ABR  Ipopt  2 
\(\phi _\text {prod}\)  bilin. prod.  ABR  CONOPT4  4 
\(\phi _\text {prod}\)  bilin. prod.  EBR  Ipopt  2 
\(\phi _\text {prod}\)  bilin. prod.  EBR  CONOPT4  8 
Table 10 lists the number of feasible solutions found by the individual reformulations. It turns out that the choice of the pseudo NCP function (Fischer–Burmeister vs. product) is not crucial here: 51 combinations using \(\phi _\text {FB}\) and 57 combinations using \(\phi _\text {prod}\) are solved to feasibility. Similarly, the choice of the specific indicator constraint type does not appear to be of great importance for entire networks (61 feasible solutions for big\(M\) constraints and 47 for bilinear product constraints).
In contrast, the choices of reformulation scheme and local solver have a strong influence. All univariate reformulation schemes (EUR, AEUR, and AAUR) yield feasible solutions significantly less often (9, 13, and 8 times) than the exact bivariate (EBR; 45 feasible solutions) and the approximate bivariate reformulation scheme (ABR; 33 feasible solutions). Concerning the tested local solvers, Ipopt (70 feasible solutions) clearly outperforms both CONOPT4 (37) and KNITRO (1).
In total, 38 instances (76 %) are solved to feasibility. The most successful combination (printed bold in Table 10) combines the Fischer–Burmeister pseudo NCP function with big\(M\) indicator constraints, the exact bivariate reformulation scheme, and the local solver CONOPT4.
The analysis of the results shows that feasibility testing for entire gas networks is possible with purely continuous models in principle. However, the used combinations of MPEC models and continuous compressor station reformulations have to be tested in more detail in order to increase the robustness of the formulations. This is a subject of future research.
5.6 Summary
A major result of our study is that local solvers applied to continuous reformulations tend be faster than global solvers applied to the mixedinteger model and its continuous reformulations. This was to be expected. Moreover, our numerical results suggest that this tendency becomes more evident for larger instances. When comparing only the continuous reformulations, the results for the larger compressor station HG indicate that reformulations with larger feasible sets are preferable and that the regularity of the formulation has a strong impact on the performance of the solution process. (Recall that the “most regular” approximate bivariate reformulation yields the best solution times on the large test set.) Finally, global solvers (especially BARON) produce more robust results, independent of the choice of model formulation.
Our results also indicate which types of indicator constraints work well with a given solver. Especially for larger instances, global solvers tend to be faster on big\(M\) formulations while local solvers perform better with bilinear products. A probable reason for this is that the behavior of global solvers is stronger influenced by nonconvexity than it is the case for local solvers. The situation changes with regard to robustness of the solution process, where the bilinear indicator constraint is clearly preferable to the big\(M\) formulation.
A second observation concerns the preferable type of pseudo NCP functions: The product formulation \(\phi _\text {prod}\) distinctly outperforms Fischer–Burmeister functions. Although the latter are quite prominent in the literature, the simple formulation using products usually produces faster and more robust formulations in practice.
Finally we can state that, for the models considered in this paper, there are no significant drawbacks concerning the quality of solutions when local solvers are used. These results are perhaps not entirely surprising for the considered class of realworld problems. However, no systematic study exists in the literature up to now.
The tendencies observed above might change when fixed boundary values are replaced with nontrivial feasible ranges. This happens, e.g., when the tested continuous reformulations of compressor stations are used in continuous models of entire networks. As expected, the resulting instances are hard, but we have seen in Sect. 5.5 that in principle they can be solved by local solvers. Nevertheless, more extensive testing of model combinations and possibly the development of further model variants are required in order to obtain more robust formulations. Both issues are subjects of future research.
6 Conclusion
In this article we have presented MINLP and GDP models for cost optimization and feasibility testing of gas compressor stations. Moreover, we have considered different types of continuous reformulation techniques from the literature and applied them to the application problem. Our computational study shows that local solvers applied to continuous reformulations can be used to replace MINLP formulations that can only be tackled by global solvers. The continuous reformulations yield comparably robust results, optimal values of almost the same quality on our test set, and tend to be solvable within shorter solution times. Together with the techniques developed in Pfetsch et al. (2015), Schmidt (2013), Schmidt et al. (2013), and Schmidt et al. (2015b), this article provides a complete continuous reformulation of the discretecontinuous problem of stationary gas transport optimization. Additionally, first promising numerical results on a largescale network underpin the practical usability of continuous reformulations for entire networks.
However, some questions remain open. We have considered a stationary and isothermal variant of the problem of compressor station optimization. Since including gas temperature as a dynamic variable mainly leads to increased nonlinearity and nonconvexity in the model, we expect that continuous reformulations tend to be even more favorable for these models. In contrast, the consideration of the transient case will also increase the amount of discrete aspects so that it is unclear which formulation will be favorable in this case. Finally, more extensive testing of suitable continuous models for entire networks is a subject of future research.
Footnotes
 1.
 2.
The detection of mathematical symmetry in this situation could be used to reduce the complexity of the corresponding station model.
 3.
The 50 randomly chosen instances are cold_1461, cold_1466, cold_1689, cold_2311, cold_2406, cold_2763, cold_3824, cold_4105, cold_712, cool_122, cool_1500, cool_1586, cool_1766, cool_1770, cool_1929, cool_2045, cool_2208, cool_2270, cool_2859, cool_3400, cool_3409, cool_3885, cool_3929, cool_4031, cool_416, cool_4192, cool_526, cool_543, cool_821, freezing_1106, freezing_1416, freezing_1599, freezing_164, freezing_2206, freezing_3078, freezing_3853, freezing_402, freezing_728, mild_1203, mild_1344, mild_1459, mild_3124, mild_92, warm_1215, warm_2356, warm_2689, warm_2718, warm_3048, warm_3235, warm_916.
Notes
Acknowledgments
This work has been supported by the German Federal Ministry of Economics and Technology owing to a decision of the German Bundestag. The responsibility for the content of this publication lies with the authors. This research has been conducted as part of the Energie Campus Nürnberg and supported by funding through the “Aufbruch Bayern (Bavaria on the move)” initiative of the state of Bavaria. We are also very grateful to Benjamin Hiller for his comments on an earlier version of this paper. Finally, we thank our former industry partner Open Grid Europe GmbH for their support.
Compliance with ethical standards
Conflicts of interest
The authors declare that they have no conflict of interest
Human and Animal Participants
This article does not contain any studies with human or animal subjects
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