Optimization of Interconnected Natural Gas and Power Systems Using Mathematical Programs with Complementarity Constraints

Abstract

The demand for thermal power generation from natural gas has increased globally due to its cleaner burning properties compared to other fossil fuels. Optimizing the gas flow through the network to meet this demand is challenging due to the nonconvex Weymouth equation constraining gas flow and nodal pressures in pipelines. Traditional methods for addressing this nonconvexity lead to significant approximation errors or high operational costs. This study poses the Weymouth constraint as a Mathematical Programming with Complementarity Constraints (MPCC) for an optimal gas flow problem. The complementarity constraints reformulate the discontinuous sign function using binary-behaving continuous variables. This MPCC-based approach avoids solving mixed-integer programming problems while enhancing the accuracy of conventional linear and second-order approximations. Testing the approach on various interconnected systems, including Colombia’s national gas transportation grid, demonstrated significant reductions in Weymouth approximation errors, thereby supporting effective optimization for interconnected networks.

1. Introduction

Natural gas has gained significant global importance as an energy source thanks to the demonstrated intrinsic connection between a nation’s economic growth and energy consumption [1]. Globally, the natural gas demand in 2020 reached 1788 billion cubic meters (bcm), with projections for 2040 rising to 2142 bcm [2]. In the Latin American context, statistics estimate a considerable increase from 96 bcm in 2020 to 148 bcm in 2040, proving a high dependence on natural gas consumption, mainly for domestic use.

As a power source, natural gas mitigates the reduced hydroelectric generation caused by low water levels during dry seasons and severe droughts in rainy seasons [3,4]. Further, the increasing awareness and emphasis on environmental responsibility solidify the preference for natural gas-fired generating units over traditional fuels such as coal and oil, yielding an interconnected gas and electric system [5]. Lower greenhouse emissions turn natural gas into an efficient and environmentally friendly power source support, positioning it favorably for climate change mitigation [6,7]. Therefore, there is a need for an interconnected system that ensures a reliable and stable power supply during challenging dry seasons and drought conditions [8] while consistently meeting the demands of residential, commercial, and industrial sectors [9,10].

Besides gas-fired power generators, natural gas systems hold other four main components that transport natural gas from production sites to end users: (i) Natural gas production fields and liquefied natural gas (LNG) regasification terminals that supply natural gas into the transmission system at specified pressures; (ii) A network of transmission pipelines for transporting natural gas over long distances; (iii) Compressors for increasing pressure at intervals along the pipeline to push the gas forward against friction; (iv) End users that consume the delivered natural gas (e.g., factories, homes, and power generators) [11]. While production fields, compressors, and end users of natural gas hold well-established representations, transmission pipelines remain complex to model due to the nonlinear relationship between the flow and the pressures at its ending nodes [12]. The Weymouth equation defines this pressure–flow relationship, which includes a nonconvex and discontinuous sign function that determines the flow direction based on the differential pressure. The nonconvexities introduce discontinuities at flow reversal points and multiple local optima in the cost function [13], yielding numerical issues and optimization instability.

2. Literature Review

Researchers in optimal gas flow have developed methodologies for dealing with the Weymouth equation challenges by balancing model complexity, accuracy, and computational tractability [14]. A straightforward approach deals with nonlinear constraints through heuristic algorithms [15]. However, these methods are sensitive to the initial conditions, leading to suboptimal solutions [16]. Another approach involves formulating the problem as a mixed-integer linear programming (MILP) model, where the Weymouth equation becomes a binary-weighted linear combination [17]. Despite inherently handling discontinuities, this MILP formulation is more computationally complex than other optimization techniques [18].

Other approaches aim to enhance global convergence and reduce computational complexity by relaxing the Weymouth constraint through tractable approximations [19]. One approach involves using a piecewise linear approximation to transform the flow–pressure relationship into linear constraints, thereby identifying redundant variables and constraints in the new problem [14]. Another approximation method, based on a Taylor series expansion, replaces the nonlinear equations with linear inequalities [20]. Additionally, an improved one-dimensional piecewise linearization method simplifies the coordinated interactive flow optimization into a more manageable MILP problem while accounting for the interconnected system dynamics [21]. Furthermore, integrating successive linearization and mixed-integer quadratic programming has enhanced computational efficiency in addressing optimal power–gas flow [22]. Although the above linearization approaches result in tractable problems, they usually offer inexact and infeasible solutions for the real world [23].

The convex relation emerges as a Weymouth approximation, enhancing the feasibility of gas-power flow solutions without compromising the computational burden. For instance, approximating Weymouth as a second-order cone (SOC) program efficiently co-optimizes the dispatch of integrated power, heat, and gas systems [24]. The difference-of-convex programming (DCP) algorithm also approximates the nonconvex power flow and Weymouth gas flow equations to fully exploit the maximum total energy supply capability [25]. Another approximation introduces an odd polynomial regression for smoothing the Weymouth equation over a predefined operating interval by matching its first and second derivatives [26]. Although previous strategies reduce the complexity and are compatible with standard solvers, the resulting approximations can significantly deviate from the actual physical behavior of pipelines.

This study introduces a new method for formulating the Weymouth equation as a Mathematical Programming with Complementarity Constraints (MPCC) to better handle the sign function at a lower computational burden than MILP or nonlinear programming. Various engineering fields have taken advantage of modeling nonconvexities and integer-like decision variables using complementarity constraints. In optimizing hydrogen pipeline networks, these constraints model nonideal flow behaviors, including flow reversals and transitions [27]. In air separation and carbon capture systems, MPCC switches the vanishing and reappearing phases in thermodynamic modules, the temperature bounds on heat exchangers serviced by chilled water, and the phase specifications in compressor models [28]. In oil production systems, formulating the pump boundary conditions as an MPCC problem for optimizing reservoirs enables simultaneous optimization with large-scale solvers [29]. Within industrial chemistry applications, the complementarity constraints integrate production scheduling and model predictive control (MPC) while satisfying the MPC controller’s Karush–Kuhn–Tucker (KKT) optimality conditions [30].

Formulating the Weymouth equation as an MPCC problem introduces continuous variables acting as binary switches that evaluate the sign function. The main contributions are summarized as follows:

• Formulating the Weymouth equation as an MPCC significantly reduces approximation errors in modeling gas flow through pipelines, particularly by handling the Weymouth equation’s nonlinearity more effectively than traditional methods.
• The complementarity constraints simplify the optimization problem and pose a computationally more efficient approach by avoiding mixed-integer programming and using continuous variables that exhibit binary-like behavior.
• The proposed approach has been validated through experiments on various interconnected systems, including convergence, reliability, and scheduling scenarios. These demonstrate its effectiveness in reducing operational costs and improving accuracy across different system scales.

The rest of this work is structured as follows: Section 3 describes the cost function for operating the system, the constraints for the power and gas sides, and their interconnection through gas-fueled generators. Section 4 describes the mathematical framework for modeling the Weymouth equation as an MPCC and its relaxation to guarantee its convergence. Section 5 compares the proposed MPCC against Taylor and SOC approximation approaches on three study cases: a 9-bus and 8-node small system, the widely studied IEEE 118-bus 48-node case, and the actual Colombian power–gas system with 96 buses and 63 nodes. Finally, Section 6 concludes this work by summarizing the main findings and proposing future research directions.

3. Formulation of Interconnected Power and Gas Systems

An interconnected system can be completely characterized by a directed graph denoted as $\left\{\mathcal{N},\mathcal{E}\right\}$, being $\mathcal{N}$ and $\mathcal{E}$ the sets of units and edges holding all power and gas components along with their interconnections. On the electrical power side, the system holds power units ${\mathcal{N}}_P\subset \mathcal{N}$, termed buses, and power edges $\mathcal{B}\subset \mathcal{E}$ or branches. The power buses comprise generators $\mathcal{G}\subset {\mathcal{N}}_P$ injecting power and users $\mathcal{D}\subset {\mathcal{N}}_P$ demanding power [31]. The branches $\mathcal{B}=\left\{b=(n,m)\mid n,m\in {\mathcal{N}}_P\right\}$ connect the buses to make the electrical power flow from the generators to the users. Although the physical power flow is alternating current, the system is accurately modeled using a linear direct current (DC) approximation. The DC model ignores reactive power flows and voltage magnitude fluctuations and approximates active power flows using linear transfer distribution factors, resulting in linear programming problems [32]. Thus, the DC modeling of power systems appropriately balances the accuracy and computational tractability for operation and planning [33]. On the natural gas side, the system denotes the units as gas nodes ${\mathcal{N}}_F\in \mathcal{N}$, including gas supply nodes or wells $\mathcal{W}\subset {\mathcal{N}}_F$, gas demand nodes or users $\mathcal{U}\subset {\mathcal{N}}_F$, and gas storage facilities $\mathcal{S}\subset {\mathcal{N}}_F$. Similarly, the adjacency edges in $\mathcal{A}=\left\{(n,m)\mid n,m\in {\mathcal{N}}_F\right\}\subset \mathcal{E}$ delineate the network structure through two kinds of gas flow elements: transport pipelines $\mathcal{P}=\left\{p=(n,m)\mid n,m\in {\mathcal{N}}_F\right\}$ and compressing stations $\mathcal{C}=\left\{c=(n,m)\mid n,m\in {\mathcal{N}}_F\right\}$, so that $\mathcal{P}\cup \mathcal{C}=\mathcal{A}$ and $\mathcal{P}\cap \mathcal{C}=\varnothing$.

Then, the optimization problem of the interconnected system seeks to minimize the operation costs for satisfying the demands of the interconnected system while encompassing the power and gas constraints. Specifically, the following cost function linearly combines the flows of power and gas through the operation costs of the interconnected system elements:

$$\begin{array}{c}\hfill \underset{\mathcal{P},\mathcal{F}}{min}\sum _{g\in \mathcal{G}}{C}_g^t{P}_g^t+\sum _{d\in \mathcal{D}}{C}_d^t{P}_d^t+\sum _{w\in \mathcal{W}}{C}_w^t{f}_w^t+\\ \hfill \sum _{p\in \mathcal{P}}{C}_p^t{f}_p^t+\sum _{c\in \mathcal{C}}{C}_c^t{f}_c^t+\sum _{u\in \mathcal{U}}{C}_u^t{f}_u^t+\\ \hfill \sum _{s\in \mathcal{S}}{C}_{s+}^t{f}_{s+}^t+\sum _{s\in \mathcal{S}}{C}_{s-}^t{f}_{s-}^t+\sum _{s\in \mathcal{S}}{C}_s^t{V}_s^t\end{array}$$

(1)

where $C_g^t$ denotes the generation cost by the g-th bus and $C_d^t$ the unsupplied power demand for the d-th user. For the natural gas system, Equation (1) integrates the costs for injecting gas into the system using the well w, for transporting gas through the pipeline p, and for the pressure boosting of compressor c, at the time instant t, namely, $C_w^t$, $C_p^t$, and $C_c^t$, respectively. $C_u^t$ denotes the penalty cost for not supplying the demanded gas to the user u. Lastly, $C_{s+}^t$, $C_{s-}^t$, and $C_s^t$ represent the costs of injecting, extracting, and storing gas at the s-th storage station. Therefore, the decision variables for the optimization problem are $P_g^t$ for the generated power, $P_d^t$ for the unsupplied power, $f_w^t$ for the inject gas flow, $f_p^t$ and $f_c^t$ for the transported gas through pipeline p and compressor c, $f_u^t$ for the unsupplied gas demand, $f_{s+}^t$, $f_{s-}^t$, and $f_s^t$ for injecting, extracting, and storing gas. Traditionally, a transported gas with a positive value of $f_p^t>0$ moves in the predefined direction, while a negative value flows in the opposite one, with no impact on the optimization process. On the other hand, compressor stations solely allow unidirectional gas flow, expressed as $f_c^t\ge 0$. By optimizing this integrated cost function while adhering to the system’s operational constraints, the proposed methodology effectively balances the demands of both energy systems, leading to a comprehensive solution that minimizes costs while ensuring reliable and efficient operation.

Optimization of the integrated cost function in Equation (1) while adhering to the system’s operational constraints must lead to a comprehensive solution balancing the demands of both energy systems while ensuring reliable and efficient operation. Three sets of operational constraints describe the within and between power and gas interplay.

The first constraint set guarantees a stable power system operation: Equation (2) ensures that the generated power $P_g^t$ lies between the technical minimum $\underline{P_g^t}$ and maximum $\overline{P_g^t}$. Equation (3) bounds the power flow through the transmission line $P_l^t$, preventing damages, such as overheating. Equation (4) models the power flow over the electrical network through the reactance-based relationship of the power flow $P_l^t$, the line susceptance $B_{nm}$, and the voltage angles ${\theta }_n,{\theta }_m$ at buses $n,m$. Equation (5) limits the unsupplied power $P_d^t$ to the user demand $\overline{P_d^t}$. Equation (6) ensures stable operating conditions within the interconnected power grid by restricting the bus voltage angles. Equation (7) defines the power balance at each bus, i.e., the total input and generated power must equal the total output and unsupplied power, being ${\mathcal{L}}_{n+}=\{(m,n^{\prime })\in \mathcal{L}:n^{\prime }=n\}$ and ${\mathcal{L}}_{n-}=\{(n^{\prime },m)\in \mathcal{L}:n^{\prime }=n\}$ the set of inflow and outflow transmission lines at the n-th bus, respectively.

$$\begin{array}{cc}\hfill \underline{P_g^t}\le P_g^t\le \overline{P_g^t}& \forall g\in \mathcal{G},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill -\overline{P_l^t}\le P_l^t\le \overline{P_l^t}& \forall l\in \mathcal{L},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill P_l^t=B_{nm}({\theta }_n-{\theta }_m)& \forall l=(n,m)\in \mathcal{L},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill 0\le P_d^t\le \overline{P_d^t}& \forall d\in \mathcal{D},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill -\overline{{\theta }_n^t}\le {\theta }_n^t\le \overline{{\theta }_n^t}& \forall n\in {\mathcal{N}}_P,\hfill \end{array}$$

(6)

$$\begin{gathered} \sum _{\begin{array}{c}l\in {\mathcal{L}}_{n+} \\ g=n\end{array}}{P}_l^t+P_g^t=\sum _{\begin{array}{c}l\in {\mathcal{L}}_{n-} \\ d=n\end{array}}{P}_l^t+P_d^t\forall n\in {\mathcal{N}}_P \end{gathered}$$

(7)

The second constraint set interconnects natural gas and electrical power systems through gas-fired power plants generating electricity, as expressed by Equation (8), where $f_n^t$ stands for the natural gas fuel consumption to generate a power $P_n^t$ at generator bus $n\in {\mathcal{N}}_I$, the heat-rate ${\mathrm{HR}}_n$ defines the generator efficiency, and the set ${\mathcal{N}}_I=\mathcal{G}\cap \mathcal{U}$ holds all the units in the interconnected system belonging to both the power generator and gas demand sets.

$$\begin{array}{cc}\hfill & f_n^t=P_n^t·{\mathrm{HR}}_n,\forall n\in {\mathcal{N}}_I,\hfill \end{array}$$

(8)

The third constraint set models the gas transportation system: Equation (9) forces each production well to inject the flow $f_w^t$ over the technical minimum $\underline{f_w^t}$ and under the maximum capacity $\overline{f_w^t}$. Equation (10) upper-bounds the gas flow through pipelines $f_p^t$ to the structural capacity $\overline{f_p^t}$. Equation (11) fixes safe operating limits for the pressure on the n-th node ${\pi }_n^t$ as $[\underline{{\pi }_n^t},\overline{{\pi }_n^t}]$. The constraint in Equation (12) asserts that the compression ratio ${\pi }_m^t/{\pi }_n^t$ cannot physically exceed the compressor’s design limitation ${\beta }_c\ge 1\forall c=(n,m)\in \mathcal{C}$, enabling the representation of different compressors by adjusting the values of ${\beta }_c\ge 1$. Equation (13) ensures that the unsupplied demand $f_u^t$ is lower than the corresponding user demand $\overline{f_u^t}$. The nodal gas balance in Equation (14) guarantees that the gas entering the node n equals the gas leaving it. Equations (15) and (16) limit the gas injection $f_{s+}$ and extraction $f_{s-}$ rates at storage facilities according to the feasible operating range determined by the currently stored volume $V_s^t$, respectively. In turn, Equation (17) balances the gas storage unit such that gas volume at operation period t$V_s^t$ equals the volume from period $V_s^{t-1}$ plus the difference between injected $f_{s+}^{t-1}$ and extracted $f_{s+}^{t-1}$ gas flow, a fundamental constraint for modeling the dynamics of gas storage over time. Lastly, Equation (18), known as the Weymouth equation, summarizes the physical behavior of gas flow through pipelines by relating the gas flow through the pipeline $f_p^t$ to the pressures at the ends of the pipeline ${\pi }_n^t,{\pi }_m^t\forall p=(n,m)\in \mathcal{P}$. The Weymouth equation defines a nonlinear, nonconvex, disjunctive flow–pressure relationship that hampers the optimization of the gas transport system.

$$\begin{array}{cc}\hfill \underline{f_w^t}\le f_w^t\le \overline{f_w^t}& \forall w\in \mathcal{W}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill -\overline{f_p^t}\le f_p^t\le \overline{f_p^t}& \forall p\in \mathcal{P}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \underline{{\pi }_n^t}\le {\pi }_n^t\le \overline{{\pi }_n^t}& \forall n\in {\mathcal{N}}_F\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\pi }_m^t\le {\beta }_c^t{\pi }_n^t& \forall c=(n,m)\in \mathcal{C}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill 0\le f_u^t\le \overline{f_u^t}& \forall u\in \mathcal{U}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \sum _{m:(m,n)\in \mathcal{A}}{f}_m^t=\sum _{m^{\prime }:(n,m^{\prime })\in \mathcal{A}}{f}_{m^{\prime }}^t& \forall n\in {\mathcal{N}}_F\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill 0\le f_{s+}^t\le V_{0s}-\underline{V_s}& \forall s\in \mathcal{S}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill 0\le f_{s-}^t\le \overline{V_s}-V_{0s}& \forall s\in \mathcal{S}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill V_s^t=V_s^{t-1}+f_{s-}^{t-1}-f_{s+}^{t-1}& \forall s\in \mathcal{S}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill sgn\left(f_p^t\right){\left(f_p^t\right)}^2=K_{nm}({\left({\pi }_n^t\right)}^2-{\left({\pi }_m^t\right)}^2)& \forall p=(n,m)\in \mathcal{P}\hfill \end{array}$$

(18)

4. Mathematical Programming with Complementarity Constraints for Weymouth Approximation

The Weymouth equation is the fundamental model for gas flow through pipelines. However, it presents a challenge for optimal interconnected operation due to its nonlinearity, which arises from the signum function determining the gas flow direction. This nonlinearity results from the complex physics of gas flow, making it challenging to find optimal solutions for gas transportation systems [34]. Traditional optimization approaches struggle to handle the nonconvex terms within the Weymouth equation. However, recent advances in optimization techniques, particularly MPCC, offer a promising solution to address this issue. MPCC specializes in handling complementarity constraints and nonconvexities, making it well-suited to tackle the intricacies of the Weymouth equation [35]. This type of formulation involves optimization problems of the form:

$$\begin{array}{cc}\hfill \mathcal{O}:& minf(x,y)\hfill \end{array}$$

(19)

$$\begin{array}{cc}& \mathrm{s}.\mathrm{t}.h_i(x,y)=0\hfill \end{array}$$

(20)

$$\begin{array}{cc}& g_j(x,y)\ge 0\hfill \end{array}$$

(21)

$$\begin{array}{cc}& 0\le G_k\left(x\right)\perp H_k\left(y\right)\ge 0\hfill \end{array}$$

(22)

where $f(x,y)$ is the cost function, $h(x,y)$ and $g(x,y)$ capture equality and inequality constraints in the optimization problem $\mathcal{O}$. Equation (22) represents the complementarity conditions, with the operator ⊥ indicating that at a solution, either x or y must be zero while the other must remain non-negative. These conditions turn MPCC into a modeling tool for scenarios with variables exhibiting complementarity relationships, such as economic equilibrium [36], variational inequalities [37], and the intricate dynamics of natural gas transportation systems [38]. To deal with the nonconvexity, this work rewrites the Weymouth equation as the following mathematical program with two complementarity constraints:

$$\begin{array}{cc}\hfill {\mathcal{O}}_W:& \underset{y_p^t}{min}-y_p^t{f}_p^t\hfill \\ & \mathrm{s}.\mathrm{t}.y_p^t{\left(f_p^t\right)}^2=K_{nm}({\left({\pi }_n^t\right)}^2-{\left({\pi }_m^t\right)}^2)\hfill \\ & -1\le y_p^t\le 1\hfill \\ & f_p^t=f_{p+}^t-f_{p-}^t\hfill \\ & 0\le f_{p+}^t\perp (y_p^t+1)\ge 0\hfill \\ & 0\le f_{p-}^t\perp (1-y_p^t)\ge 0\hfill \end{array}$$

(23)

where $f_{p+}^t\ge 0$ and $f_{p-}^t\ge 0$ hold the positive and negative components of the gas flow in the p-th pipeline at operation period t, for assessing directional flow.

Despite managing the flow direction in a continuous formulation, the program ${\mathcal{O}}_W$ misses the KKT conditions at a local minimization point because the conventional constraint qualifications for nonlinear programming, such as Linear Independence Constraint Qualification and Mangasarian-Fromovitz Constraint Qualification, are typically not satisfied in the case of MPCC [39]. Therefore, posing relaxed nonlinear programs (RNLP) deals with the numerical resolution of MPCC by introducing a positive regularization parameter $ϵ\in {\mathbb{R}}^{+}$ that simplifies the solution and properly handles the inequalities [40]. These programs typically satisfy constraint qualifications, making them more amenable to efficient optimization techniques. Relaxing MPCC ensures that inequalities are appropriately treated as inactive, particularly when $G_k\left(x\right)H_k\left(y\right)\le ϵ$, enhancing their structural integrity. Moreover, relaxed programs reliably approximate the original problem as $ϵ\to 0$ [41]. Hence, instead of working with the original problem ${\mathcal{O}}_W$, the relaxed problem ${\mathcal{O}}_{ϵ}$ is considered:

$$\begin{array}{cc}\hfill {\mathcal{O}}_{ϵ}:& \underset{y_p^t}{min}-y_p^t{f}_p^t\hfill \\ & \mathrm{s}.\mathrm{t}.y_p^t{\left(f_p^t\right)}^2=K_{nm}({\left({\pi }_n^t\right)}^2-{\left({\pi }_m^t\right)}^2)\hfill \\ & f_p^t=f_{p+}^t-f_{p-}^t\hfill \\ & -1\le y_p^t\le 1\hfill \\ & f_{p+}^t(y_p^t+1)\le ϵ\hfill \\ & f_{p-}^t(1-y_p^t)\le ϵ\hfill \end{array}$$

(24)

Theoretically, the relaxed problem offers fundamental properties that tackle challenging MPCC problems [42]. Firstly, the relaxed approach guarantees the convergence to the true MPCC solution as $ϵ\to 0$. Additionally, the boundedness of Lagrange multipliers ensures numerical stability and avoids issues with infinitely large values during optimization. Lastly, the local uniqueness of the ${\mathcal{O}}_{ϵ}$ solution under specific conditions guarantees a single and well-defined solution. Therefore, the proposed relaxed optimization problem deals with the nonconvexity in the Weymouth equation while guaranteeing the KKT conditions around $ϵ$, posing a standard optimization problem, and avoiding ambiguity in interpreting results.

5. Case Studies

The current section validates the proposed MPCC approach by comparing its performance against two well-established methods for approximating the Weymouth equation: (i) The Taylor series approach that piecewise approximates Weymouth with line segments [43] and (ii) the SOC programming that introduces a two-stage optimization, namely, flow direction estimation and cost minimization [44]. The validation contrasts Taylor, SOC, and MPCC approaches in three case studies of interconnected systems with different complexities.

The considered validation aims to quantify the inherent errors and the cost–error trade-off of the contrasted approaches to support its real-world pertinence. Therefore, this work reports two performance metrics: the cost function in Equation (1) that assesses the capacity for optimally operating an integrated system and the Weymouth error metric (${WE}_p^t\in {\mathbb{R}}^{+}$) for quantifying the required flow to guarantee equality for pipeline p at time instant t in Equation (18), as follows:

$${WE}_p^t=\left|f_p^t-{\left(K_{nm}|{\left({\pi }_n^t\right)}^2-{\left({\pi }_m^t\right)}^2|\right)}^{1/2}\right|,\forall p=(n,m)\in \mathcal{P}.$$

(25)

Hence, the ${WE}_p^t$ metric, measured in million standard cubic feet per day (MMSCFD), explains the approximations’ inherent sensitivity and validates the significance of their differences.

5.1. Case Study I: 9/8 System

The network depicted in Figure 1 interconnects a nine-bus power system and an eight-node natural gas network. The small size of case 9/8 enables fast execution, efficient analysis, and rigorous validation of the contrasted approaches. The 9/8 network also features a closed trajectory and bidirectional pipelines, allowing looped infrastructure with potential flow reversals.

To assess the performance of Weymouth approximation approaches on the 9/8 system, a Monte Carlo experiment estimates the cost function and Weymouth error distributions by solving the optimization problem for one day ($\mathcal{T}=\left\{1\right\}$) one hundred times with uniformly sampled natural gas demands. Further network parameter details can be found in the publicly available repository OptiGasFlow (https://github.com/cblancom/optigasflow, accessed on 5 April 2024). Figure 2 depicts the cost function histogram for Taylor, SOC, and MPCC approaches. Remarkably, the three histograms evidence identical distribution patterns, leading to regular solutions across approaches.

The boxplots in Figure 3 show the Weymouth approximation error distribution for each pipeline using three approaches. The error distributions, including median and interquartile range, indicate that MPCC consistently maintains accuracy throughout the network. In contrast, the widely varying errors of the Taylor and SOC approaches suggest a lack of consistency in the achieved solution. Therefore, in a small network, the proposed MPCC approach converges to identical operational costs as Taylor and SOC, even in rationing, while meeting all linear constraints and improving the Weymouth approximation.

5.2. Case II: 118/48 System

The following case simulates a complex, large-scale electric grid system, the widely studied IEEE 118 bus system [45], consisting of 54 generator buses, 9 fed by the gas system, 186 transmission lines, and 99 users, that is, $\left|\mathcal{G}\right|=54,\left|\mathcal{F}\right|=186,\left|\mathcal{D}\right|=99$. This electric grid interconnects with a 48-node natural gas system featuring 9 supply wells, 46 pipelines, eight compressor stations, and 22 user nodes through 9 connection points, i.e., $\left|\mathcal{W}\right|=9,\left|\mathcal{P}\right|=46,\left|\mathcal{C}\right|=8,\left|\mathcal{U}\right|=22,\left|\mathcal{I}\right|=9$ [46]. The network topology deliberately introduces closed flow loops to stress the solver and the constraint approximations, as do real-world systems.

Figure 4 depicts the histogram of relative cost differences for the MPCC proposal to Taylor and SOC baselines from a hundred trials of the Monte Carlo experiment and a considered operation of one day ($T=1$). It is worth noting that both baselines yielded the same cost function values. The relative difference between MPCC and the baselines is always positive, indicating that the complementarity constraint formulation consistently produces larger cost values in this system. However, the maximum difference of 6% falls within the range of real-world variations due to the dispatcher’s practical decisions in line with the actual pressure–flow relationship [47].

Contrarily to cost function analysis, results in Figure 5 reveal a significant error reduction of about seven orders of magnitude (from ${10}^1$ to ${10}^{-6}$) under the proposed complementarity constraints. As an additional benefit, MPCC exhibits a shorter error dispersion than Taylor and SOC at most of the 46 pipelines in the network. Such behavior in the 118/48 system, also evidenced in the small 9/8 case study, proves the reliability of MPCC in effectively addressing more complex network configurations and interconnected dynamics.

5.3. Case Study III: 96/63 System

The last case study focuses on the Colombian power system, a complex network comprising 96 nodes ($\left|{\mathcal{N}}_P\right|=96$), 49 generators ($\left|\mathcal{G}\right|=49$), 207 transmission lines ($\left|\mathcal{F}\right|=207$), and 80 power users ($\left|\mathcal{D}\right|=80$). From the 49 generators, 10 are thermal power plants ($\left|\mathcal{I}\right|=10$) fed by the natural gas transportation system, including 13 wells ($\left|\mathcal{W}\right|=13$), 48 pipelines ($\left|\mathcal{P}\right|=48$), 14 compressor stations ($\left|\mathcal{C}\right|=14$), and 26 consuming users ($\left|\mathcal{U}\right|=26$), yielding 63 nodes ($\left|{\mathcal{N}}_F\right|=63$). Despite its radial structure, the gas system supports bidirectional flows in its pipelines due to the highly varying demand by thermal power plants influenced by meteorological conditions: On rainy seasons, thermal power plants dramatically reduce their demand; while on dry seasons, a large amount of gas must flow to them.

Instead of estimating the distributions of the cost function and Weymouth error as in cases 9/8 and 118/48, the 96/63 case validates the Weymouth approximations in an operation case of ten consecutive days ($\left|\mathcal{T}\right|=10$) with randomly changing gas extraction costs. Such a complementary validation strategy allows the interconnected system to reduce gas transportation costs by exploiting its single storage station, extending the performance analysis to scheduling scenarios. Figure 6 illustrates the daily optimized operating cost of the integrated system over the ten-day scheduling horizon for each tested approach. The daily cost values reveal notable similarities between the Taylor series and SOC relaxations. Nonetheless, the MPCC approach yields a 2.7% more expensive solution, from 8% cheaper to 12% more expensive, with a difference standard deviation of 6%. The above results indicate that the difference between the proposed MPCC and baseline approximations is statistically negligible and will disappear after the empirical corrections.

Regarding the Weymouth approximation analysis, Figure 7 presents the error distribution and its relationship with the gas flow and the scheduled day for Taylor, SOC, and MPCC. Firstly, the error histogram in Figure 7a proves that the proposed MPCC formulation (in green) exhibits superior approximation accuracy to Taylor and SOC for most pipelines and days. Secondly, the scatter plot in Figure 7b illustrates the relationship between Weymouth error and gas pipeline flow for each approach. Note that the benchmark techniques of Taylor (blue) and SOC (orange) hold a stationary error regardless of the flow rate. In the case of MPCC (green), the larger the flow rate, the shorter the error dispersion. In addition, despite its large error dispersion at low flow rates, MPCC still delivers much lower errors than benchmark methodologies. Hence, the complementarity constraints improve the error rates of Taylor and SOC and become more reliable for higher gas flow rates.

Lastly, Figure 7c suggests independence between the Weymouth error and each scheduled day, with a stationary error distribution for all approximations. Nonetheless, MPCC holds two groups of outlying errors. The higher ones align with typical magnitudes of the benchmark techniques. The second group of errors, lying around ${10}^{-2}$, corresponds to pipelines connected to injection wells (denoted as dots in Figure 8c). Since the wells are technically regulated, their fixed injection pressure hampers the flexibility of MPCC for approximating the Weymouth equation.

The heatmaps in Figure 8c illustrate the output-to-input pressure ratio for each of the 14 compressors over the ten days of the scheduled operation. The baseline approaches of Taylor and SOC (Figure 8a,b) yield constant pressure ratios stemming from an over-relaxation of the Weymouth equation that extends the feasible region to unpractical solutions. In contrast, the MPCC approach in Figure 8c exhibits day-to-day pressure ratio changes within each compressor. The above is because the complementarity constraints closely align with the gas transport system’s real physics, restraining the range of the feasible pressure values to trade off the daily varying injection cost.

As a remark, compressor nine in Figure 8c reaches large pressure ratios on Days 0, 1, 2, 4, and 7, overlapping with the time instants with the highest approximation errors for MPCC in Figure 7c. A detailed examination of these outcomes detects that compressor nine and the outlying pipeline are the two outputs of a bifurcation, the latter being followed by an injection well. Figure 9 exemplifies that such an interconnection is the sole over the gas network. As a hypothesis, fixing the pressure at the injection well and the flow direction at compressor nine pushes the complementarity constraints to the limits and forces the compressor to augment the pressure ratio to satisfy the forthcoming branch demand.

6. Conclusions

This paper presented a novel approximation for the Weymouth constraint by representing the nonconvex pressure–flow relationship as an MPCC. The MPCC-based formulation significantly benefits the optimization problems in interconnected power and gas systems using binary-behaving continuous variables related to the flow direction, which avoids costly mixed-integer approximations. Additionally, the MPCC inherently captures the complexity in the signum function, resulting in a rigorous approximation of the Weymouth equation.

The validation compared the proposed MPCC approach against the Taylor series and SOC programming approximations on optimizing the operation of interconnected power and gas transport systems. Monte Carlo experiments validated the solution reliability in two well-known case studies, while a ten-day operation planning assessed the scheduling task in a real-world case study.

Regarding cost function, the MPCC approach demonstrated a remarkable ability to balance operational costs effectively. Results on the 9/8 system proved that MPCC converges to the exact cost of Taylor and SOC in small-scale cases. For more complex networks (cases 118/48 and 96/63), MPCC yields higher operational costs than baselines due to the more rigorous Weymouth equation modeling. Nonetheless, the cost differences among approaches lie within reasonable limits and align with the dispatcher’s empirical decisions.

In the case of Weymouth approximation, MPCC significantly outperforms Taylor and SOC in the tested cases. In the 118/48 and 96/63 systems, MPCC substantially reduces Weymouth approximation errors, often by several orders of magnitude, compared to traditional linearization and convex relaxation strategies. Such an accuracy improvement becomes crucial in large-scale, complex systems where precise approximation directly influences operational efficiency and system reliability. Hence, the introduced pressure–flow model mathematically benefits the optimization task, asserting its cost-effectiveness at various system scales.

The analysis of the scheduling task in the 96/63 Colombian interconnected system underscores the robustness and reliability of the MPCC approach. Despite the complexities of bidirectional flows and time-varying demand scenarios, MPCC maintains high accuracy levels in Weymouth approximation. Furthermore, the nearly negligible cost differences among approximation approaches establish MPCC as the most robust and reliable approach for short-term operational scheduling.

In conclusion, modeling the Weymouth equation as an MPCC improves the optimization of interconnected gas and power systems by balancing operational costs, minimizing approximation errors, and handling scheduling tasks. These findings establish strong evidence for the practical implementation of MPCC in gas transport optimization, particularly in scenarios demanding high accuracy and reliability in short-term operation scheduling.

Considering the current open issues on energy management, three future research directions may complement this study. Firstly, we propose to adapt MPCC to dynamic system constraints for validation in transient analysis scenarios. The second research direction accounts for the uncertainty in interconnected systems, mainly due to the growing share of low-inertia power sources, such as wind and solar, and potential gas transport failures. Hence, we plan to extend the proposed methodology to stochastic optimization, considering the varying parameters and power sources of interconnected systems. Lastly, we will integrate MPCC with distributed cooperative operation schemes considering multi-agent issues such as the lack of information due to privacy policies [48].