Co-Optimized Analysis and Design of Electric and Natural Gas Infrastructures

Abstract

This paper proposes and implements a long-term deterministic capacity expansion model for the co-optimization of electric and natural gas infrastructures. It determines the required investments in generation units, transmission lines and pipelines for meeting future demands, while representing electricity and natural gas flows using DC Power Flow and Weymouth equations, respectively. A Mixed Integer Nonlinear Programming (MINLP) problem is developed, from which a linearized version is derived. A 26 node integrated gas-electric system for the Eastern Region of the United States is used to demonstrate the model’s capabilities. Results show that the model provides an accurate operational representation of the integrated system, and, therefore, enhances the expansion planning process.

1. Introduction

For many decades, natural gas has had an important role in the global energy landscape. It has traditionally been used within industrial processes and as a heating alternative for residential and commercial sectors. However, over the years, its participation as a fuel for generating electricity has increased considerably. There are multiple reasons associated with this phenomenon among which are: (i) the shale gas “boom”, i.e., the significant increase of gas production due to technological developments in the extraction techniques of non-conventional resources; (ii) a reduction in the market prices as the result of the growth in the supply; (iii) the environmental benefits that combined cycle units have demonstrated when compared against other fossil-fired units; and (iv) the operational support that gas-fired units can provide for renewable integration.

Thus, it is clear that the electric sector has become an important player within the natural gas industry, which results in strong interactions between them, not only physically but also economically [1,2,3,4]. Ref. [5] presents a complete summary of different models reported in the literature, categorized by the type of analysis performed for integrated electricity and natural gas systems.

From an operational perspective, multiple research projects have been undertaken recently to analyze these interactions. A set of different methodologies and optimization problems have been proposed to quantify the flexibility that natural gas networks can provide to a power system, or to quantify the adequacy levels needed from gas systems to guarantee the reliability of a power system, under future generation scenarios considering intermittent renewable energy sources [6,7,8,9]. Coordinated scheduling strategies of electrical and natural gas resources in a day-ahead dispatch, including gas compressor operations, has also been addressed in the literature to analyze the economic and operational benefits of an integrated operation [10,11].

From a long-term planning perspective, multi-sector optimization models based on network flows were developed as an initial attempt to model these interactions by determining the combined infrastructure investment requirements [4,12,13,14]. However, these models cannot guarantee an accurate representation of electricity and natural gas flows across the transmission systems because they omit modeling of important physical laws. Ref. [15] derives a Mixed Integer Linear Programming (MILP) representation for the capacity expansion problem while considering the DC power flow equations for the transmission network to overcome this limitation. In addition, Mixed Integer Nonlinear Programming (MINLP) formulations have been derived in [16,17,18,19,20] for an integrated gas-electric representation.

This paper presents a disjunctive model using an MILP formulation that considers DC power flow and Weymouth equations in deterministic capacity expansion problems for integrated gas-electric systems. Initially, a disjunctive MINLP model is presented, which serves as the starting point for deriving a linearized representation using piecewise linear approximations. The capabilities of the proposed formulation are tested using a 26 node integrated gas-electric system simulated over 20 years.

The paper is structured as follows. Section 3 reviews the co-optimization expansion planning problem and presents a MINLP formulation for an integrated gas-electric system. Section 4 develops an initial MILP model using piecewise linear functions, while Section 5 presents an alternative, less computationally intensive formulation. Conclusions are shown in Section 6, after which the nomenclature used in the paper is provided.

2. Co-Optimized Expansion Planning Problem

A co-optimized expansion planning formulation provides simultaneous identification of two or more classes of related infrastructure decisions within a single optimization [21]. In this paper, we study the co-optimization expansion planning problem for an integrated gas-electric system. Hence, the deterministic model to be developed will aim to find the type, capacity, location, and timing of the required investments in generation units, transmission lines and pipelines that satisfy the future demands for both systems, while considering operational, security and reliability criteria.

Different representations for the electricity transmission network can be found in the literature [22], ranging from simple transportation models using network flows to more detailed formulations considering electric laws. Under steady state conditions, AC power flow equations provide an accurate representation of the electricity flows across the transmission network in planning studies.

However, this entails the need for considering a set of nonlinear/nonconvex equations in the optimization problem, with a considerable increase in their complexity. On the other hand, it is possible to use a transportation model at the expense, however, of violating Kirchhoff’s Voltage Law, possibly causing an overestimation of electric transmission capabilities under some scenarios.

Gas transmission systems can also be modeled using different representations according to the assumptions considered. The simplest approach is to represent the pipeline network using a transportation model, which, as with the electric network, could overestimate gas transmission capabilities under some scenarios. Improvements can be obtained by considering gas flow steady state equations in pipelines, such as Weymouth equations, a set of nonlinear equations commonly used in the design of gas transmission networks.

A disjunctive MINLP formulation of the co-optimization expansion planning problem for an integrated gas-electric system is presented below, where electricity and gas flows are represented using DC Power Flow equations and Weymouth equations, respectively. Equation (1) presents the objective function, which expresses the net present value of the sum of operational costs (electricity generation and transmission as well as natural gas production and transmission), investment costs (for new generation units, transmission lines and pipelines), penalties (carbon emissions and unserved demand) and salvage values (for mitigating end horizon effects in long term planning simulations) of the integrated electric-gas system. Equations (2)–(29) constitute the mathematical formulation for the electric and gas systems. Equation (17) plays a significant role in that this constraint links both systems, by modeling the amount of gas required by the combustion turbine units and the combined cycle plants to generate electricity:

$$\begin{array}{cc}\mathit{min}\zeta =\sum _{\overline{g},t}{\xi }^t\left(c_{\overline{g}t}^{fx,g}{C}_{\overline{g}}^{max}{C}_{\overline{g}t}+\sum _{m,s}{c}_{\overline{g}t}^{op,g}{P}_{\overline{g}tms}^g{h}_s\right)+\sum _{\overline{g},t,m,s,f}{\xi }^t{c}_t^{cc}{\upsilon }_{k,f}{\gamma }_{\overline{g}}{P}_{\overline{g}tms}^g{h}_s\hfill & \hfill \\ +\sum _{\overline{g},t,m,s,f}{\xi }^t{c}_{\overline{g}ftm}^{op,f\ne \mathrm{gas}}{\gamma }_{\overline{g}}{P}_{\overline{g}tms}^g{h}_s+\sum _{j,k,t,m,s}{\xi }^t{c}_{jtm}^{op,p}\eta G_{jktms}^p{h}_s+\sum _{j,t,m,s}{\xi }^t{c}_{jtm}^{d,gas}{G}_{jtms}^d{h}_s\hfill & \hfill \\ +\sum _{\begin{array}{c}\overline{g},t\\ g\in G_c\end{array}}{\xi }^t{c}_{\overline{g}t}^{in,g}{C}_{\overline{g}}^{max}{C}_{\overline{g}t}^a+\sum _{\begin{array}{c}\overline{l},t\\ l\in L_c\end{array}}{\xi }^t{c}_{\overline{l}t}^{in,tl}{Z}_{\overline{l}t}+\sum _{\begin{array}{c}\overline{p},t\\ p\in P_c\end{array}}{\xi }^t{c}_{\overline{p}t}^{in,pl}{Z}_{\overline{p}t}+\sum _{\overline{l},t,m,s}{\xi }^t\left(c_{\overline{l}t}^{op,tl}{P}_{\overline{l}tms}^{tl}+c_{\overline{l}t}^{op,tlo}{P}_{\overline{l}tms}^{{}^{\prime }tl}\right)h_s\hfill & \hfill \\ +\sum _{\overline{p},t,m,s}{\xi }^t\left(c_{\overline{p}t}^{op,pl}{G}_{\overline{p}tms}^{pl}+c_{\overline{p}t}^{op,plo}{G}_{\overline{p}tms}^{{}^{\prime }pl}\right)h_s+\sum _{j,t,m,s}{\xi }^t\tau P_{jtms}^{ls}{h}_s+\sum _{\begin{array}{c}\overline{d},t,m,s\\ d\in D_c\end{array}}{\xi }^t{c}_{\overline{d}t}^{op,cr}\left({\pi }_{j^{\prime }tms}-{\pi }_{jtms}\right)h_s\hfill & \hfill \\ +\sum _{\begin{array}{c}\overline{d},t,m,s\\ d\in D_r\end{array}}{\xi }^t{c}_{\overline{d}t}^{op,cr}\left({\pi }_{jtms}-{\pi }_{j^{\prime }tms}\right)h_s-\sum _{\overline{g},t}{\xi }^{\iota }{c}_{\overline{g}t}^{in,g}\left(1-\left(\iota -\left(t-1\right)\right)/{{\delta }_k}^g\right)C_{\overline{g}}^{max}{C}_{\overline{g}t}^a\hfill & \hfill \\ -\sum _{\overline{l},t}{\xi }^{\iota }{c}_{\overline{l}t}^{in,tl}\left(1-\left(\iota -\left(t-1\right)\right)/{\delta }^{tl}\right)Z_{\overline{l}t}-\sum _{\overline{p},t}{\xi }^{\iota }{c}_{\overline{p}t}^{in,pl}\left(1-\left(\iota -\left(t-1\right)\right)/{\delta }^{pl}\right)Z_{\overline{p}t},\hfill & \hfill \end{array}$$

(1)

subject to

$$\begin{array}{cc}{C}_{\overline{g}t}=C_{\overline{g}}^e+\sum _{i=1}^t\left(C_{\overline{g}i}^a-C_{\overline{g}i}^r\right),\forall \overline{g},t,\hfill & \hfill \end{array}$$

(2)

$$\begin{array}{cc}{C}_{\overline{g}t}^a\le C_{\overline{g}t}^{a,max},\forall \overline{g},g\in g_c,t,\hfill & \hfill \end{array}$$

(3)

$$\begin{array}{cc}{P}_{\overline{g}tms}^g\le C{C}_{\overline{g}tms}{C}_{\overline{g}}^{max}{C}_{\overline{g}t},\forall \overline{g},t,m,s,\hfill & \hfill \end{array}$$

(4)

$$\begin{array}{cc}\sum _{m,s}\left(P_{\overline{g}tms}^g{h}_s\right)\le C{F}_{\overline{g}t}{C}_{\overline{g}}^{max}{C}_{\overline{g}t}\sum _{m,s}{h}_s,\forall \overline{g},t,\hfill & \hfill \end{array}$$

(5)

$$\begin{array}{cc}\sum _{\overline{g}}F{C}_{\overline{g}t}{C}_{\overline{g}t}\ge (1+r_j)P_{jt}^{d,peak},\forall j\in J_a,t,\hfill & \hfill \end{array}$$

(6)

$$\begin{array}{cc}\sum _{\overline{l}:B_l=j}-\left(P_{\overline{l}tms}^{tl}-P_{\overline{l}tms}^{{}^{\prime }tl}\right)+\sum _{\overline{l}:E_l=j}\left(P_{\overline{l}tms}^{tl}-P_{\overline{l}tms}^{{}^{\prime }tl}\right)=P_{jtms}^d-P_{jtms}^{ls}-\sum _{\overline{g}}{P}_{\overline{g}tms}^g,\forall j\in J_a,t,m,s,\hfill & \hfill \end{array}$$

(7)

$$\begin{array}{cc}{\theta }_{B_{l}tms}-{\theta }_{E_{l}tms}=X_{\overline{l}}\left(\left(P_{\overline{l}tms}^{tl}-P_{\overline{l}tms}^{{}^{\prime }tl}\right)/S^{base}\right),\forall \overline{l}\in L_e,t,m,s,\hfill & \hfill \end{array}$$

(8)

$$\begin{array}{cc}-(1-S_{\overline{l}t})M^{tl}\le {\theta }_{B_{l}tms}-{\theta }_{E_{l}tms}-X_{\overline{l}}\left(\left(P_{\overline{l}tms}^{tl}-P_{\overline{l}tms}^{{}^{\prime }tl}\right)/S^{base}\right)\le (1-S_{\overline{l}t})M^{tl},\forall \overline{l}\in L_c,t,m,s,\hfill & \hfill \end{array}$$

(9)

$$\begin{array}{cc}{S}_{\overline{l}t}=\sum _{i=1}^t{Z}_{\overline{l}i},\forall \overline{l}\in L_c,t,\hfill & \hfill \end{array}$$

(10)

$$\begin{array}{cc}{P}_{\overline{l}tms}^{tl,min}\le -P_{\overline{l}tms}^{{}^{\prime }tl},\forall \overline{l}\in L_e,t,m,s,\hfill & \hfill \end{array}$$

(11)

$$\begin{array}{cc}{P}_{\overline{l}tms}^{tl}\le P_{\overline{l}tms}^{tl,max},\forall \overline{l}\in L_e,t,m,s,\hfill & \hfill \end{array}$$

(12)

$$\begin{array}{cc}{S}_{\overline{l}t}{P}_{\overline{l}tms}^{tl,min}\le -P_{\overline{l}tms}^{{}^{\prime }tl},\forall \overline{l}\in L_c,t,m,s,\hfill & \hfill \end{array}$$

(13)

$$\begin{array}{cc}{P}_{\overline{l}tms}^{tl}\le S_{\overline{l}t}{P}_{\overline{l}tms}^{tl,max},\forall \overline{l}\in L_c,t,m,s,\hfill & \hfill \end{array}$$

(14)

$$\begin{array}{cc}{G}_{jtms}^{p,t}\le G_{jtms}^{p,max},\forall j\in J_a,t,m,s,\hfill & \hfill \end{array}$$

(15)

$$\begin{array}{cc}\sum _{j\in J_a}{G}_{jktms}^p=\sum _{\overline{g}}\left({\displaystyle \frac{{\gamma }_{\overline{g}}}{\eta }}\right)P_{\overline{g}tms}^g,\forall k\in K_g,t,m,s,\hfill & \hfill \end{array}$$

(16)

$$\begin{array}{cc}\sum _{\overline{d}:B_d=j}-\left(G_{\overline{d}tms}^{cr}-G_{\overline{d}tms}^{{}^{\prime }cr}\right)+\sum _{\overline{d}:E_d=j}\left(G_{\overline{d}tms}^{cr}-G_{\overline{d}tms}^{{}^{\prime }cr}\right)=G_{jtms}^d+\sum _{k\in K_g}{G}_{jktms}^p-G_{jtms}^{p,t},\forall j\in J_a,t,m,s,\hfill & \hfill \end{array}$$

(17)

$$\begin{array}{cc}\sum _{\overline{d}:B_d=j}-\left(G_{\overline{d}tms}^{cr}-G_{\overline{d}tms}^{{}^{\prime }cr}\right)+\sum _{\overline{d}:E_d=j}\left(G_{\overline{d}tms}^{cr}-G_{\overline{d}tms}^{{}^{\prime }cr}\right)-\sum _{\overline{p}:B_p=j}\left(G_{\overline{p}tms}^{pl}-G_{\overline{p}tms}^{{}^{\prime }pl}\right)\hfill & \hfill \\ +\sum _{\overline{p}:E_p=j}\left(G_{\overline{p}tms}^{pl}-G_{\overline{p}tms}^{{}^{\prime }pl}\right)=0,\forall j\in J_p,t,m,s,\hfill & \hfill \end{array}$$

(18)

$$\begin{array}{cc}{\pi }_{B_{p}tms}-{\pi }_{E_{p}tms}-Y_{\overline{p}}^{}{G}_{\overline{p}tms}^{pl}\left|G_{\overline{p}tms}^{pl}\right|=0,\forall \overline{p}\in P_e,t,m,s,\hfill & \hfill \end{array}$$

(19)

$$\begin{array}{cc}-(1-S_{\overline{p}t})M^{pl}\le {\pi }_{B_{p}tms}-{\pi }_{E_{p}tms}-Y_{\overline{p}}^{}{G}_{\overline{p}tms}^{pl}\left|G_{\overline{p}tms}^{pl}\right|\le (1-S_{\overline{p}t})M^{pl},\forall \overline{p}\in P_c,t,m,s,\hfill & \hfill \end{array}$$

(20)

$$\begin{array}{cc}{S}_{\overline{p}t}=\sum _{i=1}^t{Z}_{\overline{p}i},\forall \overline{p}\in P_c,t,\hfill & \hfill \end{array}$$

(21)

$$\begin{array}{cc}{G}_{\overline{p}tms}^{pl,min}\le -G_{\overline{p}tms}^{{}^{\prime }pl},\forall \overline{p}\in P_e,t,m,s,\hfill & \hfill \end{array}$$

(22)

$$\begin{array}{cc}{G}_{\overline{p}tms}^{pl}\le G_{\overline{p}tms}^{pl,max},\forall \overline{p}\in P_e,t,m,s,\hfill & \hfill \end{array}$$

(23)

$$\begin{array}{cc}{S}_{\overline{p}t}{G}_{\overline{p}tms}^{pl,min}\le -G_{\overline{p}tms}^{{}^{\prime }pl},\forall \overline{p}\in P_c,t,m,s,\hfill & \hfill \end{array}$$

(24)

$$\begin{array}{cc}{G}_{\overline{p}tms}^{pl}\le S_{\overline{p}t}{G}_{\overline{p}tms}^{pl,max},\forall \overline{p}\in P_c,t,m,s,\hfill & \hfill \end{array}$$

(25)

$$\begin{array}{cc}{\pi }_{jtms}\le {\pi }_{j^{\prime }tms},\forall \overline{d}\in D_c,t,m,s,\hfill & \hfill \end{array}$$

(26)

$$\begin{array}{cc}{\pi }_{j^{\prime }tms}\le W_{\overline{d}}{\pi }_{jtms},\forall \overline{d}\in D_c,t,m,s,\hfill & \hfill \end{array}$$

(27)

$$\begin{array}{cc}{\pi }_{j^{\prime }tms}\le {\pi }_{jtms},\forall \overline{d}\in D_r,t,m,s,\hfill & \hfill \end{array}$$

(28)

$$\begin{array}{cc}{\pi }_{jtms}\le W_{\overline{d}}{\pi }_{j^{\prime }tms},\forall \overline{d}\in D_r,t,m,s,\hfill & \hfill \end{array}$$

(29)

$$\begin{array}{cc}-1.57\le {\theta }_{jtms}\le 1.57,\forall j\in J_a,t,m,s,\hfill & \hfill \end{array}$$

(30)

$$\begin{array}{cc}{\pi }_{jtms}^{min}\le {\pi }_{jtms}\le {\pi }_{jtms}^{max},\forall j,t,m,s,\hfill & \hfill \end{array}$$

(31)

$$\begin{array}{cc}{C}_{\overline{g}t},P_{\overline{g}tms}^g,P_{jtms}^{ls},P_{\overline{l}tms}^{tl},P_{\overline{l}tms}^{{}^{\prime }tl}\ge 0,\hfill & \hfill \end{array}$$

(32)

$$\begin{array}{cc}{G}_{jtms}^{p,t},G_{jktms}^p,G_{\overline{p}tms}^{pl},G_{\overline{p}tms}^{{}^{\prime }pl},G_{\overline{d}tms}^{cr},G_{\overline{d}tms}^{{}^{\prime }cr}\ge 0,\hfill & \hfill \end{array}$$

(33)

$$\begin{array}{cc}\mathit{Integer}\mathit{variables}:C_{\overline{g}t}^a,C_{\overline{g}t}^r,\hfill & \hfill \end{array}$$

(34)

$$\begin{array}{cc}\mathit{Binary}\mathit{variables}:S_{\overline{l}t},Z_{\overline{l}t},S_{\overline{p}t},Z_{\overline{p}t}.\hfill & \hfill \end{array}$$

(35)

Integer variables are used for modeling generation additions and retirements; therefore, Equation (2) allows determination of the number of units available at every time step. Equation (3) imposes an upper bound for new generation investments. Equations (4) and (5) limit power and electricity outputs from the units using capacity credit and capacity factor parameters, respectively. In particular, Equation (4) accounts for the tendency of each generator to operate within certain ranges during certain time periods, while Equation (5) accounts for the tendency of each generator to produce over a year a certain fraction of the energy it would produce if it is continuously operated at its capacity during that time frame. Additionally, Equation (6) enforces a reliability criterion by requiring a minimum amount of capacity reserves in each region using a peak power demand value.

Equations (7)–(14) represent the electric system transmission network using a disjunctive approach as proposed in [15]. Equation (7) enforces power balance in each area. Equations (8) and (9) represent the DC power flow equations for the existing and candidate transmission lines, respectively. Equation (10) tracks the investments in transmission lines in every time step. Equations (11)–(14) constrain the amount of power that can be transferred across each transmission line in the electric system. A sufficiently large parameter $M^{tl}$ is required in Equation (9). Although not essential, computational benefit can result from choosing this parameter according to the rationale given in [23] as explained in [24].

Natural gas production is represented using Equations (15) and (16). Equation (15) imposes upper bound limits for gas production levels while Equation (16) enforces a global balance between the amount of gas produced with generation purposes and the gas consumed by each type of generation technology.

Equations (17)–(29) define a disjunctive model for the pipeline network of the natural gas system. In this formulation, every natural gas transmission link includes one or more pipelines (considering existing and candidate) which are coupled with compression and reduction stations located at the start and end nodes, respectively, as illustrated in Figure 1. By using these new elements, pressure variables are adequately controlled across the pipeline network, increasing not only the system operational flexibility but also the accuracy of the investment decisions.

Equations (17) and (18) enforce gas balances in area and pipeline nodes, respectively, while Equations (19) and (20) represent Weymouth equations for existing and candidate pipelines. Similar to the disjunctive model proposed for the electric transmission system, Equation (21) tracks the investments in pipelines, while Equations (22)–(25) limit the amount of gas that can be transferred using the pipeline network. Once again, a large parameter $M^{pl}$ must be used in Equation (20) to relax the pressure differences across the candidate pipelines when these have not been built yet.

Equations (26)–(29) describe the operation of the compression and reduction stations as a function of the pressure variables at the terminal points of the pipelines. For compression stations, the pressure at the receiving node has to be equal to or greater than the pressure at the sending node. In addition, the former cannot exceed the latter scaled by the compression ratio. For reduction stations, the pressure at the sending node has to be equal to or greater than the pressure at the receiving node, and the latter has to be at least equal to the former scaled by the inverse of the compression ratio.

Finally, Equations (30)–(35) establish upper and lower limits for some of the decision variables, impose non-negativity requirements, and define the required binary and integer variables. The proposed model is a MINLP formulation. To facilitate its solution, we approximate it with an MILP formulation as described in the following sections.

3. Co-Optimized Expansion Planning Problem Using Piecewise Linear Functions

The MINLP problem described in the previous section is challenging to solve. We do so by replacing its nonlinear functions with piecewise linear ones that reasonably capture the nonlinearities associated with the original problem. This enables us to take advantage of the maturity of the algorithms developed to solve MILP problems. There are multiple approaches for linearizing bivariate nonlinear functions, one of which is described as follows. Consider a bivariate nonlinear function $h:\mathsf{\Omega }\subseteq {\Re }^2\to \Re$ with $\mathsf{\Omega }=\left\{\left(x,y\right)\setminus x_a\le x\le x_b,y_a\le y\le y_b\right\}$. Partitioning $\mathsf{\Omega }$ as $i\in {\mathsf{\Lambda }}_x=\{1,\dots ,k_x\}$, and $j\in {\mathsf{\Lambda }}_y=\{1,\dots ,k_y\}$, it is possible to represent the value of any point $(x,y)$ enclosed inside the rectangle ${\Delta }_{ij}=\{(x,y)/x_i\le x\le x_{i+1},y_j\le y\le y_{j+1}\}$ using a convex combination of the coordinates associated with its corners. Moreover, it is also possible to approximate h using piecewise linear functions defined for each of multiple rectangles in the domain $\mathsf{\Omega }$. Following this procedure, the surface defined by the equation $h(x,y)=0$ is approximated using a set of rectangular planes joined together by their corners.

In general, Equations (36)–(40) can be used to approximate constraints of the form $h(x,y)=0$ (where h is a nonlinear function), in optimization problems using Special Ordered Set Type 2 (SOS2) variables [25]:

$$\begin{array}{cc}x=\sum _{j\in {\mathsf{\Lambda }}_x}^{}{\lambda }_i^x{x}_i,y=\sum _{j\in {\mathsf{\Lambda }}_y}^{}{\lambda }_j^y{y}_j,\hfill & \hfill \end{array}$$

(36)

$$\begin{array}{cc}h(x,y)\approx \sum _{i\in {\mathsf{\Lambda }}_x}^{}\sum _{j\in {\mathsf{\Lambda }}_y}^{}{\lambda }_{ij}^h{h}_{ij},\hfill & \hfill \end{array}$$

(37)

$$\begin{array}{cc}\sum _{i\in {\mathsf{\Lambda }}_x}^{}{\lambda }_i^x=1,\sum _{j\in {\mathsf{\Lambda }}_y}^{}{\lambda }_j^y=1,\hfill & \hfill \end{array}$$

(38)

$$\begin{array}{cc}\sum _{i\in {\mathsf{\Lambda }}_x}^{}{\lambda }_{ij}^h={\lambda }_i^y\dots \forall j\in {\mathsf{\Lambda }}_y,\sum _{j\in {\mathsf{\Lambda }}_y}^{}{\lambda }_{ij}^h={\lambda }_i^x\dots \forall i\in {\mathsf{\Lambda }}_x,\hfill & \hfill \end{array}$$

(39)

$$\begin{array}{cc}{\lambda }_i^x,{\lambda }_j^y,{\lambda }_{ij}^h\ge 0,{\lambda }_i^x,{\lambda }_j^y\mathit{SOS}\mathit{2}\mathit{variables}.\hfill & \hfill \end{array}$$

(40)

We applied this approach to approximate the Weymouth equations, which are bivariate nonlinear functions relating gas flows to pressure values at the terminals of a pipeline. The MINLP problem presented in Section 2 can be reformulated as an MILP model by replacing Equations (19) and (20) with Equations (41)–(49):

$$\begin{array}{cc}{\pi }_{jtms}=\sum _i\lambda {\pi }_{jtmsi}{\mathsf{\Pi }}_{ji},\forall j,t,m,s,\hfill & \hfill \end{array}$$

(41)

$$\begin{array}{cc}{G}_{\overline{p}tms}^{pl}-G_{\overline{p}tms}^{{}^{\prime }pl}=\sum _i\sum _{i^{\prime }}\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}{g}_{\overline{p}tmsi{i}^{\prime }}^{val},\forall \overline{p},t,m,s,\hfill & \hfill \end{array}$$

(42)

$$\begin{array}{cc}\sum _i\lambda {\pi }_{jtmsi}=1,\forall j,t,m,s,\hfill & \hfill \end{array}$$

(43)

$$\begin{array}{cc}\sum _{i^{\prime }}\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}=\lambda {\pi }_{jtmsi},\forall \overline{p}\in P_e,t,m,s,i,\hfill & \hfill \end{array}$$

(44)

$$\begin{array}{cc}\sum _i\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}=\lambda {\pi }_{jtms{i}^{\prime }},\forall \overline{p}\in P_e,t,m,s,i^{\prime },\hfill & \hfill \end{array}$$

(45)

$$\begin{array}{cc}-(1-S_{\overline{p}t})M^{pl}+\lambda {\pi }_{jtmsi}\le \sum _{i^{\prime }}\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}\le \lambda {\pi }_{jtmsi}+(1-S_{\overline{p}t})M^{pl},\forall \overline{p}\in P_c,t,m,s,i,\hfill & \hfill \end{array}$$

(46)

$$\begin{array}{cc}-(1-S_{\overline{p}t})M^{pl}+\lambda {\pi }_{jtms{i}^{\prime }}\le \sum _i\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}\le \lambda {\pi }_{jtms{i}^{\prime }}+(1-S_{\overline{p}t})M^{pl},\forall \overline{p}\in P_c,t,m,s,i^{\prime },\hfill & \hfill \end{array}$$

(47)

$$\begin{array}{cc}\lambda {\pi }_{jtmsi},\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}\ge 0,\hfill & \hfill \end{array}$$

(48)

$$\begin{array}{cc}\lambda {\pi }_{jtmsi}\mathit{SOS}\mathit{2}\mathit{variables}.\hfill & \hfill \end{array}$$

(49)

Equations (41)–(49) constitute a disjunctive MILP capacity expansion model, representing the natural gas flows using piecewise linear functions via a partition of the pressure variables in the pipeline network. Equations (41) and (42) correspond with Equations (36)–(37) which are linear combinations for calculating pressures and gas flows in the pipeline network, provided the unknown coefficients $\lambda {\pi }_{jtmsi}$ and $\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}$. Equations (43)–(45) correspond with Equations (38) and (39) and establish a set of relations that parameters $\lambda {\pi }_{jtmsi}$ and $\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}$ must satisfy in order to be able to adequately define the set of piecewise linear functions. Equations (46) and (47) enforce Weymouth equations for candidate pipelines. Recalling that $S_{\overline{p}t}$ is a binary variable used to track if candidate pipeline $\overline{p}$ has or has not been installed until period t, therefore, when $S_{\overline{p}t}=1$, Equations (46) and (47) reduce to Equations (44) and (45), and enforce the Weymouth equation for $\overline{p}$. However, if $S_{\overline{p}t}=0$, Equations (46) and (47) reduce to Equations (50) and (51), which in conjunction with Equations (24), (25) and (42) will set variables $\lambda g_{\overline{p}tmsi{i}^{\prime }}^{pl}$ to zero for $\overline{p}$, while $\lambda {\pi }_{jtmsi}$ and $\lambda {\pi }_{jtms{i}^{\prime }}$ still can take nonzero values:

$$\begin{array}{cc}-M^{pl}+\lambda {\pi }_{jtmsi}\le \sum _{i^{\prime }}\lambda g_{dtmsi{i}^{\prime }}^{pl}\le \lambda {\pi }_{jtmsi}+M^{pl},\hfill & \hfill \end{array}$$

(50)

$$\begin{array}{cc}-M^{pl}+\lambda {\pi }_{jtms{i}^{\prime }}\le \sum _i\lambda g_{dtmsi{i}^{\prime }}^{pl}\le \lambda {\pi }_{jtms{i}^{\prime }}+M^{pl}.\hfill & \hfill \end{array}$$

(51)

Therefore, it is possible to approximate the MINLP problem presented in Section 2 with an MILP by using Equations (41)–(49) instead of Equations (19) and (20).

4. Alternative Formulation Using Piecewise Linear Functions

Although the MILP formulation presented in Section 3 is a novel approximation for solving the co-optimized expansion planning problem for an integrated electric-gas system, it is also a computationally intensive one (it requires an SOS2 variable for each $\lambda {\pi }_{jtmsi}$ coefficient), and therefore for large co-optimization problems, particularly ones with large number of candidate transmission lines (and therefore large number of integer variables), it requires significant computation just to find a feasible solution. We address this by developing a different MILP approximation.

It is possible to reduce the number of SOS2 variables required by representing the gas flow terms in the Weymouth equations directly as piecewise linear functions [26], as presented in Equations (52)–(54). This means that Equations (21)–(25) in conjunction with Equations (52)–(58) will constitute an alternative disjunctive formulation for a pipeline network; however, it is also a less accurate representation because of the linear approximation used in Equation (53):

$$\begin{array}{cc}{G}_{\overline{p}tms}^{pl}-G_{\overline{p}tms}^{{}^{\prime }pl}=\sum _i\lambda g_{\overline{p}tmsi}^{pl}{g}_{\overline{p}tmsi}^{val},\forall \overline{p},t,m,s,\hfill & \hfill \end{array}$$

(52)

$$\begin{array}{cc}{\mathsf{\Gamma }}_{\overline{p}tms}^{pl}=\sum _i\lambda g_{\overline{p}tmsi}^{pl}{\left(g_{\overline{p}tmsi}^{val}\right)}^2,\forall \overline{p},t,m,s,\hfill & \hfill \end{array}$$

(53)

$$\begin{array}{cc}\sum _i^{}\lambda g_{\overline{p}tmsi}^{pl}=1,\forall \overline{p},t,m,s,\hfill & \hfill \end{array}$$

(54)

$$\begin{array}{cc}{\pi }_{B_{p}tms}-{\pi }_{E_{p}tms}=Y_{\overline{p}}^{}{\mathsf{\Gamma }}_{\overline{p}tms}^{pl},\forall \overline{p}\in P_e,t,m,s,\hfill & \hfill \end{array}$$

(55)

$$\begin{array}{cc}-(1-S_{\overline{p}t})M^{pl}\le {\pi }_{B_{p}tms}-{\pi }_{E_{p}tms}-Y_{\overline{p}}^{}{\mathsf{\Gamma }}_{\overline{p}tms}^{pl}\le (1-S_{\overline{p}t})M^{pl},\forall \overline{p}\in P_c,t,m,s,\hfill & \hfill \end{array}$$

(56)

$$\begin{array}{cc}\lambda g_{\overline{p}tmsi}^{pl}\ge 0,\forall \overline{p},t,m,s,\hfill & \hfill \end{array}$$

(57)

$$\begin{array}{cc}\lambda g_{\overline{p}tmsi}^{pl}\mathit{SOS}\mathit{2}\mathit{variables}.\hfill & \hfill \end{array}$$

(58)

5. Results

5.1. Model Results

The 26 node electric system described in [27] is modified and extended by creating a more disaggregated dataset for the generation plants, transmission lines and pipelines, using the additional information resources provided in [28] to represent an integrated gas-electric system for the region of the Eastern Interconnection in the United States, which in turn is used to demonstrate the capabilities of the proposed formulation in a 20-year simulation. The optimization problem is formulated using the software GAMS, and it is optimized using the solver CPLEX.

A high renewable scenario is created by imposing carbon emission costs to fossil fuel generation, restricting new investments in nuclear plants and allowing significant developments of wind farms. Final generation capacity values are summarized in

Table 1. Final generation capacity values (GW).

Region	Generation Technologies
	Coal	Gas	Hydro	Nuclear	Others	Wind
ENT	0.7	24.6	1.9	5.3	0.3	0.1
FRCC	0.5	28.3	0.1	24.4	0.9	0.0
HQ	0.0	0.0	4.8	0.0	0.0	0.0
IESO	0.0	3.6	9.1	11.5	1.5	5.0
MAPP_CA	0.0	2.7	7.2	0.0	0.0	0.6
MAPP_US	0.0	2.1	2.2	0.5	0.2	9.9
MISO_IN	0.6	6.4	0.1	0.0	0.1	11.5
MISO_MI	0.0	24.9	2.0	1.9	0.5	16.0
MISO_MO-IL	1.8	6.9	1.1	2.2	0.5	28.3
MISO_W	0.0	16.6	0.5	2.3	0.5	136.2
MISO_WUMS	0.5	9.8	0.3	1.6	0.2	3.3
NB	0.0	0.0	0.0	0.0	0.0	0.0
NE	0.2	7.1	0.2	1.3	0.1	17.9
NEISO	0.0	10.5	6.3	4.6	1.6	4.9
NonRTO_Midwest	0.0	11.5	0.9	0.0	0.1	0.1
NYISO_A-F	0.0	1.8	5.8	3.2	0.6	6.7
NYISO_G-I	0.0	2.7	0.0	2.0	0.3	0.2
NYISO_J-K	0.0	14.7	0.0	0.0	0.6	0.0
PJM_E	0.0	19.7	0.7	8.5	0.7	8.6
PJM_ROM	0.5	7.9	3.6	5.0	0.7	18.1
PJM_ROR	2.2	72.1	5.4	20.0	1.5	19.1
SOCO	0.5	35.0	6.7	14.5	0.5	0.0
SPP_N	0.5	11.3	0.0	1.2	0.1	41.6
SPP_S	1.0	20.2	2.6	0.0	0.1	43.5
TVA	1.0	19.6	6.9	7.9	0.3	0.0
VACAR	1.3	23.9	4.8	13.8	0.4	30.0

, respectively, aggregated under six different technology groups (biomass, geothermal, landfill gas, solar, and oil/wood steam turbines are grouped under the label Others). Large investments in gas and wind technologies can be observed in the simulation results. New gas plants are built close to the new shale areas, or to areas connected to them using pipelines. Wind generation is developed strongly in the Midwest region, mainly due to the high capacity factor values of the technology in this area.

Initial transmission capacities and new investments in lines and pipelines are presented in

Table 2. Transportation initial capacity and investments.

Link	Lines (GW)		Pipes (MMcf/h)
	Initial	Invest.	Initial	Invest.
ENT to MISO_MO-IL	2.5	0.0	265.6	63.0
ENT to SOCO	2.4	0.0	312.5	0.0
ENT to SPP_N	7.3	0.0	NA	NA
ENT to SPP_S	4.3	2.0	208.3	0.0
ENT to TVA	3.0	0.0	265.6	0.0
FRCC to SOCO	3.7	0.0	215.0	0.0
HQ to NEISO	NA	NA	8.3	0.0
IESO to MAPP_CA	0.3	0.0	187.5	0.0
IESO to MISO_MI	3.1	4.0	125.0	63.0
IESO to MISO_W	0.2	0.0	NA	NA
IESO to NEISO	NA	NA	31.3	21.0
IESO to NYISO_A-F	2.2	0.0	NA	NA
MAPP_CA to MAPP_US	0.4	0.0	156.3	0.0

and

Table 3. Transportation initial capacity and investments (continuation).

Link	Lines (GW)		Pipes (MMcf/h)
	Initial	Invest.	Initial	Invest.
MAPP_CA to MISO_W	2.0	0.0	93.8	0.0
MAPP_US to MISO_W	2.6	0.0	156.3	0.0
MAPP_US to NE	2.0	0.0	NA	NA
MISO_IN to MISO_MI	5.5	0.0	41.7	0.0
MISO_IN to MISO_MO-IL	5.0	24.0	156.3	0.0
MISO_IN to NonRTO_Midwest	4.8	0.0	NA	NA
MISO_IN to PJM_ROR	1.0	12.0	187.5	0.0
MISO_MI to MISO_MO-IL	NA	NA	104.2	0.0
MISO_MI to MISO_WUMS	0.3	0.0	NA	NA
MISO_MI to PJM_ROR	1.4	2.0	41.7	0.0
MISO_MO-IL to MISO_W	4.0	34.0	NA	NA
MISO_MO-IL to MISO_WUMS	NA	NA	62.5	0.0
MISO_MO-IL to NE	NA	NA	62.5	0.0
MISO_MO-IL to PJM_ROR	1.2	0.0	NA	NA
MISO_MO-IL to SPP_N	4.0	2.0	114.6	0.0
MISO_MO-IL to TVA	4.0	10.0	NA	NA
MISO_W to MISO_WUMS	1.7	6.0	93.8	0.0
MISO_W to NE	3.6	0.0	133.3	0.0
MISO_W to PJM_ROR	19.8	0.0	NA	NA
MISO_W to SPP_N	3.2	0.0	NA	NA
MISO_WUMS to PJM_ROR	1.6	0.0	NA	NA
NE to SPP_N	1.9	2.0	NA	NA
NEISO to NYISO_A-F	0.6	0.0	93.8	0.0
NEISO to NYISO_G-I	0.6	0.0	NA	NA
NEISO to NYISO_J-K	1.0	0.0	NA	NA
NonRTO_Midwest to PJM_ROR	NA	NA	156.3	42.0
NonRTO_Midwest to TVA	2.4	0.0	166.7	0.0
NYISO_A-F to NYISO_G-I	5.3	2.0	20.8	0.0
NYISO_A-F to NYISO_J-K	NA	NA	20.8	42.0
NYISO_A-F to PJM_ROM	2.0	0.0	187.5	0.0
NYISO_G-I to NYISO_J-K	6.1	2.0	NA	NA
NYISO_G-I to PJM_E	1.5	0.0	NA	NA
NYISO_J-K to PJM_E	0.3	0.0	NA	NA
PJM_E to PJM_ROM	8.0	0.0	125.0	0.0
PJM_ROM to PJM_ROR	8.0	0.0	250.0	0.0
PJM_ROR to TVA	2.5	0.0	NA	NA
PJM_ROR to VACAR	3.0	0.0	125.0	189.0
SOCO to TVA	3.2	10.0	NA	NA
SOCO to VACAR	3.0	0.0	156.0	126.0
SPP_N to SPP_S	4.0	2.0	NA	NA
TVA to VACAR	0.9	0.0	NA	NA

(an NA label means that the transmission link is not represented either as an existing or as a candidate element in the model). Large investments in transmission lines to move the electricity produced by the new wind farms from the Midwest region to the main load centers located in the East Coast are reported in the results, as well as significant builds on new pipelines to move shale gas resources to regions with high gas generation capacities.

A high level and aggregated design for an integrated gas-electric transmission network according to the co-optimization results is depicted in Figure 2. Final generation capacities aggregated by super regions (as described in

Table 4. Super regions’ description.

Super Regions	Areas
Canada	HQ, IESO, MAPP_CA
MISO	MISO (IN, MI, MO-IL, W, WUMS)
North	NEISO, NYISO (A-F, G-I, J-K)
PJM	PJM (E, ROM, ROR)
West	MAPP_US, NE, SPP (N, S)
South	ENT, FRCC, NonRTO Midwest, SOCO, TVA, VACAR

) for three of the six generation group technologies (coal, gas and wind) are also presented. The figure illustrates strong correspondence between the required investments in generation, transmission lines and pipelines. This design proposes a robust electric transmission grid configured as a ring between MISO and PJM to move the large amounts of electricity produced in the new wind farms. It also requires a robust pipeline network connecting the regions South, PJM and North to transport the shale gas in the south-north direction.

5.2. Comparative Results

Table 5. Cost comparison (dollars).

Costs		Model 1	Model 2	Model 3
O&M	Generation	$9.850\times {10}^{11}$	$9.684\times {10}^{11}$	$1.008\times {10}^{12}$
	Lines	$2.147\times {10}^{10}$	$2.008\times {10}^{10}$	$1.562\times {10}^{10}$
	Pipes	–	$5.318\times {10}^9$	$1.075\times {10}^{10}$
Investment	Generation	$7.953\times {10}^{11}$	$7.815\times {10}^{11}$	$7.834\times {10}^{11}$
	Lines	$4.536\times {10}^{10}$	$4.677\times {10}^{10}$	$1.994\times {10}^{10}$
	Pipes	–	$4.382\times {10}^9$	$1.341\times {10}^{10}$
Other	Emissions	$1.438\times {10}^{11}$	$1.458\times {10}^{11}$	$1.745\times {10}^{11}$
	End Horizon	$-4.56\times {10}^{11}$	$-4.52\times {10}^{11}$	$-4.58\times {10}^{11}$
Total Cost		$1.535\times {10}^{12}$	$1.521\times {10}^{12}$	$1.568\times {10}^{12}$

,

Table 6. Final capacity comparison.

Capacity	Unit	Model 1	Model 2	Model 3
Lines	GW	272	272	266
Pipes	MMcf/h	–	4456	4871
Generation	GW	1055	1049	1014
Coal	GW	9	9	11
Gas	GW	397	395	384
Hydro	GW	75	75	73
Nuclear	GW	122	124	132
Offshore Wind	GW	–	–	55
Onshore Wind	GW	442	436	347
Others	GW	11	11	12

and

Table 7. Simulation parameters’ comparison.

Parameter	Model 1	Model 2	Model 3
Simulation time (hh:mm:ss)	1:35:06	1:57:57	197:54:10
Relative gap	0.001757	0.001871	0.014033

compare the results from three different long-term capacity expansion models: a Pipes and Bubbles approach (also known as a transportation model) for an electric system (Model 1); a Pipes and Bubbles approach for an integrated electric-gas system (Model 2); and the disjunctive model for an integrated electric-gas system developed throughout this paper (Model 3). The comparison between the results obtained for models 1 and 2 explains the impact of considering the pipeline network in the co-optimization problem; similarly, the effect of modeling the steady state flow equations for electricity and natural gas is pointed out by comparing models 2 and 3.

The inclusion of the pipeline network in Model 2 allows the decrease of the total cost of the co-optimization problem when compared with Model 1. This reduction is mainly due to a better utilization of the available gas resources between areas, and the decrease of the O&M costs for the generation plants because of a more economical selection of the investments. This is not necessarily the best environmental choice. Notice that, under this investment plan, some wind farms are replaced by gas-fired units, which marginally increase the penalties associated with carbon dioxide emissions.

As expected, Model 3 has the highest total cost. From a mathematical point of view, this phenomenon is explained by the increased number of constraints. From an operational point of view, the addition of the steady state flow equations limits the amount of electricity and natural gas that can be sent through the transmission links, requiring more investments in generation and transmission, and increasing the O&M costs. Therefore, in order to optimize system operation under the new constraints, some of the investments in new wind farms and gas-fired plants must be replaced by other generation technologies, and some new transmission lines are replaced by new pipelines.

Finally, simulation times for Model 3 are considerably high. This is due to the computation requirements when modeling the Weymouth equations as a set of piecewise linear approximations, especially for candidate pipelines.

6. Conclusions

The co-optimized analysis and design of electric and natural gas infrastructures enable identification of less costly investment alternatives, highlighting the relevance of developing new planning procedures and tools that guarantee a systematic approach for the integrated system. This issue is particularly important during a time of major expansions and even more, considering the longevity of the infrastructure to be built.

Additional modeling characteristics could be addressed to improve the problem formulation and the simulation results (e.g., losses in the transportation networks), while managing adequately the trade-off between results’ accuracy and computational requirements. Sensitivity analysis of the different models could also be considered as future research in order to identify relevant model parameters.

In our work, we have developed the computational tools to facilitate gas-electric expansion planning. However, there will also need to be coordinating bodies and procedures developed because, at this point in time, the organizations that plan and build natural gas infrastructure are completely different from the organizations that plan and build electric infrastructure. Finally, because these infrastructures are interregional, decision-making processes will benefit from coordination at the national level.