Derivation and Generation of Path-Based Valid Inequalities for Transmission Expansion Planning

# Derivation and Generation of Path-Based Valid Inequalities for Transmission Expansion Planning

J. Kyle Skolfield Laura M. Escobar111Her work is supported by the Brazilian institutions CAPES, CNPq (Grant NO. 142150/2015-0) and São Paulo Research Foundation–FAPESP (Grant NO. 2015/21972-6). Adolfo R. Escobedo School of Computing, Informatics, and Decision Systems Engineering (CIDSE), Arizona State University, Tempe, Arizona Electrical Engineering Department, São Paulo State University (UNESP), Ilha Solteira, São Paulo, Brazil
###### Abstract

This paper seeks to solve the long-term transmission expansion planning problem more effectively by reducing the solution search space and the computational effort. The proposed methodology finds and adds cutting planes based on structural insights about bus angle-differences along paths. Several theorems are proposed which show the validity of these cutting planes onto the underlying mathematical formulations. The path-based bus angle-difference constraints, which tighten the relaxed feasible region, are used in combination with branch-and-bound to find lower bounds on the optimal investment of the transmission expansion planning problem. This work also creates an algorithm that automates the process of finding and applying the discussed valid inequalities, resulting in significantly reduced testing and computation time. The algorithm is implemented in Python, using the solver CPLEX to add constraints and solve the transmission expansion problem. This paper uses two different-sized systems to illustrate the effectiveness of the proposed framework: a modified IEEE 118-bus system and the Polish 2383-bus system.

###### keywords:
OR in energy, Mathematical Modeling, Mixed-integer Linear Programming, Transmission Expansion Planning, Valid Inequalities.
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## Nomenclature

Sets :

• Buses

• Corridors

• Lines in each corridor

Parameters:

• Cost of line in corridor

• Per unit cost of generation at bus

• Number of established lines in corridor

• Max number of lines in corridor

• Maximum limit of power generation at bus

• Active power demand at bus

• Maximum bus angle-difference magnitude

• Capacity of candidate line in corridor

• Capacity of existing line in corridor

• Reactance of line in corridor

• Susceptance of line in corridor

• Maximum limit for index in corridor

• Large number (big-M) used in the disjunctive constraints

• Scaling factor to align generation and expansion costs

Continuous Variables:

• Active power flow in existing line in corridor

• Active power flow in candidate line in corridor

• Active power output of generator in bus

• Voltage angle at bus

Binary Variables:

• Decision to construct the candidate line in corridor

## 1 Introduction

### 1.1 Background

The objective of the Transmission-network Expansion Planning (TEP) problem is to find the least costly investment options in new transmission devices required to ensure proper power system operations into the future Garver (1970). Optimizing this problem is important because the transmission network belongs to the so-called heavy technologies, which are both expensive and difficult to withdraw or relocate once they are installed Dominguez (2017). Inadequate long-term planning can lead to low service quality, excessive oversizing, inefficient systems with high operating costs, and delays in the expansion of electricity markets. While new systems are growing in size and the demands imposed on them are increasing, deregulation and other challenges have made meeting those requirements ever more difficult Lumbreras and Ramos (2016). Hence, it is critical to obtain solutions that maximize cost efficiency to enable the incorporation of more avant-garde technologies into the smart grid. For these reasons, it is necessary to devise new planning methodologies that can effectively deal with the associated combinatorial difficulties of the underlying TEP optimization models.

In its standard form, TEP consists of linear and non-linear functions that include continuous variables (e.g., voltage angles, power flows, etc.) and integer variables (decisions to, e.g., add lines to the network). Accordingly, TEP is a non-convex, mixed-integer nonlinear programming problem. It is NP-complete, which makes its solution generally intractable Latorre et al. (2003). This is exacerbated by the fact that in large-scale systems, the number of network components and associated restrictions can number in the hundreds or thousands. That is, the size and/or topology of the transmission network and the inclusion of discrete variables for representing possible transmission investments lead to a combinatorial explosion of potential solutions. Due to these complications, TEP cannot practically be solved using standard optimization techniques, in general. Different modeling techniques and algorithms have been proposed to expedite solution times (e.g., Haghighat and Zeng (2018); Da Silva et al. (2001); Cabrera et al. (2018); Choi et al. (2006); Wickramarathna and Wickramaarachchi (2006)). Exact methods require larger calculation times when compared to those required by metaheuristic techniques such as Tabu Search Gallego et al. (1998); García-Martínez et al. (2015) and Genetic Algorithms Gallego et al. (1998); de Oliveira et al. (2005), among others. However, the latter techniques generally do not provide formal optimality guarantees. In small- and medium-sized systems, the ideal solution can be found using methods such as branch-and-bound or branch-and-cut when a disjunctive integer linear programming model approximation is utilized Bahiense et al. (2001); Sousa and Asada (2011); Di et al. (2013). Such methods provide formal guarantees, but they are demanding computationally. Such methods also include decomposition techniques, such as hierarchical Benders decomposition (e.g. Romero and Monticelli (1994); Binato et al. (2001); Haffner et al. (2001)). Additionally, recent work has used Bender’s decomposition techniques to solve generation and transmission expansion planning together Jenabi et al. (2015). The valid inequalities presented in this paper can be seen as a complementary technique for solution time reduction to these exact methods.

### 1.2 Aim and Contributions

This work considers a mixed-integer programming version of the static TEP problem, which consists of a single investment period occurring at the beginning of the planning horizon. The choice of this model helps illustrate the computational intractability of TEP even for this simplified context. Moreover, it highlights the potential of the fundamental insights introduced herein to be extended to a variety of more complex TEP models with a similar core structure (e.g. Binato et al. (2001), Ploussard et al. (2017), Vinasco et al. (2011), etc.). Explicitly, this work derives and implements a set of theoretical contributions for detecting and including structural information on the underlying network which is relevant to any DCOPF-based model that incorporates the linear relationship between bus angle-differences and power flows (i.e. constraints) into the constraint set. This structural information is common to many power system formulations, so the insights presented in this paper may be applied to aid in solving a variety of problem classes. Such insights are captured via the concept of valid inequalities, which represent one of the most effective solution techniques and are a highly active research area in mathematical programming Nemhauser and Wolsey (1988).

In addition to these theoretical contributions, this work also provides techniques for applying the theory in the form of a heuristic algorithm used to help find the more effective candidate valid inequalities (also referred to herein as cuts). These techniques are then used to perform computational experiments that show the effectiveness of the proposed valid inequalities in reducing the solution time of two benchmark instances. While their effectiveness is shown only for single investment static TEP, the reduction in solution time would be amplified in, for example, stochastic programming approaches to TEP, where many scenarios need to be solved and can use the same collection of valid inequalities since they are based only on the structure of the network.

Other papers have explored the structural insights based on bus angle-differences, which serve as the inspiration of this work. In particular, in Escobar and Romero. (2017), angular cuts were introduced and applied in an ad hoc manner, applying only a subset of the classes of valid inequalities introduced in this paper (in particular, only from Theorem 1 and Theorem 2, which had not been proven). It lacked the systematic and theoretical contributions presented in this paper. This paper automates and extends the process of that work as well as formally establishes the validity of those two classes of valid inequality plus two additional classes.

The structure of the paper is as follows: Section 2 introduces the disjunctive model used for modeling TEP in this work. Section 3 presents the key insights and intuition for deriving and generating the valid inequalities. Section 4 contains the main contribution of this work, the theorems which prove the validity of the discussed cuts. Section 5 presents numerical results from testing the application of these theorems to three different test cases, and Section 6 summarizes the conclusions drawn from these results.

## 2 Modeling Framework

The nonlinear ACOPF model for TEP can be transformed into a mixed-integer linear model with bilinear equations Zhang (2013). This model is transformed into a disjunctive model with binary variables, which is always possible using a large enough disjunctive coefficient (big-M). In the disjunctive model (DM), a binary variable is considered for each candidate line, which converts the original mixed-integer non-linear program into a mixed-integer linear program (MILP). The DCOPF-based model, which is appropriate for TEP due to its focus on long-term, steady-state planning Lumbreras and Ramos (2016), is as follows.

The objective function (1) is to minimize the joint cost of investments in new lines, with investment considered to be performed at the beginning of the planning horizon, and generation costs, weighted by a factor to make generation costs and planning costs comparable as in Mínguez et al. (2018):

 min∑(i,j)∈ΩKij∑k=1cij,kyij,k+∑n∈Bσcngn. (1)

Here, is the cost of each line in corridor and represents the decision to add the candidate line in corridor . When the binary variable , the th candidate line is added in corridor . Additionally, is the maximum number of binary variables considered in corridor , and is the set of expansion corridors in the expansion plan. The set of constraints is as follows:

 \mathrlap∑(l,i)∈Ω⎛⎜⎝ω0ij∑k=1P0li,k+K∑k=1Pli,k⎞⎟⎠−∑(i,l)∈Ω⎛⎜⎝ω0ij∑k=1P0il+K∑k=1Pil,k⎞⎟⎠+gn=dn ∀n∈B (2) −¯¯¯¯P0ij,k≤P0ij,k≤¯¯¯¯P0ij,k ∀(i,j)∈Ω,k∈{1…ω0ij} (3) −¯¯¯¯Pij,k≤Pij,k≤¯¯¯¯Pij,k ∀(i,j)∈Ω,k∈{1…Kij} (4) --1bij,kP0ij,k−(θi−θj)=0 ∀(i,j)∈Ω,k∈{1…ω0ij} (5) \mathrlap--M(1−yij,k)≤--1bij,kPij,k−(θi−θj)≤M(1−yij,k) ∀(i,j)∈Ω,k∈{1…Kij} (6) Kij∑k=1yij,k+ω0ij≤¯ωij ∀(i,j)∈Ω (7) gn≤¯¯¯gn ∀n∈B (8) −¯¯¯θ≤θi−θj≤¯¯¯θ ∀(i,j)∈Ω (9) yij,k∈{0,1} ∀(i,j)∈Ω,k∈{1…Kij} (10) gn≥0,θnunr. ∀n∈B (11) P0ij,k,Pij,kunr. ∀(i,j)∈Ω,k∈{1…Kij} (12)

The constraints start with (2), which interrelates the active power flows that arrive at and leave bus through both existing and candidate lines and the demand and supply of active power at bus . (3) represents the limit of active power flow through the current network in corridor , where is the power flow in existing line . (4) represents the limit of the active power flow through the candidate lines in corridor , while (5) and (6) show the link between the active power flows of a corridor and the bus angle-difference between adjacent buses and . Equations (5) and (6) both represent Kirchhoff’s second law, either for each of existing lines or each of candidate lines to be added to the transmission system, respectively. (6) becomes an active constraint within the model when the decision variable takes the value of 1, i.e. when that candidate line is built. Otherwise, the big-M parameter ensures that (6) is extraneous for the model. Finding the best value for the big-M parameter is itself an NP-hard problem Binato et al. (2001). Therefore, the value used in this model is , which has been shown to be effective in Vinasco et al. (2011). (7) limits the number of candidate lines allowed in each corridor. (8) presents the limits of the active power supply for the generators, where any bus with no generator is assumed to have . (9) enforces the maximum bus angle-difference for adjacent bus-pairs , i.e. those bus-pairs connected by a corridor. Finally, (10), (11) and (12) give the variable domains.

## 3 Motivating the derivation and generation of path-based valid inequalities

Due to the combinatorial explosion of TEP, it is not possible to find an optimal solution for large-scale systems using standard, off-the-shelf algorithms. The computational difficulty of the problem is related directly to the size of the system to be analyzed. However, other factors increase computational difficulty, including the connectivity of the buses or how well the system is enmeshed. This leads to the “Braess Paradox,” according to which a more efficient system can be obtained when removing lines from the transmission system O’Neill et al. (2005).

To solve NP-complete problems, it is often useful to investigate the structural characteristics of a particular instance. This knowledge can be highly valuable when it comes to designing effective solution methods Nemhauser and Wolsey (1988). One key application of this knowledge is to derive valid inequalities (VIs): additional problem constraints that preserve the original solution space but may otherwise reduce an associated relaxed solution space , where . Formally, for the set , the coefficient vector , and the constant , the inequality is called a valid inequality for if it is satisfied by all points (i.e., herein, is the TEP solution space). Because the solution of MILP typically proceeds by solving a sequence of linear relaxations, adding structurally useful VIs as cutting planes can reduce the number of such linear problems solved in a branch-and-bound framework, thus decreasing the computational time necessary to solve the overall problem Nemhauser and Wolsey (1988). The proposed method seeks to provide mechanisms that reduce the size of the solution space by incorporating the structural information of TEP that can eliminate unpromising settings of decision variables.

The structural insights derived in this work stem from the relationships between the bus angle and flow decision variables that characterize DCOPF-based transmission system models. Specifically, if there is an existing line with index in corridor , with , a angular VI relating the difference between and (the angles of adjacent buses ) can be obtained through (the flow along the line), as follows:

 θi−θj=−1bij,kPij,k=xij,kPij,k≤xij,k¯Pij,k, (13)

where and are the line reactance and flow capacity, respectively. The right hand side of this inequality is referred to henceforth as a capacity-reactance product and may be useful for improving angular VIs as presented here. This VI is a direct result of (4)-(6). The present work leverages such adjacent-bus VIs to derive formal restrictions on non-adjacent buses and on buses connected via multiple paths in the network. That is, the TEP model provides only simple angular constraints for the buses that are directly connected via a transmission line. However, by forming a single path connecting adjacent buses in the transmission network, these VIs can be combined into more insightful path-based constraints relating the initial bus angle and the terminating bus angle of said path and the corresponding flow restrictions of each corridor along the path. Even stronger restrictions may be obtained from the combination of VIs along parallel paths—two otherwise disjoint paths which share initial and terminating buses—by taking the tighter of the separate bus angle-difference restrictions or, equivalently, flow restrictions. An example application of these insights is illustrated in Figure 1 via a stylized bus-line diagram consisting of bus set , corridor set , and single lines between each pair of buses with reactances and capacities . For this simple example, and for all future numerical examples, we assume there can be at most one existing line and at most one candidate line per corridor. This allows us to drop the third index to increase visual clarity.

In Figure 1, three path-based VIs (adjacent to each transmission line) are obtained by considering the capacity-reactance products of every pair of buses in the network (see (13)). Moreover, by combining two of these VIs, a tighter VI for bus angles and is obtained (see the boxed expression). It is important to remark that this constraint would be valid even in the absence of a direct transmission line between and , i.e. if it were an expansion corridor. In larger networks, many such VIs can be constructed, which may or may not tighten the model’s simple bus angle-difference constraints. Indeed, in electric systems with high mesh levels, the number of parallel paths can increase exponentially Kavitha et al. (2009). Consequently, it may be prohibitive to identify and verify the strength of each possible VI for large-size systems. Instead, it will be expedient to identify the most effective of these constraints and to provide data-driven insights through the use of relaxation models that are easier to solve.

We use the above ideas to generate a set of structurally useful VIs based on single paths and parallel paths that may appear in the solution to TEP. To this end, we make use of three relaxed models. By solving a subset of these models, each of which takes significantly less time to solve than the full MILP, we can generate a set of structural backbones. These are solution patterns that suggest single paths and parallel paths that are more likely to occur than others in the solution to the original problem. In particular, we consider adding a VI based on any single path or collection of parallel paths which flows in the same direction in the solution to each of a combination of relaxation models. The technique of using these relaxation models in this way will be denoted the low-effort heuristic, first implemented in a non-algorithmic way in Escobar and Romero. (2017). The models used are: the linear model, where the restriction on the binary variables is relaxed, allowing them to be continuous within the interval ; the transportation model where the restriction that flows on all lines obey (5) and (6) are relaxed; and the hybrid model, which is similar to the transportation model, but in which only (6) is relaxed.

## 4 Path-based Angular Valid Inequalities Derivation Theorems

This section will introduce the theorems which are the main contribution of this work. For this purpose, a graph with candidate lines (dotted lines) and existing lines (continuous lines) is presented in Figure 2. An example of these lines can be seen between buses and , where there is one candidate line and one existing line. This graph will be used to illustrate an application of each theorem.

We say then that is an established corridor of if ; otherwise we say that is an expansion corridor. To better clarify instances when we must distinguish individual lines for each corridor along a path, we introduce the term to denote any vector of valid line-indices available for each corridor along a path . Then, for ease of presentation we refer for example to , where encapsulates a valid setting of element of vector , i.e. . Specifically, in each upcoming theorem and proof, whenever is used as a line index, it is shorthand for when there is no ambiguity. Likewise, a fixed line will be used as shorthand for when there is no ambiguity. Additionally, because these problems traditionally specify corridors from a lower index bus to a higher index bus, we define , where if and if . Define and analogously for .

### 4.1 Single path over established corridors

#### Theorem 1

Let represent a directed path over established corridors in . Set coefficient vector as,

 πj={∑|ρ|n=1xin−1in,k––⋅¯¯¯¯Pin−1in,k––,if j=0sgn(ij−1−ij),otherwise, (14)

where is fixed for each corridor , but may vary between corridors. Then the following two-sided inequality is valid for TEP for any :

 −π0≤|ρ|∑j=1πjxij−1ij,k~Pij−1ij,k≤π0. (15)

Proof. According to (5), the flow along any fixed, existing line of corridor is given by:

 ~P0ij−1ij,k––=sgn(ij−ij−1)bij−1ij,k––(θij−θij−1) (16)

where . Hence, the bus angle-difference for consecutive bus-pairs in can be written as:

 θi1−θi0=sgn(i0−i1)xi0i1,k––~P0i0i1,k––=π1xi0i1,k––~P0i0i1,k––,θi2−θi1=sgn(i1−i2)xi1i2,k––~P0i1i2,k––=π2xi1i2,k––~P0i1i2,k––,⋮θi|ρ|−θi|ρ|−1=sgn(i|ρ|−1−i|ρ|)⋅xi|ρ|−1i|ρ|,k––⋅~P0i|ρ|−1i|ρ|=π|ρ|xi|ρ|−1i|ρ|,k––⋅~P0i|ρ|−1i|ρ|,k––

When these equations are summed, this creates a telescoping effect on the left-hand side, which yields the following bus angle-difference equation for the starting and ending buses in :

 θi|ρ|−θi0 =|ρ|∑j=1πjxij−1ij,k––~P0ij−1ij,k–– (17) ≤|ρ|∑j=1∣∣πjxij−1ij,k––~P0ij−1ij,k––∣∣ (18) ≤|ρ|∑j=1xij−1,ij⋅¯¯¯¯Pij−1ij,k––=π0 (19)

where the latter inequality is obtained by adding the rightmost inequalities from (3). By a similar argument we have that,

 |ρ|∑j=1πjxij−1ij,k––~P0ij−1ij,k–– ≥−|ρ|∑j=1∣∣πjxij−1ij,k––~P0ij−1,ij∣∣ (20) ≥−|ρ|∑j=1xij−1,ij⋅¯¯¯¯Pij−1ij,k–– (21) =−π0, (22)

Since every corridor considered has at least one existing line to select and fix as , and (16) holds for any line in corridor , we have established the validity of (15).

As an example using Figure 2, the path is an established path, which creates the example type 1 two-sided VI:

On the same note, in Figure 2 the path creates the example type 1 two-sided VI:

### 4.2 Parallel paths over established corridors

#### Theorem 2

Let represent alternative directed paths over established corridors in with the same starting/ending buses but with non-overlapping intermediate buses; that is, , , and for with . Setting coefficient vectors according to (14) for each path , the following two-sided inequalities are valid for TEP for any :

 −min{πk0}mk=1 ≤|ρr|∑j=1πrjxirj−1irj,kr~Pirj−1irj,kr ≤min{πk0}mk=1for r=1,…,m. (23)

Proof. Since paths and share the same starting/ending buses, this gives that , or equivalently,

 |ρr|∑j=1πrjxirj−1irj,k––~P0irj−1irj,k––=|ρr′|∑j=1πr′jxir′j−1ir′j,k––~P0ir′j−1ir′j,k––, for 1≤r, r′≤mwithr≠r′

according to the respective telescoped bus angle-difference equations of the starting and ending buses associated with each path (e.g., see (17)). Thus, the proof is completed by joining together the two-sided inequalities,

 −πr0≤|ρr|∑j=1πrjxirj−1irj,k––~Pirj−1irj,k––≤πr0for r=1,…,m,

each of which is valid due to Theorem 1.

Continuing the example from subsection IV.A, in Figure 2, creates an established parallel path with . Assuming that path is the path with low capacity-reactance product creates the example type 2 two-sided VI:

### 4.3 Single path over established and expansion corridors

#### Theorem 3

Let represent a directed path over established and expansion corridors in . Additionally, let refer to any fixed line in each established corridor, and in expansion corridors to the candidate line with the maximum capacity-reactance product. By including auxiliary variable and the logic-enforcing constraints,

 z ≥|ρ|∑j=1Ijyij−1ij,k−|ρ|∑j=1Ij+1∀^kρ (24) z ≤Kij∑k=1yij−1ij,kfor j=1..|ρ| s.t. Ij=1 (25)

where is used as shorthand for the indicator function (i.e. to identify expansion corridors). Then the following two-sided inequality is valid for TEP for any :

 −M(1−z)−π0 ≤|ρ|∑j=1πjxij−1ij,k~Pij−1ij,k ≤π0+M(1−z), (26)

where coefficient vector is defined as in (14), and is sufficiently large so that constraint (26) becomes redundant for TEP when .

Proof. The telescoped bus angle-difference equation (17) can be written if and only if corridors are each serviced by transmission lines (i.e., all consecutive bus-pairs must be connected). Since indicates that there are existing lines servicing corridor , it is necessary to ensure only that whenever , an investment in some line must be made, i.e. some where . In other words, when contains established and expansion corridors, (15) is valid only if line investments are made within every expansion corridor (see (24)); and it is invalid when no line investments are made in at least one of the expansion corridors traversed (see (25)). Because any combination of candidate lines may be built in an expansion corridor and there are no existing lines present, we must pick the most generous bound on from the potential candidate lines, hence the specific choice of for expansion corridors. Notice that this logic is reflected through (24)-(26).

In Figure 2, a new single path, is created when line is added where which creates the type 3 VIs:

### 4.4 Parallel paths over established and expansion corridors

#### Theorem 4

Let represent alternative directed paths over established and expansion corridors in with the same starting/ending buses but with non-overlapping intermediate buses. Additionally, let refer to any fixed line in each established corridor, and in expansion corridors to the candidate line with the maximum capacity-reactance product. By including auxiliary variable and the logic-enforcing constraints,

 z ≥m∑r=1|ρr|∑j=1Irjyirj−1irj,k−m∑r=1|ρr|∑j=1Irj+1∀^kρ (27) z ≤Kij∑k=1yirj−1,irj,k% for j=1..|ρr|,r=1..m s.t. Irj=1 (28)

where is used analogously as shorthand for the indicator function , the following two-sided inequalities are valid for TEP for any :

 −M(1−z)−min{πk0}mk=1≤|ρr|∑j=1πrjxirj−1irj,k~Pirj−1irj,k ≤min{πk0}mk=1+M(1−z)for r=1,…,m. (29)

where coefficient vectors are defined according to (14) for each path , is defined as in Theorem 3, and is sufficiently large so that all constraints specified by (4.4) become redundant for TEP when .

Proof. The rationale for obtaining this result is a straightforward extension of the proofs of Theorems 2 and 3.

Continuing the example from subsection IV.C, in Figure 2 the new parallel path is created by adding line . Assuming is the parallel path with lower capacity-reactance product creates the type 4 VIs:

## 5 Tests and Results

The structure of the experiment and its implementation are as follows: First, the low-effort heuristic method, explained in section 3, is applied. The solution flows from the chosen relaxations are then analyzed on the same graph to find single paths of same-direction flows of maximum length using a breadth-first search algorithm. For larger instances, the maximum length of each path and maximum number of paths starting from each bus are capped to prevent memory issues. Paths with the same initial and final bus are combined to form parallel paths. Once all or, in the case of the particularly large instances, the maximum allowed number of single paths and parallel paths are found, cuts are added to the model from those lists in a random order. It should be noted that in each of the tested instances, all candidate lines for a given branch, , have identical properties. When this is the case, we can enforce the additional set of symmetry-breaking constraints , , since each line is interchangeable. First, testing is performed on a modified version of the IEEE 118-bus system Christie (2000) in order to showcase the potential for the effectiveness of the proposed path-based VIs in a relatively simple and easily replicable context. This system is also used to detail the distribution of cuts applied from each theorem. Then, testing is performed on the Polish 2383-bus system in order to show their effectiveness in a more realistically sized and designed instance. The algorithm is implemented in Python and solves the DM using CPLEX version 12.8.0.0. All tests are run on a Dell OptiPlex 7050 with eight Intel i7-7700 processors at 3.60GHz and 64 GB of RAM.

### 5.1 IEEE 118-Bus System

The effectiveness of the new theorems was first tested using a modified 118-bus test case. This system is relatively simple to solve, and in fact the unmodified 118-bus instance already satisfies demand without constructing any additional lines. To tailor this instance for TEP and add some computational difficulty, we consider the possibility that up to 7 candidate lines with the same characteristics as the existing lines in that corridor may be added, similar to what was done with the Southern Brazilian 46-bus system in Escobar and Romero. (2017). Additionally, all line ratings have been reduced to 60% in order to create congestion in the original network, as in Zhang (2013). Finally, we have chosen at random ten existing lines to remove from the system. These lines are branches 33, 37, 46, 52, 62, 66, 87, 104, 145, and 157. This allows for all four theorems to be applied to this test case. The resulting system has 118 buses, 54 generators, and 186 corridors, of which 176 possess existing lines, allowing up to 7 candidate lines to be built per corridor which results in 1302 binary decisions.

When adding valid inequalities to our model, we used CPLEX’s user cut option Studio (2012). This was to address the fact that adding such a large number of cuts as linear constraints directly via CPLEX produced inconsistent and often large solution times. The number of such cuts added is shown in detail in Table 1, for all potential combinations of relaxation models and broken down according to which theorem was used to generate the cut. In this table as in future tables, TR refers to the transportation relaxation, HR to the hybrid relaxation, and LR to the linear relaxation. The user cut option allows CPLEX to implement only those inequalities it deems most beneficial at each node of the branch-and-bound process Ostrowski et al. (2012). Although this option potentially increases computation time, it produced significantly decreased overall solve times and more consistent results from repeated trials.

Table 2 summarizes the complete results from adding all possible VIs to the 118-bus instance and then solving. For this table and for future tables, N/A refers to the time spent solving the model with no VIs added. Additionally, the column Relax Time refers to the total time spent solving the subset of relaxation models, the column Path Search refers to the total time finding all paths and parallel paths from overlaying the solutions of the relaxation models on the network, and the column Solution refers to the time spent solving the original problem after adding all possible VIs. Finally, the C+P+R column is the total time spent on this whole process. Note that all times are in seconds and refer to the average runtime.

We can see from this table that solving the modified 118-bus instance without adding any VIs took 743.79 seconds. In comparison, the best total solve time including finding and implementing all VIs took 203.37 seconds, a roughly 3.6x improvement. This time comes from solving only both the transportation relaxation and the hybrid relaxation. However, note that the lowest solve time disregarding the time spent finding paths and finding relaxations, was only 95.07 seconds and came from solving all three relaxations, which is nearly an 8x improvement. This is significant because, for the relatively simple 118-bus case, a significant amount of the total solve time of our algorithm is spent finding paths and solving relaxations. These reduced times help show the potential of solving multiple relaxation models, rather than simply the linear relaxation. Additionally, as the size and complexity of the problem scales up, the time spent solving relaxation models and finding paths comprises significantly less of the overall computational effort, which will be shown in the next subsection.

### 5.2 Polish 2383-Bus System

We use the Polish 2383-bus system adapted for TEP in Mínguez et al. (2018). This system has 2383 buses, 327 generators, and 3020 total corridors. Of those corridors, 120 have no existing line and allow for one candidate line to be built, while the remaining 2900 corridors do not allow for any expansion. Thus there are 120 binary decisions to be made. Due to the size of this instance, additional testing restrictions were introduced. Limits were placed on the path-finding algorithm, permitting only 1000 paths to be found per starting bus and allowing only paths of 20 buses or fewer in length. Although there are fewer binary variables in this instance than in the modified 118-bus system described above, the overall system is much more complex and representative of a realistic-size system.

As illustrated in Table 3, solving any subset of relaxation models and adding a number of VIs generated from overlaying solutions to those relaxation models significantly reduces the time spent solving TEP for the 2383-bus system. Due to the time restrictions implemented in the path-finding algorithm, this is only a subset of all possible paths from which to generate VIs; however, we remark that the addition of all such VIs may be impractical from a computational standpoint, due to the exponential growth in the number of possible paths on which to base them. In this case, solving only the transportation relaxation produced the greatest reduction in both total solution time and in solution time not including time spent searching for paths and solving relaxations. While the original instance took 16271.94 seconds to solve without adding any valid inequalities, this greatest reduction took only 2347.2 seconds, which is approximately an 8x speed up. These results show the effectiveness of the proposed VIs even on systems of realistically large size. Additionally, solving only the linear relaxation results in only a roughly 2x reduction in total solution time. In fact, solution times that use the linear relaxation showed the least improvement generally. This suggests that solving multiple relaxation models, rather than just the traditional linear one, can produce significant improvements to solution algorithms for TEP.

## 6 Conclusions and Future Work

This work presents a new mathematical framework and an algorithm that uses a mixed-integer linear programming model, valid inequalities, and a low-effort heuristic method for solving TEP. The objective is to reduce the total computational effort of planning. This work is a significant improvement of the preliminary studies carried out in Escobar and Romero. (2017), in which the solutions were found after manual analysis of the test system, creation of cuts using two of the valid inequalities introduced in this paper (specifically from Theorems 1 and 2), which at that time had not been proven, and tests made with different cut combinations. However, this work automates each step of the process and formally establishes the validity of four types of valid inequalities.

Computational tests show the effectiveness of the four presented theorems in generating valid inequalities which are effective in reducing the solution time of TEP. They also suggest how to best apply the theorems for use in solving multiple test cases, as well as how they may be of use in the solution of more realistic, larger scale problems. Additionally, the results demonstrate different options for the implementation of these valid inequalities that offer distinct trade-offs in efficiency in the various stages of the solution process, which provides options for approaching instances of varying size and expected computational effort.

In future work, we will perform a polyhedral study on the strength of the proposed VIs and we will conduct further studies to determine the most effective use of the presented theorems for particular instances. In particular, as the size of a system increases, the number of possible paths, and thus the number of possible valid inequalities, increases at an exponential rate. Finding and adding all these inequalities takes significant computational time, and the sheer number added does not necessarily improve the performance of solving via CPLEX. Thus, additional testing is planned to determine how to select an ideal subset of single path and parallel path inequalities to help decrease total solution time, particularly in large systems. Fine tuning of the implementation such as this will allow us to solve more complex problems, such as the L-1 reliability on TEP Escobar et al. (2018) and planning with uncertainty due to renewables as well as incorporating new technology such as FACTS devices.

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