1. Introduction
The integrated electric utility, owning both generation and transmission, traditionally develops generation plans first and then plans the transmission necessary to support those generation plans, an approach consistent with the fact that generation investments are normally far more costly than the needed transmission investments. Today, however, there are organizations that own transmission only. There are also service companies, called regional transmission organizations in the U.S., that own no infrastructure, but are responsible for coordinating the planning function. Both of these organizations must solve the transmission expansion planning (TEP) problem under uncertainty regarding technology, quantity, location, and timing of generation investments [
1,
2,
3,
4]. Robust and stochastic optimization approaches are two major modeling techniques for transmission planning under uncertainty, which have their own strengths and limitations.
Robust TEP maximizes the performance of the power system under the worst case scenario, which is dependent on the transmission plan [
5,
6,
7,
8,
9,
10]. Most solution techniques for robust TEP consist of two iterative steps: one step identifies the worst case scenario for a proposed transmission plan, whereas the other proposes the most robust transmission plan against a pool of worst case scenarios. A major advantage of robust TEP is its robustness against worst case scenarios, which is especially desirable for enhancing long-term resiliency of power system infrastructures. The iterative algorithm also allows the model to efficiently identify worst case scenarios from an enormous solution space with a large number of (or even infinitely many) possible scenarios. On the other hand, this approach often tends to be over-conservative by focusing on the possibility and ignoring the probability of worst case scenarios.
Stochastic TEP addresses uncertainty with a very different philosophy. It maximizes the average performance of the power system under all scenarios, weighted by their probabilities of occurrence. As such, the optimal stochastic transmission plan finds a good balance between the (positive or negative) impact and the likelihood of all scenarios [
11,
12,
13,
14,
15,
16]. However, this approach also faces a dilemma. On the one hand, due to the numerous sources of uncertainty and long planning horizon, a large number of scenarios is necessary to realistically represent the complexity and uncertainty of the TEP problem. On the other hand, however, the computation time of most solution techniques is very sensitive to the number of scenarios, and many algorithms become intractable for even a few dozen scenarios. To address this dilemma, scenario generation and reduction techniques have been proposed, which attempt to identify a small set of high-quality scenarios that represent the whole set of scenarios [
17,
18,
19,
20,
21].
An alternative approach is to use an iterative stochastic TEP model, which attempts to combine the advantages of robust and stochastic TEP approaches and to overcome their limitations. A major limitation of the stochastic TEP approach with scenario generation and reduction is the assumption that the set of selected scenarios will be a good representation of the whole set of scenarios for all transmission plans. This is analogous to identifying one scenario as the “worst case scenario” for all transmission plans in the robust TEP approach. The iterative model in robust TEP acknowledges the need for re-identifying a worst case scenario for each new transmission plan, until the pool of worst case scenarios is sufficiently inclusive of all worst case scenarios. Similarly, the iterative stochastic TEP model re-identifies a new set of high-quality scenarios for each new transmission plan until the pool of high-quality scenarios is sufficiently representative of the whole set of scenarios. A formal definition of high-quality scenarios is given in
Section 2.2.
Figure 1 highlights the conceptual differences among robust TEP, stochastic TEP, and the iterative stochastic TEP. The scope of this paper is the highlighted component for selecting high-quality scenarios.
Scenario generation and reduction has been a topic of great interest in the power systems literature. Most existing methods use clustering [
19,
22,
23,
24] or sampling methods to reduce the number of scenarios from a randomly generated initial set. In a recent review article, Park et al. [
25] compared four methods for scenario reduction using a two-stage stochastic transmission planning model, including random sampling, importance sampling [
26], the distance-based method [
17,
27,
28], iterative scenario reduction approaches, and stratified scenario sampling [
25]. They used these methods to reduce a whole set of 20 scenarios to smaller subsets and compared their pros and cons. Other methods also include extended and improved initial-center-refined and weighted k-means [
21], data-driven [
29], forward selection [
20], moment-based [
18,
30,
31], and objective-based [
32,
33] approaches.
The proposed scenario selection approach in this paper differs from previous methods for scenario generation and reduction in three major ways. First, we generate scenarios with explicit consideration of temporal and spatial correlations, including generation investment and retirement decisions in response to demand, fuel cost, and transmission capacity. Second, we use the Kantorovich distance of social welfare distributions to assess the quality of the selected scenarios. Third, a different subset of high-quality scenarios is selected for each transmission plan candidate. In comparison, most existing methods in the literature ignore the correlations among the scenarios, select a subset of scenarios based on their similarity rather than their implications on social welfare, and use the same subset of scenarios for all transmission plans.
2. Model Formulation
In this section, we describe and explain the models that we propose to identify a small set of high-quality scenarios for a given transmission plan. In the following, the variables and parameters used in the model are described.
Sets and indices:
| Set of planning years; |
| Set of load blocks; |
| Set of buses; |
| Set of transmission lines; |
| Set of generation technologies; |
| Set of existing renewable generators at bus b in year y; |
| Set of existing non-renewable generators at bus b in year y; |
| Set of existing generators at bus b in year y: ; |
| Set of candidate renewable generators at bus b in year y; |
| Set of candidate non-renewable generators at bus b in year y; |
| Set of candidate generators at bus b in year y: ; |
| Set of renewable generators at bus b in year y: ; |
| Set of existing and candidate generators at bus b in year y: ; |
| Set of existing generators of technology h at bus b in year y. |
Parameters:
r | Discount rate, in %; |
| Cost of load-shedding, in $/MWh; |
| Operating duration of load block t, in hours; |
| Heat rate for generator g, in Mbtu/MWh; |
| Fuel cost of generator g’s technology in year y, in $/Mbtu; |
| Fixed Operation and Maintenance (O&M) cost for generator g’s technology, in $/MW; |
| Amortized annual investment cost for generator g’s technology, in $/MW; |
| Variable O&M cost for generator g’s technology, in $/MWh; |
| Variable generation cost for generator g in year y: , in $/MWh; |
| Demand at bus b for load block t in year y, in MW; |
| Monetary value of energy consumption at bus b for load block t in year y, in $/MWh; |
| Susceptance of transmission line , in Siemens; |
| Capacity of transmission line in year y, in MW; |
| Capacity factor of generator g for load block t, in %; |
| Production capacity of generator g, in MW; |
| Renewable portfolio requirement in year y, in %. |
Decision variables:
| Power production from generator g at bus b for load block t in year y, in MW; |
| Power flow through transmission line for load block t in year y, in MW; |
| Load shedding at bus b for load block t in year y, in MW; |
| Voltage angle at bus b for load block t in year y, in radians; |
| Locational marginal price at bus b for load block t in year y, in $/MWh; |
| Social welfare of energy producers and consumers in the power system, in $; |
| Binary variable indicating whether the generator g at bus b exists in year y () or not (). For an existing generator , means the generator has retired; and for a candidate generator , means the generator has been added to the existing generation capacity. |
2.1. Motivating Example
Suppose there are 10 scenarios
and 10 transmission plans
, and we try to identify the optimal transmission plan. The number of scenarios and plans is limited to 10 for illustration purposes, but the methods are applicable to situations where there are millions or even infinitely many scenarios and plans. With the values in the table representing social welfare of the transmission plans under different scenarios,
Table 1 illustrates the solution process of the robust TEP approach.
Table 2 illustrates the solution process of the iterative stochastic TEP approach for the same example, in which two high-quality scenarios were selected to represent the whole set of 10 scenarios for each transmission plan. The algorithm proposed in
Section 3.3 was used to select these scenarios and assign their probabilities.
The optimal transmission plan for robust TEP approach is determined in
Table 3 as
, which resulted in a higher social welfare than all other transmission plans under their respective worst case scenarios. For iterative stochastic TEP approach, the optimal transmission plan is determined in
Table 3 as
, which maximized the weighted average social welfare. The third column of
Table 3 calculates the average performance of all transmission plans under all scenarios, which shows that the true optimal solution to the stochastic TEP problem was indeed
. This example illustrated how the iterative stochastic TEP approach was able to identify the optimal (or close to optimal) solution by selecting a subset of high-quality scenarios chosen specific to each transmission plan.
2.2. Definition of High-Quality Scenarios
For a given transmission plan, a scenario includes three elements: demand for all buses, fuel cost for all generation technologies, and generation capacity for all generators, throughout the entire planning horizon. Such a definition of a scenario captures the temporal and spatial trajectory of the power system, which we view as evolving as a function of a given transmission plan.
High-quality scenarios selected using our proposed approach satisfy two requirements. First, the correlations between generation capacity and other elements (i.e., demand and fuel cost) are reflected. This is because generation investments and retirement decisions are made by decentralized and for-profit generation companies, which have been found to be sensitive to demand and fuel cost. Second, the probabilistic distance between the distribution of social welfare resulting from high-quality scenarios and that from all scenarios is small (the smaller the distance, the higher the quality).
2.3. Estimation of Generation Investment and Retirement
For given demand and fuel cost elements of a scenario, we propose to estimate the generation capacity element of the scenario by solving a simplified generation expansion planning (GEP) problem, which has the following features.
We cast the GEP problem as a bilevel optimization model, in which the upper level maximizes the net present value of the investment by determining the investment of new generation capacity and the retirement of existing generators, whereas the lower level computes the optimal power flow (used by the upper level to calculate the revenue of power generation) as a result of the GEP decisions from the upper level. Similar models have been used in other studies such as [
7,
34].
It uses a single upper level decision maker for all generation investment and retirement decisions in all buses. This is a simplifying approximation of the power systems, in which investment and retirement decisions are made by multiple generation companies in a decentralized manner. Similar approximations have been used in other studies such as [
18].
The GEP model identifies generation investments based on profit maximization accounting for fixed and variable operational costs, investment costs, and load shedding costs, subject to renewable portfolio standards and generation adequacy requirements, attributes common to other studies [
18,
35,
36,
37,
38].
The bilevel optimization model of the GEP problem is formulated as the following optimization model in Equations (
1)–(
4), where the upper level decision variable
x represents generation investment and retirement decisions
, and the lower level decision variable
p represents both dispatch decisions
and locational marginal prices
.
The upper level objective function
is to maximize the profit of generation companies, which can be estimated as:
The first term in the objective function is the estimated revenue from selling power generation at the locational marginal prices. The second term in the objective function is the O&M cost for existing and new generators, and the third term is the investment cost for new generators. All future cost terms are discounted to the current year to calculate the net present value. All fixed and variable investment costs are amortized so that investments towards the end of planning horizon would not be disincentivized, since only part of the investment cost appropriated for the planning horizon is calculated to offset the benefit of the investment.
The upper level constraints in Equations (
2) and (
3) are defined as follows:
Here, Constraints in Equations (
6) and (
7) allow the retirement of existing generators and investment of new generators; Constraint in Equation (
8) imposes a renewable portfolio standard type of requirement on the minimal percentage of generation capacity being renewable; Constraint in Equation (
9) imposes a generation adequacy requirement, where the total generation capacity for each year must exceed 120% of predicted peak demand; and Constraint in Equation (
10) is the definition of binary decision variables.
Constraint in Equation (
4) defines the lower level problem, which takes generation investment and retirement decisions from the upper level as input and solves the optimal power flow problem to determine power dispatches and locational marginal prices throughout the planning horizon. This lower level problem can be defined as follows, which needs to be solved for all
and
.
The lower level objective function in Equation (
11) is to minimize the production cost and load shedding cost. Constraint in Equation (
12) enforces Kirchhoff’s current law, which requires nodal balance of power generation, in-flow, out-flow, demand, and load shedding; Constraint in Equation (
13) defines the lower and upper bounds of load shedding; Constraint in Equation (
14) enforces Kirchhoff’s voltage law, which computes power flows through transmission lines based on voltage angles and susceptance of the power network; Constraint in Equation (
15) defines the bounds of voltage angles; Constraint in Equation (
16) limits the power flow within transmission capacity; and Constraint in Equation (
17) limits the power production within existing generators’ capacity, which depends on generation investment and retirement decisions.
2.4. Definition of Social Welfare
Social welfare is a commonly used objective for transmission planning [
39,
40,
41,
42,
43,
44,
45]. We measure the quality of selected scenarios based on the similarity between social welfare distributions resulting from the selected and whole set of scenarios, rather than the similarity of the elements of the scenarios. For a given transmission plan under a given scenario, the social welfare (
) is defined as the summation of producers’ surplus and consumers’ surplus:
Here, the first term is the objective function of the GEP model, which represents the producers’ surplus; the second and third terms are consumers’ surplus, which includes monetary valuation of energy consumption less prices of energy and economic cost of load shedding. All cost and benefit terms in the social welfare definition are discounted to Year 0, so that it reflects the net present value of all transactions throughout the planning horizon.
2.5. The Kantorovich Distance
We use the Kantorovich distance [
46] to measure the probabilistic distance between the two distributions of social welfare resulting from the selected high-quality scenarios and those from the whole set of scenarios:
Here, the notations are defined as follows:
: the whole set of scenarios;
: the high-quality set of scenarios;
: probability of scenario s;
: social welfare under scenario s.
5. Conclusions
The main contributions of this paper included a scenario selection method for iterative transmission planning, which had three salient features: (1) correlations between generation capacity and demand, fuel cost, and transmission capacity were explicitly captured; (2) high-quality scenarios were selected to minimize the Kantorovich distance of social welfare distributions between the selected and the whole set of scenarios; and (3) the set of high-quality scenarios was specific for each transmission plan, which enabled this approach to interact iteratively with a stochastic TEP model.
A case study was conducted to demonstrate the proposed approach, in which a 169-bus network was used to represent the U.S. Eastern and Western Interconnections. When compared with three other methods for scenario selection, our algorithm was found to provide a more accurate estimation of the economic value of transmission plans.
The proposed approach was not without its limitations and caveats. For example, the GEP model was a simplified estimation of the actual decision-making process, which involved far more realistic and complex constraints such as multiple decision makers and risk hedging constraints. We also acknowledge that there are numerous other sources of uncertainty in realistic transmission planning projects, which were not explicitly taken into consideration in the proposed approach, such as investment cost, growth rate of distributed energy resources, renewable energy production, energy policies, etc. Besides Kantorovich distance, other probabilistic distances could also be used as the selection criterion for high-quality scenarios. Moreover, the social welfare evaluation could be extended to include environmental and reliability benefits of the power system. Co-optimization of GEP and TEP is another potential topic for future research.