A Stochastic Multi-agent Optimization Model for Energy Infrastructure Planning under Uncertainty in An Oligopolistic Market

Abstract

This paper presents a mathematical model for analyzing long-term infrastructure investment decisions in a deregulated electricity market, such as the case in the United States. The interdependence between different decision entities in the system is captured in a network-based stochastic multi-agent optimization model, where new entrants of investors compete among themselves and with existing generators for natural resources, transmission capacities, and demand markets. To overcome computational challenges involved in stochastic multi-agent optimization problems, we have developed a solution method by combining stochastic decomposition and variational inequalities, which converts the original problem to many smaller problems that can be solved more easily.

Appendices

Appendix 1: Small Example Illustrating Flow Conservation Constraint (3b)–(3e)

1.1 Network Structure

Fig. 9

Small example structure

1.2 OD and Path Information

Table 7 Small example structure

1.3 Network Flow

1.4 Incidence Matrix

Appendix 2: Subroutine Pseudocode

Appendix 3: Data for Example 2

Table 8 Capacity cost data

Table 9 Generation cost data

Table 10 Demand function parameters d _b and d _a (demand function is d=−d _a ∗w + d _b )

Table 11 Transmission capacity c _t (link transmission cost function is ϕ _t =10∗[1+(v/c _t )⁴)]

Appendix 4: Calculation of $\protect \frac {\partial \rho _{k^{\prime }}}{\partial {g_{i}^{j}}}$

${\partial \rho _{k^{\prime }}}/{\partial {g_{i}^{j}}}$ can be computed from the ISO’s optimization problem (3a), where ρ is the dual variables of constraint (3a) and g is parameters. ${\partial \rho _{k^{\prime }}}/{\partial {g_{i}^{j}}}$ is essentially the derivative of dual variables with respect to right-hand side constants. We use the standard notations for convex optimization with linear constraints:

$$\begin{array}{@{}rcl@{}} \min_{\boldsymbol{x}} && f(\boldsymbol{x}) \end{array} $$

(18a)

$$\begin{array}{@{}rcl@{}} \text{s.t.} \quad (\boldsymbol{\lambda}) && A\boldsymbol{x}=\boldsymbol{b} \end{array} $$

(18b)

where f(x) is a convex function and $\boldsymbol {x} \in \mathbb {R}^{n}, \boldsymbol {\lambda }, \boldsymbol {b} \in \mathbb {R}^{m}$, to illustrate the calculating process and our goal is to calculate the Jacobian matrix J _λ (b).

Lagrangian of problem (18b) is $\mathcal {L} = f(\boldsymbol {x}) - \boldsymbol {\lambda }^{T}(A\boldsymbol {x}-\boldsymbol {b})$. The optimality conditions of problem (18b) is:

$$\begin{array}{@{}rcl@{}} \nabla f(\boldsymbol{x}) - A^{T}\boldsymbol{\lambda} &=& \boldsymbol{0} \end{array} $$

(19a)

$$\begin{array}{@{}rcl@{}} A\boldsymbol{x}-\boldsymbol{b} &=& \boldsymbol{0} \end{array} $$

(19b)

Take implicit derivatives of Eq. (4) with respect to b :

$$\begin{array}{@{}rcl@{}} \nabla^{2}_{\boldsymbol{x}} f(\boldsymbol{x}^{\ast}(\boldsymbol{b})) J_{\boldsymbol{x}}(\boldsymbol{b}) - A^{T} J_{\boldsymbol{\lambda}}(\boldsymbol{b}) &=& \boldsymbol{0} \end{array} $$

(20a)

$$\begin{array}{@{}rcl@{}} AJ_{\boldsymbol{x}}(\boldsymbol{b})-\boldsymbol{I} &=& \boldsymbol{0} \end{array} $$

(20b)

The unknown variables in Eq. 20a are two Jacobian matrices, J _x (b) and J _λ (b). The total number of equations is equal to the total number of variables in Eq. 20a, which is n m + m ². Therefore, as long as these equations are consistent and independent, J _λ (b) can be calculated uniquely. Clearly, if function f(x) is in quadratic form of x, $\nabla ^{2}_{\boldsymbol {x}} f(\boldsymbol {x})$ then only involves constants, in which case J _λ (b) can be computed analytically. For general form of f(x), one can solve J _λ (b) numerically; one can take an initial guess of b based on historical data and then solve (20a) and Algorithm 2 iteratively until J _λ (b) converged.