Microgrid investment under uncertainty: a real option approach using closed form contingent analysis

Abstract

The traditional net present value approach to investment in microgrid assets does not take into account the inherent uncertainties in fuel prices, cost of technology, and microgrid load profile. We propose a real option approach to microgrid investment, which includes solar photovoltaic (PV) and gas-fired generation assets. Likewise the (n, m) exchange literature in real option analysis, we examine cases with interdependency and independency of fuel price and the cost of PV technology. This work, however, makes a major contribution by the way of introducing a new parameter, which is defined as the elasticity of the option value to prices and is used in the formulation of closed form solutions. We further extend the (1, 1) exchange problem here to include operational flexibility of microgrid, such that optimal switching between investment, suspension and re-activation can be examined.

Appendix: More on solution approach

Four equations associated to suspension and re-activation option can be solved separately to obtain $B_1,\, K_2,\, C_M$ and $C_R$:

$$\begin{aligned}&B_1 C_M^{\lambda _1 } -K_2 C_M^{\lambda _2 } -365\times \frac{\textit{Cap}_\textit{GF} \times \epsilon _\textit{GF} }{\delta _C }C_M +365\times \frac{\textit{Cap}_\textit{GF} \times C_e }{r}+\frac{M}{r}-E_M =0 \nonumber \\&\quad B_1 \lambda _1 C_M^{\lambda _1 -1} -K_2 \lambda _2 C_M^{\lambda _2 -1} -365\times \frac{\textit{Cap}_\textit{GF} \times \epsilon _\textit{GF} }{\delta _C }=0 \nonumber \\&\quad B_1 C_R^{\lambda _1 } -K_2 C_R^{\lambda _2 } -365\times \frac{\textit{Cap}_\textit{GF} \times \epsilon _\textit{GF} }{\delta _C }C_R +365\times \frac{\textit{Cap}_\textit{GF} \times C_e }{r}+\frac{M}{r}-R=0 \nonumber \\&\quad B_1 \lambda _1 C_R^{\lambda _1 -1} -K_2 \lambda _2 C_R^{\lambda _2 -1} -365\times \frac{\textit{Cap}_\textit{GF} \times \epsilon _\textit{GF}}{\delta _C }=0 \end{aligned}$$

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These equations are nonlinear in terms of thresholds $C_M$ and $C_R$; however, it can be proved that a unique solution exists (see Dixit and Pindyck 1994). There are three other boundary conditions of new investment explained earlier in the value of idle project.

Value matching condition:

$$\begin{aligned} F\left( {C^{{*}},I^{{*}}} \right) =V_\textit{ActiveMG} \left( {C^{{*}}} \right) -I_\textit{GF} -\textit{Cap}_\textit{PV} I^{{*}} \end{aligned}$$

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and smooth pasting conditions:

$$\begin{aligned} \frac{\partial F}{\partial C}\bigg |_{C^{{*}}}= & {} \frac{\partial \left( {V_\textit{ActiveMG} -I_\textit{GF} -\textit{Cap}_\textit{PV} I} \right) }{\partial C}\bigg |_{C^{{*}}} \nonumber \\ \frac{\partial F}{\partial I}\bigg |_{I^{{*}}}= & {} \frac{\partial \left( {V_\textit{ActiveMG} -I_\textit{GF} -\textit{Cap}_\textit{PV} I} \right) }{\partial I}\bigg |_{I^{{*}}} \end{aligned}$$

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Three other unknowns, namely, $A_2,C^{*}$ and $I^{*}$ are determined using these three equations. We use numerical schemes to solve these equations.