# Investment Incentives in Competitive Electricity Markets
## 6 Mathematical Formulation
### Bi-Level Model Formulation of Markets
\begin{array}{l}\textbf{max}\ \sum\limits_{s}\sum\limits_{t}\text{WDB}\sum \limits_{w}\text{WSW}\\ \left[\text{FCP}\sum\limits_{F_{1}}\overline{Z}(F_{1})\text{FCV}\right].\end{array}  \tag{9}\]  
The term of Equation (9) is associated with the expected profits of the strategic investor obtained by selling a pre-determined part of the production in a firm contractual market. The FCV is the percentage of the capacity of the new unit that can be purchased from the strategic GENCO, in terms of their firm contract, which is considered as a pre-requisite. In our research case study, the FCV is considered to be 10% for stability purposes:  
\[ \begin{array}{l}\textbf{max}\ \sum\sum\limits_{t}\text{WDB}\sum\limits_{w} \text{WSW}\\ \text{CPP}\sum\overline{Z}(F_{1})(1-\text{FCV})(1-\text{F.O.R}^{\text{SN}})\\ +\text{CPP}\sum\overline{Z}(F_{2})(1-\text{FCV})(1-\text{F.O.R}^{\text{SE}}). \end{array}  \tag{10}\]  
The term of Equation (10) is the expected profit obtained from the capacity payments:  
\[\overline{Z}(F_{1})=\sum\limits_{h}V(Q_{1})\times V(Q_{2}), \tag{11}\]  
\[\sum\limits_{h}V(Q_{2})=1:V(Q_{2})\in\{0,1\}. \tag{12}\]  
The term of Equations (11) and (12) are the conditional statements for setting investment options only available to discrete blocks. For all these numerical expressions, they impose the condition that only one technology is binding and determines the new technology to be installed at each bus of the system:  
\[\text{IC}(\overline{Z}(F_{1}))<\text{IB}, \tag{13}\]  
\[L(F)\geq 0:\forall F_{1,2}. \tag{14}\]  
In addition, to comply with the limitations of cost margins and positive loop implementation to reduce redundancy, the investment budget limit is represented by terms Equation (13), and the term Equation (14) makes sure that offers to strategic GENCO for its new and existing units are also always positive values, numerically. The market clearing solutions are represented by the objective function Equation (15) along with terms of Equations (16)-(19). For all the mathematical expressions defined, all dual variables are indicated at the relevant terms of constraints by a following notation of \"colon\" to mark the dual input of variables. The unique feature of the above mathematical expression of objective functions is that both elastic and inelastic conditions can be considered. If demand is elastic to the price, the objective in the second-level maximizes the social welfare, numerically expressed by the adjoining term of:  
\[Z(F_{4})L(F_{4})-\sum\limits_{F_{1}}^{F_{3}}Z(F)L(F). \tag{15}\]  
However, if demand is inelastic to the price, the objective in the second-level minimizes the operation cost, numerically estimated by the summation and multiplication of GENCOs production and operation cost of GENCOs unit  
\[\sum\limits_{F_{1}}^{F_{3}}Z(F)L(F). \tag{16}\]The optimization of the objective function in the second-level solution is obtained by considering Equation (15):  
\[ \begin{array}{l}\underline{Z}(F)\leq Z(F)\leq\overline{Z}(F)-(\overline{Z}(F) \text{FCV})|_{\forall F_{1}}:\\ (\underline{\text{Dual}}^{z}(F),\overline{\text{Dual}}^{z}(F)).\end{array}  \tag{17}\]  
Equation (17) imposes power bounds for the blocks of generation constraints, the power flow and the angle bounds:  
\[\sum_{F_{1}}^{F_{3}}Z(F\{n\})+\overline{Z}(F_{1})\text{FCV}-Z(F_{4}\{n\})-Z(F_ {6}\{n\}):\ L(F_{5}). \tag{18}\]  
Equation (18) represents the energy balance at each bus, being the associated dual variables LMPs or nodal prices for global variable. For the above, when the demand is inelastic to the energy price, the \(Z(F_{4})=\overline{Z}(F_{4})\) satisfies the condition in Equation (17) and \(Z(F_{6})\) takes the form:  
\[ \begin{array}{l}Z(F_{6}\{m,n\})=L(F_{6}\{m,n\})(Z(F_{5}\{m\}))-Z(F_{5}\{n\}): \\ \text{Dual}^{\text{PF}}(F).\end{array}  \tag{19}\]  
Equation (19) defines the power flow through transmission lines using a loss-free DC model. In addition,  
\[Z(F_{6}\{n=1\})=0:\ \text{Dual}^{\text{SN}}. \tag{20}\]  
Equation (20) regulates the voltage angle at each bus for the reference bus.