The linear uncertain inverse demand functions are computed from the price elasticity between the data of sales and the average price of electricity to ultimate customers for 2014–2015 together with uncertain variables. Here, the uncertain variable depicts some uncertain factors in the electricity power market. Those uncertainties are caused by a certain natural disasters such as hurricanes, tornadoes, seaquakes, earthquakes, etc., and the uncertain demands of consumption sectors.
The linear uncertain inverse demand function of the electricity power at sectors
\(j_1,j_2\) are, respectively, represented by
$$\begin{aligned} p_{1}({\varvec{x}},\xi _1)=713{,}079.9804-0.4178(x_{11}+x_{21}+\xi _1), \end{aligned}$$
and
$$\begin{aligned} p_{2}({\varvec{x}},\xi _{2})=264{,}749.8419-0.1164(x_{12}+x_{22}+\xi _{2}), \end{aligned}$$
where
\({\varvec{x}}=({\varvec{x}}_{1},{\varvec{x}}_{2})^{T}=((x_{11},x_{12})^{T},(x_{21},x_{22})^{T})^{T},\)\(\xi _{1}\) is a linear uncertain variable with the uncertainty distribution
\({\mathcal {L}}(-100{,}100),\) while
\(\xi _{2}\) is a normal uncertain variable with uncertainty distribution
\({\mathcal {N}}(100,10).\) Moreover,
\(\xi _{1}\) and
\(\xi _{2}\) are independent of each other. Denote
\(\varvec{\rho }=(\rho _{1},\rho _{2})^{T}\) as the transmission price vector, where
\(\rho _{1}\) and
\(\rho _{2}\) are the unit transmission prices delivered to two sectors. For convenience, the operating expense of the production is scaled up as USD/GWh, the consumption of the electricity power is weighted as GWh. Given the decision variable
\({\varvec{x}}_{-1}={\varvec{x}}_{2}\) and the transmission price vector
\(\varvec{\rho },\) the uncertain profit function of utility
\(U_1\) is
$$\begin{aligned} \begin{aligned} \theta _{1}({\varvec{x}},\varvec{\xi })&=x_{11} p_{1}({\varvec{x}},\xi _1)+x_{12} p_{2}({\varvec{x}},\xi _2)\\&\quad -\,25{,}710(x_{11}+x_{12})\\&\quad -\,\rho _{1}x_{11}-\rho _{2}x_{12}\\&=x_{11}\cdot \left[ 687{,}369.9804-\rho _{1}\right. \\&\quad \left. -\,0.4178(x_{11}+x_{21}+\xi _1)\right] \\&\quad +\,x_{12}\cdot \left[ 227{,}477.4968-\rho _{2}\right. \\&\quad \left. -\,0.1164(x_{12} +x_{22}+\xi _2)\right] . \end{aligned} \end{aligned}$$
The corresponding constraint set is
$$\begin{aligned}&K_{1}({\varvec{x}}_{-1})=K_{1}({\varvec{x}}_{2})\\&\quad =\left\{ \begin{array}{ll} &{}x_{11}\geqslant 0,x_{12}\geqslant 0,\\ (x_{11},x_{12})^{T}:&{}x_{11}+x_{21}\leqslant 984{,}837.6,\\ &{}x_{12}+x_{22}\leqslant 1{,}066{,}907.4,\\ \end{array} \right\} . \end{aligned}$$
Then the problem
\(\beta -OPT({\varvec{x}}_{-1},\varvec{\rho })\) with
\(\beta =0.75\) for utility
\(U_1\) is
$$\begin{aligned} {\left\{ \begin{array}{ll} \text{ maximize }\; \theta _{1}({\varvec{x}},\varvec{\xi })_{\sup }(0.75)\\ \text{ subject } \text{ to }\; {\varvec{x}}_{1}\in K_{1}(x_{-1}). \end{array}\right. } \end{aligned}$$
From the inverse uncertainty distributions of the linear and the normal uncertain variables, together with Theorem
1, it leads to the following result.
$$\begin{aligned} \begin{aligned}&\theta _{1}({\varvec{x}},\varvec{\xi })_{\sup }(0.75)\\&\quad =x_{11}\cdot \left[ 687{,}369.9804-0.4178\left( x_{11}+x_{21}\right. \right. \\&\qquad \left. \left. +250\right) -\rho _{1}\right] +x_{12} \cdot \left[ 227{,}489.8419\right. \\&\qquad -0.1164\left. \left( x_{12}+x_{22}+100\right. \right. \\&\qquad \left. \left. +\dfrac{10\sqrt{2}}{\pi }\ln 3\right) -\rho _{2}\right] \\&\quad =x_{11}\cdot \left[ 687{,}265.5304-0.4178\left( x_{11} +x_{21}\right) -\rho _{1}\right] \\&\qquad +x_{12}\cdot \left[ 227{,}477.4968-0.1164(x_{12}+x_{22}) -\rho _{2}\right] \\&\quad ={\tilde{\theta }}_{1}({\varvec{x}},0.75). \end{aligned} \end{aligned}$$
The gradient of
\({\tilde{\theta }}_{1}({\varvec{x}},0.75)\) is
$$\begin{aligned} \nabla \tilde{\theta _{1}}({\varvec{x}},0.75)=(MR_{11}(x_{11},0.75)), MR_{12}(x_{12},0.75))^{T}, \end{aligned}$$
where
$$\begin{aligned} {\left\{ \begin{array}{ll} &{}MR_{11}(x_{11},0.75)=\dfrac{\partial {{\tilde{\theta }}}_{1}(x,0.75)}{x_{11}}\\ &{}=687{,}265.5304-0.4178(2x_{11}+x_{12})-\rho _{1},\\ &{}MR_{12}(x_{12},0.75)=\dfrac{\partial {{\tilde{\theta }}}_{1}(x,0.75)}{x_{12}}\\ &{}=227{,}477.4968-0.1164(x_{11}+2x_{12})-\rho _{2},\\ \end{array}\right. } \end{aligned}$$
(14)
We do not intend to write down the uncertain profit function of utility
\(U_2,\) its
\(0.75-\)optimistic value
\({\tilde{\theta }}_{2}({\varvec{x}},0.75),\) and its gradient
\(\nabla {\tilde{\theta }}_{1}({\varvec{x}},0.75)\) because they can be reached analogously. Denote
$$\begin{aligned} {{\tilde{K}}}_1= & {} \{{\varvec{x}}_{1}=(x_{11},x_{12})^{T}:x_{1}\geqslant 0, \;\text{ i.e., }\; x_{11},x_{12}\geqslant 0\},\\ {{\tilde{K}}}_2= & {} \{{\varvec{x}}_{2}=(x_{21},x_{22})^{T}:x_{2}\geqslant 0, \;\text{ i.e., }\; x_{21},x_{22}\geqslant 0\},\\ \bar{K}= & {} \left\{ \begin{array}{ll} ({\varvec{x}}_{1},{\varvec{x}}_{2})^{T}: &{}x_{11}+x_{21}\leqslant 984{,}837.6,\\ &{}x_{12}+x_{22}\leqslant 1{,}066{,}907.4,\\ \end{array} \right\} . \end{aligned}$$
Let
$$\begin{aligned} K=(\tilde{K_1}\times \tilde{K_2})\cap \bar{K}. \end{aligned}$$
Denote
\(F({\varvec{x}},0.75)=(\nabla {\tilde{\theta }}_{1}({\varvec{x}},0.75), \nabla {\tilde{\theta }}_{2}({\varvec{x}},0.75))^{T}, {\varvec{x}}=({\varvec{x}}_1,{\varvec{x}}_2)^{T},\) it follows from Theorem
2 that the variational inequality of
\(0.75-OPT\) is to find an
\({\varvec{x}}^{*}\in K\) such that
$$\begin{aligned} F({\varvec{x}}^{*},0.75)^{T}({\varvec{x}}^{*}-{\varvec{x}})\geqslant 0,\;\text{ for } \text{ all }\;{\varvec{x}}\in K. \end{aligned}$$
In other words, it is
$$\begin{aligned}&\nabla {\tilde{\theta }}_{1}({\varvec{x}}^{*},0.75)^{T}({\varvec{x}}_{1}^{*}-{\varvec{x}}_1) +\nabla {\tilde{\theta }}_{2}({\varvec{x}}^{*},0.75)^{T}({\varvec{x}}_{2}^{*}-{\varvec{x}}_2)\geqslant 0,\\&\quad \text{ for } \text{ all }\;x\in K, \end{aligned}$$
which is also
$$\begin{aligned} \sum _{i=1}^{2}\sum _{j=1}^{2}MR_{ij}(x_{ij}^{*},0.75)(x_{ij}^{*}-x_{ij})\geqslant 0 \;\text{ for } \text{ all }\;x_{ij}\in K.\nonumber \\ \end{aligned}$$
(15)
Furthermore, the variational inequality problem (
15) is equivalent to the following linear programming problem:
$$\begin{aligned}&\max _{x_{ij}\geqslant 0,i,j=1,2} \quad MR_{11}(x_{11}^{*},0.75)x_{11}+ MR_{12}(x_{12}^{*},0.75)x_{12}\\&\qquad +MR_{21}(x_{21}^{*},0.75)x_{21}+MR_{22}(x_{22}^{*},0.75)x_{22} \end{aligned}$$
subject to
$$\begin{aligned} {\left\{ \begin{array}{ll} &{}x_{11}+x_{21}\leqslant 984{,}837.6,\\ &{}x_{12}+x_{22}\leqslant 1{,}066{,}907.4.\\ \end{array}\right. } \end{aligned}$$
(16)
Let
\(\lambda _{1},\lambda _{2}\) be the dual variables of the constraints above, then the corresponding dual problem of (
16) is
$$\begin{aligned} \min _{\lambda _{1},\lambda _{2}\geqslant 0} \quad&984{,}837.6 \lambda _{1}+1{,}066{,}907.4\lambda _{2} \end{aligned}$$
subject to
$$\begin{aligned}&{\left\{ \begin{array}{ll} &{}\lambda _{1}-MR_{11}(x^{*}_{11},0.75)\geqslant 0,\\ &{}\lambda _{2}-MR_{12}(x^{*}_{12},0.75)\geqslant 0,\\ &{}\lambda _{1}-MR_{21}(x^{*}_{21},0.75)\geqslant 0,\\ &{}\lambda _{2}-MR_{22}(x^{*}_{22},0.75)\geqslant 0,\\ \end{array}\right. } \end{aligned}$$
(17)
The complementarity slackness implies that
$$\begin{aligned} {\left\{ \begin{array}{ll} &{} \lambda _{1}=MR_{11}(x^{*}_{11},0.75)=MR_{21}(x^{*}_{21},0.75),\\ &{} \lambda _{2}=MR_{12}(x^{*}_{12},0.75)=MR_{22}(x^{*}_{22},0.75),\\ \end{array}\right. } \end{aligned}$$
(18)
for any solution
\(x^{*}_{ij}>0,i,j=1,2.\)
It follows from (
14) and its counterparts of the electric utility
\(U_2,\) together with (
18), that
$$\begin{aligned} \begin{aligned} x^{*}_{11}&=506{,}241.2031,x^{*}_{12}=583{,}067.1021,\\ x^{*}_{21}&=478{,}596.397,x^{*}_{22}=483{,}840.298. \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} \lambda _{1}^{*}&=64{,}292.8065-\rho _{1},\\ \lambda _{2}^{*}&=35{,}408.8247-\rho _{2} \end{aligned} \end{aligned}$$
(19)
Let
\(\lambda _{i}^{*}=0,i=1,2.\) It follows from the remark of Theorem
2 that the equilibrium prices of the corresponding transmission links are
$$\begin{aligned} \begin{aligned} \rho _{1}^{*}&=64{,}292.8065,\\ \rho ^*_{2}&=35{,}408.8247.\\ \end{aligned} \end{aligned}$$
(20)
Expression (
20) reveals that the transmission prices of the residential and commercial sectors, respectively, are 64, 292.8065 and 35, 408.8247 USD/GWh. Both are close to the delivery prices 6.59, 4.38 Cents/KWh (that is 65, 900, 43, 800 USD/GWh) reported by the US Energy Information Administration. This application indicates that our model is not only effective but also can be used as a way to deal with the transmission pricing problem through a confidence level
\(\beta \) under uncertainties.