tutorial-4: clarifications and improvement of nomenclature

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 ... ... @@ -177,8 +177,7 @@ Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, t \begin{enumerate}[(a)] \item Calculate the price of electricity at each bus, the production of each generator, the flow on the line, and the value of any KKT multipliers for the following cases: of each generator, and the flow on the line for the following cases. You may also calculate the values of any KKT multiplier as a bonus. \begin{enumerate}[(i)] \item The line between buses A and B is disconnected. \item The line between buses A and B is in service and has an unlimited capacity. ... ...
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 ... ... @@ -158,76 +158,82 @@ Assume that the company is maximising its net surplus for a given electricity pr & = U'(d) - \pi - \m_{max} + \m_{min} \label{eq:2stat} \end{align} Note that it does not matter whether you pull in the constant of the right-hand side of the respective constraint. The marginal utility curve is $U'(d) = 70 - 6d$ [\euro/MWh]. At $\pi = 5$, the demand would be determined by $5=70-6d$, i.e. $d =$\pi = 5$and if the demand were unconstrained, the demand would be determined by$5=70-6d$, i.e.$d = 65/6 = 10.8333$, which is above the consumption limit$d_{max} = 10$. Therefore the optimal demand is$d^* = 10$, the upper limit is binding$\mu_{max} \geq 0$and the lower limit is non-binding$\mu_{min} = 0$.$d_{max} = 10$. Therefore the optimal demand is$d^* = 10$, the upper limit is binding such that$\mu_{max} \geq 0$and the lower limit is non-binding such that$\mu_{min} = 0$. To determine the value of$\mu_{max}$we use \eqref{eq:2stat} to get$\m_{max} = U'(d^*) - \pi = U'(10) - 5 = 5$.$\m_{max} = U'(d^*) - \pi + \mu_{min} = U'(10) - 5 + 0= 5$. \begin{shaded} \item Suppose now the electricity price is$\pi = 60$~\euro/MWh. What are the optimal demand$d^*$,$\mu_{max}$and$\mu_{min}$now? \end{shaded} At$\pi = 60$, the demand would be determined by$60=70-6d$, i.e.$d = 10/6 = 1.667$, which is below the consumption limit$d_{min} = 2$. Therefore the optimal demand is$d^* = 2$, the upper limit is non-binding$\mu_{max} = 0$and the lower limit is binding$\mu_{min} \geq 0$. At$\pi = 60$, the demand would be determined by$60=70-6d$, i.e.$d = 10/6 = 1.667$, which is below the consumption limit$d_{min} = 2$. Therefore the optimal demand is$d^* = 2$, the upper limit is non-binding such that$\mu_{max} = 0$and the lower limit is binding such that$\mu_{min} \geq 0$. To determine the value of$\mu_{min}$we use \eqref{eq:2stat} to get$\m_{min} = \pi - U'(d^*) = 60 - U'(2) = 2$.$\m_{min} = \pi - U'(d^*) + \mu_{max} = 60 - U'(2) + 0 = 2$. \end{enumerate} %=============== ====================================================== \paragraph{Solution IV.2 \normalsize (Economic dispatch in a single bidding zone).}~\\ %===================================================================== Consider an electricity market with two generator types, one with the cost function$C_1(G_1)=c_1G_1$with variable cost$c_1 = 20\emwh$, capacity$K_1 = 300\mw$and a dispatch rate of$G_1$~[MW], and another with the cost function$C_2(G_2)=c_2G_2$with variable cost$c_2=50\emwh$, capacity$K_2=400\mw$and a dispatch rate of$G_2$~[MW]. The demand has utility function$U(D) = 8000D - 5D^2$~[\euro/h] for a consumption rate of$D$~[MW]. Consider an electricity market with two generator types, one with the cost function$C_1(g_1)=c_1g_1$with variable cost$c_1 = 20\emwh$, capacity$G_1 = 300\mw$and a dispatch rate of$g_1$~[MW], and another with the cost function$C_2(g_2)=c_2g_2$with variable cost$c_2=50\emwh$, capacity$G_2=400\mw$and a dispatch rate of$g_2$~[MW]. The demand has utility function$U(d) = 8000d - 5d^2$~[\euro/h] for a consumption rate of$d~[MW]. \begin{enumerate}[(a)] \begin{shaded}\item What are the objective function and constraints required for an optimisation problem to maximise short-run social welfare in this market?\end{shaded} The optimisation problem has objective function: The optimisation problem has the objective function: \begin{equation*} \max_{D,G_1,G_2}\left[ U(D) - C_1(G_1) - C_2(G_2) \right] = \max_{D,G_1,G_2} \left[8000D-5D^2 - c_1G_1 - c_2G_2 \right] \max_{d,g_1,g_2}\left[ U(d) - C_1(g_1) - C_2(g_2) \right] = \max_{d,g_1,g_2} \left[8000d-5d^2 - c_1g_1 - c_2g_2 \right] \end{equation*} with constraints: \begin{align*} D - G_1 - G_2 & = 0 \leftrightarrow \l \\ G_1 & \leq K_1 \leftrightarrow \bar{\m}_1 \\ G_2 & \leq K_2 \leftrightarrow \bar{\m}_2 \\ -G_1 & \leq 0 \leftrightarrow \ubar{\m}_1 \\ -G_2 & \leq 0 \leftrightarrow \ubar{\m}_2 d - g_1 - g_2 & = 0 \leftrightarrow \l \\ g_1 & \leq G_1 \leftrightarrow \bar{\m}_1 \\ g_2 & \leq G_2 \leftrightarrow \bar{\m}_2 \\ -g_1 & \leq 0 \leftrightarrow \ubar{\m}_1 \\ -g_2 & \leq 0 \leftrightarrow \ubar{\m}_2 \end{align*} \begin{shaded}\item Write down the Karush-Kuhn-Tucker (KKT) conditions for this problem.\end{shaded} Stationarity gives forD$: Stationarity gives for$d$: \begin{equation*} \frac{\d U}{\d D} - \l = 8000 - 10D - \l = 0 \frac{\d U}{\d d} - \l = 8000 - 10d - \l = 0 \end{equation*} Stationarity for$G_1$gives: Stationarity for$g_1$gives: \begin{equation*} -\frac{\d C_1}{\d G_1} + \l - \m_1 = -c_1+ \l - \bar{\m}_1 + \ubar{\m_1} = 0 -\frac{\d C_1}{\d g_1} + \l - \m_1 = -c_1+ \l - \bar{\m}_1 + \ubar{\m_1} = 0 \end{equation*} Stationarity for$G_2$gives: Stationarity for$g_2$gives: \begin{equation*} -\frac{\d C_2}{\d G_2} + \l - \m_2 = -c_2+ \l - \bar{\m}_2 + \ubar{\m_2} = 0 -\frac{\d C_2}{\d g_2} + \l - \m_2 = -c_2+ \l - \bar{\m}_2 + \ubar{\m_2} = 0 \end{equation*} Primal feasibility is just the constraints above. Dual feasibility is$\bar{\m}_i,\ubar{\m}_i \geq 0$and complementary slackness is$\bar{\m}_i(G_i-K) = 0$and$\ubar{\m}_i G_i = 0$for$i=1,2$. Primal feasibility is just the generator limits above in (a). Dual feasibility is$\bar{\m}_i,\ubar{\m}_i \geq 0$and complementary slackness is$\bar{\m}_i(G_i-K) = 0$and$\ubar{\m}_i G_i = 0$for$i=1,2$. \begin{shaded}\item Determine the optimal rate of production of the generators and the value of all KKT multipliers. What is the interpretation of the respective KKT multipliers?\end{shaded} The marginal utility at the full output of the generators,$K_1 + K_2 = $700~MW is$U'(700) = 8000 - 10\cdot700 = 1000$\euro/MWh, The marginal utility at the full output of the generators,$G_1 + G_2 = $700~MW is$U'(700) = 8000 - 10\cdot700 = 1000$\euro/MWh, which is higher than the costs$c_i$, so we'll find optimal rates$G_1^* = K_1$and$G_2^* = K_2$and$D^* = K_1+K_2$. This means$\l = U'(K_1+K_2) = 1000$\euro/MWh, which is the market price. Because$g_1^* = G_1$and$g_2^* = G_2$and$d^* = G_1+G_2$. This means$\l = U'(G_1+G_2) = 1000$\euro/MWh, which is the market price. Because the lower constraints on the generator output are not binding, from complementary slackness we have$\ubar{\m}_i = 0$. The upper constraints are binding, so$\bar{\m}_i \geq 0$. From stationarity$\bar{\m}_i = \l - c_i$, which is the increase in social welfare if Generator$i$\l - c_i + \ubar{\m}_i$, which is the increase in social welfare if Generator $i$ could increase its capacity by a marginal amount. $$\bar{\m}_1=1000-20=980 \text{\euro/MWh}$$ $$\bar{\m}_2=1000-50=950 \text{\euro/MWh}$$ \end{enumerate} %=============== ====================================================== ... ... @@ -251,8 +257,10 @@ Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, t \begin{enumerate}[(a)] \begin{shaded}\item Calculate the price of electricity at each bus, the production of each generator, the flow on the line, and the value of any KKT multipliers for the following cases:\end{shaded} of each generator, and the flow on the line for the following cases. You may also calculate the values of any KKT multiplier as a bonus.\end{shaded} The price of electricity is the value of the dual variable at the nodal balance equation. Use the following nomenclature: price $\lambda_{A/B}$, generation $G_{A/B}$, flow $F_{AB}$. \begin{enumerate}[(i)] \begin{shaded}\item The line between buses A and B is disconnected.\end{shaded} ... ...
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