A Merchant Mechanism for Electricity Transmission Expansion

Abstract

The analysis of incentives for electricity transmission expansion is not easy. Beyond economies of scale and cost sub-additivity externalities in electricity transmission are mainly due to “loop flows” that come up from complex network interactions. The effects of loop flows imply that transmission opportunity costs are a function of the marginal costs of energy at each location. Power costs and transmission costs depend on each other since they are simultaneously settled in electricity dispatch. Loop flows imply that certain transmission investments might have negative externalities on the capacity of other (perhaps distant) transmission links (see Bushnell and Stoft 1997). Moreover, the addition of new transmission capacity can sometimes paradoxically decrease the total capacity of the network (Hogan 2002a).

This paper was originally published as: Kristiansen T, Rosellón J (2006) A merchant mechanism for electricity transmission expansion. J Regul Econ 29(2):167–193. We are grateful to William Hogan for very useful suggestions and discussions. We also thank Ross Baldick, Ingo Vogelsang, and an anonymous referee for insightful comments. All remaining errors are our own. Part of this research was carried out while Kristiansen was a Fellow, and Rosellón a Fulbright Senior Fellow, at the Harvard Electricity Policy Group, Center of Business and Government, John F. Kennedy School of Government of Harvard University. Rosellón acknowledges support from the Repsol-YPF-Harvard Kennedy School Fellows Program and the Fundación México en Harvard, as well as from CRE and Conacyt.

Appendices

Appendix 1

6.1.1 Solution to Program (6.30)

The Lagrangian of the problem is:

$$ \begin{aligned} & L(a,\hat{t},{\delta_{12 }},\lambda, \varOmega )={b_{12 }}a{\delta_{12 }}+\gamma \left( {C_{12}^{+}-{T_{12 }}-(a+\hat{t}){\delta_{12 }}} \right) \\ & -\theta ({p_{12 }}{\delta_{12 }}-\lambda {\delta_{12 }})-\zeta (\lambda ({C_{12 }}-{T_{12 }}-\hat{t}{\delta_{12 }})) \\ & +\varepsilon ({C_{12 }}-{T_{12 }}-\hat{t}{\delta_{12 }})+\varphi \left( {1-\delta_{12}^2} \right)+\kappa a+\pi \lambda \\ \end{aligned} $$

(6.37)

where $\gamma, \theta, \zeta, \varepsilon, \varphi, \kappa,$ and $\pi$ are the multipliers associated with the respective constraints.

At optimality the Kuhn-Tucker conditions are:

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial a}}={b_{12 }}{\delta_{12 }}-\gamma {\delta_{12 }}=0, $$

(6.38)

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \varOmega )}}{{\partial {\delta_{12 }}}}=a{b_{12 }}-(\hat{t}+a)\gamma -({p_{12 }}-\lambda )\theta \\ & +\lambda \zeta \hat{t}-\varepsilon \hat{t}-2{\delta_{12 }}\varphi =0, \\ \end{aligned} $$

(6.39)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \hat{t}}}=-\gamma {\delta_{12 }}+\lambda \zeta {\delta_{12 }}-\varepsilon {\delta_{12 }}=0, $$

(6.40)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \lambda }}={\delta_{12 }}\theta -({C_{12 }}-{T_{12 }}-\hat{t}{\delta_{12 }})\zeta =0, $$

(6.41)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \gamma }}=\left( {C_{12}^{+}-{T_{12 }}-(a+\hat{t}){\delta_{12 }}} \right)=0, $$

(6.42)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \theta }}=-({p_{12 }}{\delta_{12 }}-\lambda {\delta_{12 }})=0, $$

(6.43)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \zeta }}=-\lambda ({C_{12 }}-{T_{12 }}-\hat{t}{\delta_{12 }})=0, $$

(6.44)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \varepsilon }}=({C_{12 }}-{T_{12 }}-\hat{t}{\delta_{12 }})=0, $$

(6.45)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \varphi }}=\left( {1-\delta_{12}^2} \right)=0, $$

(6.46)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \kappa }}=a>0,\quad \kappa =0, $$

(6.47)

$$ \frac{{\partial L(a,\hat{t},{\delta_{12 }},\lambda, \Omega )}}{{\partial \pi }}=\lambda >0,\quad \pi =0, $$

(6.48)

$$ \gamma, \varepsilon \ge 0 $$

(6.49)

Equation (6.46) gives ${\delta_{12 }}=1$. Equation (6.38) gives $\gamma ={b_{12 }}$. Equation (6.43) gives $\gamma ={b_{12 }}$, (6.40) $\zeta =\gamma /{p_{12 }}={b_{12 }}/{p_{12 }}$ ($\varepsilon$ is zero because the constraint is redundant), and (6.41) $\theta =0$. From this it follows (6.39) that $\varphi =0$ Furthermore (6.44) gives $\hat{t}={C_{12 }}-{T_{12 }}$. Equation (6.42) implies that $a=C_{12}^{+}-{T_{12 }}-\hat{t}=C_{12}^{+}-{C_{12 }}$.

6.1.2 Solution to Program (6.32)

The Lagrangian of the problem is:

$$ \begin{aligned} & L(a,\hat{t},\delta, \lambda, \varOmega )= \\ & a({b_{13 }}{\delta_{13 }}+{b_{23 }}{\delta_{23 }})+{\gamma_1}\left( {{C_{13 }}-\frac{2}{3}({T_{13 }}+(\hat{t}+a){\delta_{13 }}} \right)-\frac{1}{3}({T_{23 }}+(\hat{t}+a){\delta_{23 }})) \\ & +{\gamma_2}\left( {{C_{12 }}-\frac{1}{3}({T_{13 }}+(\hat{t}+a){\delta_{13 }}} \right)+\frac{1}{3}({T_{23 }}+(\hat{t}+a){\delta_{23 }})) \\ & -\zeta (\lambda ({C_{13 }}-({T_{13 }}+\hat{t}{\delta_{13 }})) \\ \end{aligned} $$

(6.50)

$$ \begin{aligned} & +\varepsilon ({C_{13 }}-({T_{13 }}+\hat{t}{\delta_{13 }})) \\ & +\varphi \left( {1-\delta_{13}^2-\delta_{23}^2} \right)+\kappa a+\pi \lambda \\ \end{aligned} $$

where ${\gamma_1}\;\mathrm{ and}\;{\gamma_2}$ are the Lagrange multipliers associated with transmission capacity on the lines 1–3 and 1–2 in the expanded network, respectively. $\zeta$ is the multiplier associated with the Kuhn-Tucker condition of transmission capacity in the pre-expansion network for line 1–3. This line has the Lagrange multipliers $\lambda$ associated with it before expansion. $\varepsilon$ is the investor’s marginal value of transmission capacity in the pre-expansion network when allocating incremental FTRs. The normalization condition has the multiplier $\varphi$ and the non-negativity conditions have the associated multipliers $\kappa$ and $\pi$. The first order conditions are:

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial a}}=({b_{13 }}{\delta_{13 }}+{b_{23 }}{\delta_{23 }})-\left( {\frac{2}{3}{\delta_{13 }}+\frac{1}{3}{\delta_{23 }}} \right){\gamma_1} \\ & -\left( {\frac{1}{3}{\delta_{13 }}-\frac{1}{3}{\delta_{23 }}} \right){\gamma_2}=0, \\ \end{aligned} $$

(6.51)

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial {\delta_{13 }}}}=a{b_{13 }}-\frac{2}{3}(\hat{t}+a){\gamma_1}-\frac{1}{3}(\hat{t}+a){\gamma_2} \\ & +\zeta \lambda \hat{t}-\varepsilon \hat{t}-2\varphi {\delta_{13 }}=0, \\ \end{aligned} $$

(6.52)

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial {\delta_{23 }}}}=a{b_{23 }}-\frac{1}{3}(\hat{t}+a){\gamma_1}+\frac{1}{3}(\hat{t}+a){\gamma_2} \\ & -2\varphi {\delta_{23 }}=0, \\ \end{aligned} $$

(6.53)

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial \hat{t}}}=-\left( {\frac{2}{3}{\delta_{13 }}+\frac{1}{3}{\delta_{23 }}} \right){\gamma_1}-\left( {\frac{1}{3}{\delta_{13 }}-\frac{1}{3}{\delta_{23 }}} \right){\gamma_2} \\ & +{\delta_{13 }}\zeta \lambda -{\delta_{13 }}\varepsilon =0, \\ \end{aligned} $$

(6.54)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \lambda }}=-\zeta ({C_{13 }}-{T_{13 }}-\hat{t}{\delta_{13 }})=0, $$

(6.55)

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial {\gamma_1}}}={C_{13 }}-\frac{2}{3}({T_{13 }}+(\hat{t}+a){\delta_{13 }}) \\ & -\frac{1}{3}({T_{23 }}+(\hat{t}+a){\delta_{23 }})=0, \\ \end{aligned} $$

(6.56)

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial {\gamma_2}}}={C_{12 }}-\frac{1}{3}({T_{13 }}+(\hat{t}+a){\delta_{13 }}) \\ & +\frac{1}{3}({T_{23 }}+(\hat{t}+a){\delta_{23 }})=0, \\ \end{aligned} $$

(6.57)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \zeta }}=-\lambda ({C_{13 }}-({T_{13 }}+\hat{t}{\delta_{13 }}))=0, $$

(6.58)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \varepsilon }}=({C_{13 }}-{T_{13 }}-\hat{t}{\delta_{13 }})=0, $$

(6.59)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \varphi }}=1-\delta_{13}^2-\delta_{23}^2=0, $$

(6.60)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \kappa }}=a>0,\quad \kappa =0, $$

(6.61)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \pi }}=\lambda >0,\quad \pi =0, $$

(6.62)

The solution for the first order conditions is given by:

$$ \begin{gathered} {\delta_{13 }}=\frac{{(1/3{\gamma_1}-1/3{\gamma_2})}}{{{{{\left( {{{{(2/3{\gamma_1}+1/3{\gamma_2}-\zeta \lambda )}}^2}+{{{(1/3{\gamma_1}-1/3{\gamma_2})}}^2}} \right)}}^{1/2 }}}} \hfill \\ {\delta_{23 }}=\frac{{-(2/3{\gamma_1}+1/3{\gamma_2}-\zeta \lambda )}}{{{{{\left( {{{{(2/3{\gamma_1}+1/3{\gamma_2}-\zeta \lambda )}}^2}+{{{(1/3{\gamma_1}-1/3{\gamma_2})}}^2}} \right)}}^{1/2 }}}} \hfill \\ \end{gathered} $$

$$ \begin{aligned} & a=\frac{{{C_{12 }}}}{{{\delta_{13 }}}} \\ & \hat{t}=\frac{{({C_{13 }}-{T_{13 }})}}{{{\delta_{13 }}}} \\ \end{aligned} $$

$$ {\gamma_1}=\frac{{({b_{13 }}+B{b_{23 }}+{\gamma_2}(B/3-1/3))}}{(2/3+B/3) } $$

$$ \begin{aligned} & {\gamma_2}=\frac{1}{(1-B-AB+A)}\left[ {{b_{13 }}(1+3A-B-2A-AB)} \right. \\ & +{b_{23 }}(B+3AB-{B^2}-2A-A\left. {B)} \right] \\ \end{aligned} $$

$$ \zeta \lambda =(1+A){\gamma_1}-A({b_{13 }}+{b_{23 }}) $$

with

$$ \begin{aligned} & A=\frac{{{C_{12 }}}}{{({C_{13 }}-{T_{13 }})}} \\ & B=\frac{1}{(1+A)}\frac{{({C_{13 }}-2{C_{12 }}-{T_{23 }})}}{{({C_{13 }}-{T_{13 }})}} \\ \end{aligned} $$

Appendix 2

This appendix derives the power transfer distribution factors (PTDFs) for the three-node network with two parallel lines, and where all lines have identical reactance. The net injection (or net generation) of power at each bus is denoted P _i . We have the following relationship between the net injection, the power flows P _ij and phase angles ${\theta_i}$ (Wood and Wollenberg 1996):

$$ {P_i}=\sum\limits_j {{P_{ij }}=\sum\limits_j {\frac{1}{{{x_{ij }}}}(} } {\theta_i}-{\theta_j}) $$

where ${x_{ij }}$ is the line inductive reactance in per unit.

We can write the power flow equations as:

$$ \left[ \begin{gathered} {P_1} \hfill \\ {P_2} \hfill \\ {P_3} \hfill \\ \end{gathered} \right]=\left[ \begin{gathered} 2\quad -1\quad -1 \hfill \\ -1\quad 2\quad -1 \hfill \\ -1 - 1\quad\;\;\,2 \hfill \\ \end{gathered} \right]\,\left[ \begin{gathered} {\theta_1} \hfill \\ {\theta_2} \hfill \\ {\theta_3} \hfill \\ \end{gathered} \right] $$

The matrix is called the susceptance matrix. The matrix is singular, but by declaring one of the buses to have a phase angle of zero and eliminating its row and column from the matrix, the reactance matrix can be obtained by inversion. The resulting equation then gives the bus angles as a function of the bus injection:

$$ \left[ \begin{gathered} {\theta_2} \hfill \\ {\theta_3} \hfill \\ \end{gathered} \right]=\left[ \begin{gathered} 2/3\quad 1/3 \hfill \\ 1/3\quad 2/3 \hfill \\ \end{gathered} \right]\,\left[ \begin{gathered} {P_2} \hfill \\ {P_3} \hfill \\ \end{gathered} \right] $$

The PTDF is the fraction of the amount of a transaction from one node to another node that flows over a given line. PTDF _ij,mn is the fraction of a transaction from node m to node n that flows over a transmission line connecting node i and node j. The equation for the PTDF is:

$$ PTD{F_{ij,mn }}=\frac{{{x_{im }}-{x_{jm }}-{x_{in }}+{x_{jn }}}}{{{x_{ij }}}} $$

where x _ij is the reactance of the transmission line connecting node i and node j and x _im is the entry in the ith row and the mth column of the bus reactance matrix. Utilizing the formula for the specific example network gives:

$$ \begin{aligned} & PTD{F_{12,13 }}=1/3,\,\,\,PTD{F_{13,13 }}=2/3,\,\,PTD{F_{23,13 }}=1/3, \\ & PTD{F_{12,23 }}=-1/3,\,\,\,PTD{F_{13,23 }}=1/3,\,\,PTD{F_{23,23 }}=2/3 \\ & PTD{F_{21,13 }}=-1/3,\,\,\,PTD{F_{21,23 }}=1/3 \\ \end{aligned} $$

Appendix 3

6.3.1 Transmission Investment That Does Not Change PTDFs

An example on an investment that does not change the PTDFs of the network is shown in Fig. 6.5 where there is an expansion of line 1–3 from 900 to 1,000 MW transmission capacity. The associated feasible expansion FTR set is shown in Fig. 6.6. We observe that whatever feasible FTRs that existed before expansion, none of these will become infeasible after the expansion.

Fig. 6.5

Three-node network with expansion in one line

Fig. 6.6

Feasible expansion FTR set

6.3.2 Transmission Investment That Does Change PTDFs

Figure 6.7 shows a three-node network where a line is inserted in parallel with an existing line between the nodes 2 and 3. Inserting a parallel line with identical reactance as the existing line halves the total reactance between nodes 2 and 3. As a result the PTDFs of the expanded network change.

$$ PTD{F_{12,13 }}=1/3\;\mathrm{and}\;PTD{F_{13,13 }}=2/3 $$

change to

$$ PTD{F_{12,13 }}=0.4\;\mathrm{and}\;PTD{F_{13,13 }}=0.6. $$

Furthermore, the inserted line has identical transmission capacity to the existing one so that the total transmission capacity is doubled between the buses 2 and 3. However, the simultaneous interaction of the reactances and transmission capacities changes the feasible expansion FTR set as illustrated in Fig. 6.8. Then some of the pre-existing FTRs may become infeasible.

Fig. 6.7

Three-node network where a line is inserted in parallel with an existing line

Fig. 6.8

Feasible expansion FTR set