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.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

6.1 Introduction

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.¹ 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).

The welfare effects of an increment in transmission capacity are analyzed by Léautier (2001). The welfare outcome of an expansion in the transmission grid depends on the weight in the welfare function of the generators’ profits relative to the consumers’ utility weight. Incumbent generators in load pockets are not in general the best agents to carry out transmission expansion projects. Even though an increase in transmission capacity might allow them to increase their revenues due to increased access to new markets and higher transmission charges, such gains are usually overcome by the loss of their local market power.

The literature on incentives for long-term expansion of the transmission network is scarce. The economic analysis of electricity markets has been reduced to short-run issues, and has typically assumed that transmission capacity is fixed (see Joskow and Tirole 2003). However, transmission capacity is random in nature, and it jointly depends on generation investment.

The way to solve transmission congestion in the short run is well known. In a power flow model, the price of transmission congestion is determined by the difference in nodal prices (see Hogan 1992, 2002b). Yet, there is no consensus with respect to the method to attract investment to finance the long-term expansion of the transmission network, so as to reconcile the dual opposite incentives to congest the network in the short run, and to expand it in the long run. Incentive structures proposed to promote transmission investment range from a “merchant” mechanism, based on long-term financial transmission right (LTFTR) auctions (as in Hogan 2002a), to regulatory mechanisms that charge the transmission firm the social cost of transmission congestion (see Léautier 2000; Vogelsang 2001; Joskow and Tirole 2002).

In practice, regulation has been used in England, Wales and Norway to promote transmission expansion, while a combination of planning and auctions of long-term transmission rights has been tried in the Northeast of the U.S. A mixture of regulatory mechanisms and merchant incentives is alternatively used in the Australian market.

In this paper we develop a merchant model to attract investment to small-scale electricity transmission projects based on LTFTR auctions. Locational prices give market players incentives to initiate transmission investments. FTRs provide transmission property rights, since they hedge the market player against future price differences. Our model further develops basic conditions under which FTRs and locational pricing provide incentives for long-term investment in the transmission network.

In meshed networks, a change in network capacity might imply negative externalities on transmission property rights. Then, in the process of allocation of incremental FTRs, the system operator must reserve certain unallocated FTRs so that the revenue adequacy of the transmission system is preserved. In order to deal with this issue, we develop a bi-level programming model for allocation of long-term FTRs and apply it to different network topologies.

The structure of the paper is as follows. In Sect. 6.2 we carry out an analytical review on the relevant literature on electricity transmission expansion. In Sect. 6.3 we develop our model. We first introduce FTRs and the feasibility rule, and then address the rationale for FTR allocation and efficient investments. We develop general optimality conditions as well. In Sect. 6.4, we carry out applications of our model to a radial line, and to a three-node network. Next, we describe the welfare implications in Sect. 6.5. In Sect. 6.6 we provide concluding comments.

6.2 Literature Review

There exist some hypotheses on structures for transmission investment: the market-power hypothesis, the incentive-regulation hypothesis, and the long-run financial-transmission-right hypothesis. The first approach seeks to derive optimal transmission expansion from the power-market structure of power generators, and takes into account the conjectures of each generator regarding other generators’ marginal costs due to the expansion (Sheffrin and Wolak 2001; Wolak 2000, and The California ISO and London Economics International 2003). The generators’ bidding behaviors are estimated before and after a transmission upgrade, and a real-option analysis is used to derive the net present value of transmission and generation projects together with the computation of their joint probability.

The model shows that there are few benefits of transmission expansion until added capacity surpasses a certain threshold that, in turn, is determined by the possibility of induced congestion by the strategic behavior of generators with market power. The generation market structure then determines when transmission expansion yield benefits. Additionally, many small upgrades of the transmission grid are preferable to large greenfield projects when cost uncertainty is added to the model.

The contribution of this method is that it models the existing interdependence of transmission investment and generation investment within a transportation model with no network loop flows. However, as pointed out by Hogan (2002b), the use of a transportation model in the electricity sector is inadequate since it does not deal with discontinuities in transmission capacity implied by the multidimensional character of a meshed network.

The second method for transmission expansion is a regulatory alternative that relies on a “Transco” that simultaneously runs system operation and owns the transmission network. The Transco is regulated through benchmark regulation or price regulation so as to provide it with incentives to invest in the development of the grid, while avoiding congestion. Léautier (2000), Grande and Wangensteen (2000), and Harvard Electricity Policy Group (2002) discuss mechanisms that compare the Transco performance with a measure of welfare loss due to its activities. Joskow and Tirole (2002) propose a surplus-based mechanism to reward the Transco according to the redispatch costs avoided by the expansion, so that the Transco faces the complete social cost of transmission congestion.

Another regulatory alternative is a two-part tariff cap proposed by Vogelsang (2001) that addresses the opposite incentives to congest the existing transmission grid in the short run, and to expand it in the long run. Incentives for investment in expansion of the network are achieved through the rebalancing of the fixed part and the variable part of the tariff. This method tries to deepen into the analysis of the cost and production functions for transmission services, which are not very well understood in the economics literature. Nonetheless, to achieve this goal Vogelsang needs to define an output (or throughput) for the Transco. As argued in the FTR literature (Bushnell and Stoft 1997; Hogan 2002a, b), this task is very difficult since the average physical flow through a meshed transmission network is not well defined.

The third approach is a “merchant” one based on LTFTR auctions by an independent system operator (ISO). This method deals with loop-flow externalities in that, to proceed with line expansions, the investor pays for the negative externalities it generates. To restore feasibility, the investor has to buy back sufficient transmission rights from those who hold them initially, or the ISO retains some unallocated transmission rights (proxy awards) during the LTFTR auction to protect unassigned rights while simultaneous feasibility of the system protects the rights of the existing FTR holders. This is the core of an LTFTR auction (see Hogan 2002a).

Joskow and Tirole (2003) criticize the LTFTR approach. They argue that the efficiency results of the short-run version of the FTR model rely on perfect-competition assumptions, which are not real for transmission networks. Moreover, defining an operational FTR auction is technically difficult² and, according to these authors, the FTR analysis is static (a contradiction with the dynamics of transmission investment). Joskow and Tirole analyze the implications of eliminating the perfect competition assumptions of the FTR model.

First, market power and vertical integration might impede the success of FTR auctions. Prices will not reflect the marginal cost of production in regions with transmission constraints. Generators in constrained regions will then withdraw capacity in order to increase their prices, and will overestimate the cost-saving gains from investments in transmission.³

Second, lumpiness in transmission investment makes the total value paid to investors through FTRs less than the social surplus created. The large and lumpy nature of major transmission upgrades requires long-term contracts before making the investment, or temporal property rights for the incremental investment.

Third, contingencies in electricity transmission impede the merchant approach to really solve the loop-flow problem. Moreover, existing transmission capacity and incremental capacity are stochastic. Even in a radial line, realized capacity could be less than expected capacity and the revenue-adequacy condition would not be met. Even more, the initial feasible FTR set can depend on random exogenous variables.

Fourth, an expansion in transmission capacity might negatively affect social welfare (as shown by Bushnell and Stoft 1997).

Fifth, a moral hazard “in teams” problem arises due to the separation of transmission ownership and system operation in the FTR model. For instance, an outage can be claimed to be the consequence of poor maintenance (by the transmission owner) or of negligent dispatch (by the system operator).⁴ Additionally, there is no perfect coordination of interdependent investments in generation and transmission, and stochastic changes in supply and demand conditions imply uncertain nodal prices. Likewise, there is no equal access to investment opportunities since only the incumbent can efficiently carry out deepening transmission investments.

Hogan (2003) responds to the above criticisms by arguing that LTFTRs only grant efficient outcomes under lack of market power, and non-lumpy marginal expansions of the transmission network. Furthermore, Hogan argues that regulation has an important role in fostering large and lumpy projects, and in mitigating market power abuses.

As argued by Pérez-Arriaga et al. (1995), revenues from nodal prices typically recover only 25 % of total costs. LTFTRs should then be complemented with a fix-price structure or, as in Rubio-Oderiz and Pérez-Arriaga (2000) a complementary charge that allows the recovery of fixed costs.⁵ This fact is recognized by Hogan (1999) who states that complete reliance on market incentives for transmission investment is undesirable. Rather, Hogan (2003) claims that merchant and regulated transmission investments might be combined so that regulated transmission investment is limited to projects where investment is large relative to market size, and lumpy so that it only makes sense as a single project as opposed as to many incremental small projects.

Hogan also responds to contingency concerns.⁶ On one hand, only those contingencies outside the control of the system operator could lead to revenue inadequacy of FTRs, but such cases are rare and do not represent the most important contingency conditions. On the other hand, most of remaining contingencies are foreseen in a security-constrained dispatch of a meshed network with loops and parallel paths. If one of “n” transmission facilities is lost, the remaining power flows would still be feasible in an “n − 1” contingency constrained dispatch.

Hogan (2003) also assumes that agency problems and information asymmetries are part of an institutional structure of the electricity industry where the ISO is separated from transmission ownership and where market players are decentralized. However, he claims that the main issue on transmission investment is the decision of the boundary between merchant and regulated transmission expansion projects. He argues that asymmetric information should not necessarily affect such a boundary.

Hogan (2002a) finally analyzes the implications of loop flows on transmission investment raised by Bushnell and Stoft (1997). He analytically provides some general axioms to properly define LTFTRs so as to deal with negative externalities implied by loop flows. We next present a model that develops the general analytical framework suggested by Hogan (2002a).

6.3 The Model

Assume an institutional structure where there are various established agents (generators, Gridcos, marketers, etc.) interested in the transmission grid expansion. Agents do not have market power in their respective market or, at least, there are in place effective market-power mitigation measures.⁷ Also assume that transmission projects are incrementally small (relative to the total network) and non-lumpy so that the project does not imply a relatively large change in nodal-price differences. However, although projects are small, they might change or not the power transfer distribution factors (PTDFs) of the network.⁸

Under an initial condition of non-fully allocation of FTRs in the grid, the auctioning of incremental LTFTRs should satisfy the following basic criteria in order to deal with possible negative externalities associated with the expansion.

An LTFTR increment must keep being simultaneously feasible (feasibility rule).

• 1. An LTFTR increment remains simultaneously feasible given that certain currently unallocated rights (or proxy awards) are preserved.

• 2. Investors should maximize their objective function (maximum value).

• 3. The LTFTR awarding process should apply both for decreases and increases in the grid capacity (symmetry).

The need for proxy awards arises whenever there is less than full allocation of the capacity of the existing grid. This occurs prominently during a transition to an electricity market when there is reluctance to fully allocate the existing grid for all future periods. Hence FTRs for the existing grid are short term (this period), but investors in grid expansion seek long term rights (next period). Full allocation of the existing grid seems necessary but not sufficient for defining and measuring incremental capacity. Hogan explains though that defining proxy awards is a difficult task. We next address this issue in a formal way in the context of an auction model designed to attract investment for transmission expansion.

6.4 The Power Flow Model and Proxy Awards

Consider the following economic dispatch model⁹:

$$ \begin{aligned} & \mathop{\mathrm{Max}}\limits_{{Y,u\in U}}B(d-g) \\ & s.\,t. \\ \end{aligned} $$

(6.1)

$$ Y=d-g, $$

(6.2)

$$ L(Y,u)+{\tau^T}Y=0 $$

(6.3)

$$ K(Y,u)\le 0 $$

(6.4)

where $d$ and $g$ are load and generation at the different locations. The variable $Y$ represents the real power bus net loads, including the swing bus $S({Y^T}=({Y_s},{{\bar{Y}}^T})).B(d-g)$ is the net benefit function,¹⁰ and $\tau$ is a unity column vector, ${\tau^T}=(1,1,\ldots,1)$ All other parameters are represented in the control variable u. The objective equation (6.1) includes the maximization of benefit to loads and the minimization of generation costs. Equation (6.2) denotes the net load as the difference between load and generation. Equation (6.3) is a loss balance constraint where $L(Y,u)$ is a vector which denotes the losses in the network. In (6.4) $K(Y,u)$, is a vector of power flows in the lines, which are subject to transmission capacity limits. The corresponding multipliers or shadow prices for the constraints are $(P,{\lambda_{ref }},{\lambda_{tran }})$ for net loads, reference bus energy (or loss balance) and transmission constraints, respectively.¹¹

The locational prices P are the marginal generation cost or the marginal benefit of demand, which in turn equals the reference price of energy plus the marginal cost of losses and congestion. With the optimal solution $({d^{*}},{g^{*}},{Y^{*}},{u^{*}})$ and the associated shadow prices, we have the vector of locational prices as:

$$ {P^T}=\nabla C({g^{*}})=\nabla B({d^{*}})={\lambda_{ref }}{\tau^T}+{\lambda_{ref }}\nabla {L_Y}({Y^{*}},{u^{*}})+\lambda_{tran}^T\nabla {K_Y}({Y^{*}},{u^{*}}) $$

(6.5)

If losses¹² are ignored, only the energy price at the reference bus and the marginal cost of congestion contribute to set the locational price.

FTR obligations¹³ hedge market players against differences in locational prices caused by transmission congestion.¹⁴ FTRs are provided by an ISO, and are assumed to redistribute the congestion rents. The pay-off from these rights is given by:

$$ FTR=({P_j}-{P_i}){Q_{ij }} $$

(6.6)

where P _j is the price at location j, P _i is the price at location i, and Q _ij is the directed quantity injected at point i and withdrawn at point j specified in the FTR. The FTR payoffs can take negative, positive or zero values.

A set of FTRs is said to be simultaneously feasible if the associated set of net loads is simultaneously feasible, that is if the net loads satisfy the loss balance and transmission capacity constraints as well as the power flow equations given by:

$$ \begin{aligned} & Y=\sum\limits_k {t_k^f} \\ & L(Y,u)+{\tau^t}Y=0, \\ & K(Y,u)\leqslant 0 \\ \end{aligned} $$

(6.7)

where $\sum\limits_k {t_k^f}$ is the sum over the set of point-to-point obligations.¹⁵

If the set of FTRs is simultaneously feasible and the system constraints are convex,¹⁶ then the FTRs satisfy the revenue adequacy condition in the sense that equilibrium payments collected by the ISO through economic dispatch will be greater than or equal to payments required under the FTR forward obligations.¹⁷

Assume now investments in new transmission capacity. The associated set of new FTRs for transmission expansion has to satisfy the simultaneous feasibility rule too. That is, the new and old FTRs have to be simultaneously feasible after the system expansion. Assume that T is the current partial allocation of long-term FTRs, then by assumption it is feasible $(K(T,u)\le 0)$. Suppose there is to be a total possible incremental award, and that a fraction of the possible awards is reserved as proxy awards for the existing grid with the remainder provided to the incremental investor as representing the proportion that could only be awarded as a result of the investment. Let a be the scalar amount of incremental FTR awards, and $\hat{t}$ the scalar amount of proxy awards. Furthermore let $\delta$ be directional vector¹⁸ such that $a\delta$ is the MW amount of incremental FTR awards, and $\hat{t}\delta$ is the MW amount of proxy awards between different locations. Any incremental FTR award $a\delta$ should comply with feasibility rule in the expanded grid. Hence we must have ${K^{+}}(T+a\delta, u)\le 0$, where K ⁺ corresponds to capacity of the expanded grid.

When certain currently unallocated rights (proxy awards) $\hat{t}\delta$ in the existing grid must be preserved, combined with existing rights they sum up to $T+\hat{t}\delta$.¹⁹ Then K ⁺ should also satisfy simultaneous feasibility so that $K(T+\hat{t}\delta, u)\le 0$, ${K^{+}}(T+a\delta, u)\le 0$, and ${K^{+}}(T+\hat{t}\delta +a\delta, u)\le 0$ for incremental awards $a\delta$ _.

A question then arises regarding the way to best define proxy awards. One possibility is to define them as the “best use” of the current network along the same direction as the incremental awards.²⁰ This includes both positive and negative incremental FTR awards. The best use in a three-node network may be thought of as a single incremental FTR in one direction or a combination of incremental FTRs defined by the directional vector $\delta$, depending on the investor preference. Hogan (2002a) suggests two ways of defining “best use”:

$$ \begin{aligned} & \mathrm{Preset}\,\,\mathrm{proxy}\,\,\mathrm{preferences}\,\,(p) \\ & \hat{y}=T+\hat{t}\delta, \\ & \hat{t}\in \arg\,\,\mathop{\max}\limits_t\left\{ {\hat{t}p\delta |K(T+t\delta )\leqslant 0} \right\} \\ \end{aligned} $$

(6.8)

or,

$$ \begin{aligned} & \mathrm{Investor}\,\,\mathrm{preferences}\,\,(\beta (\mathrm{a}\delta )) \\ & \hat{y}=T+\hat{t}\delta, \\ & \hat{t}\in \arg\,\,\mathop{\min}\limits_{{\mathop{{K(T+t\delta )\leqslant 0}}\limits^t}}\left\{ {\mathop{\max}\limits_{{a\geqslant 0}}\left\{ {\beta (a\delta )|{K^{+}}(T+t\delta +a\delta )\leqslant 0} \right\}} \right\} \\ \end{aligned} $$

In the preset proxy formulation the objective is to maximize the value (defined by prices p) of the proxy awards given the pre-existing FTRs, and the power flow constraints in the pre-expansion network. In the investor preference formulation the objective is to maximize the investor’s value (defined by the bid functions for different directions, $\beta (a\delta )$) of incremental FTR awards given the proxy and pre-existing FTRs and the power flow constraints in the expanded network, while simultaneously calculating the minimum proxy scalar amount that satisfies the power flow constraints in the pre-expansion network.

We will use as a proxy protocol the first definition. We next analyze the way to use this protocol to carry out an allocation of LTFTRs that stimulates investment in transmission.

6.5 The Auction Model

Assume the preset proxy rule is used to derive prices that maximize the investor preference $\beta (a\delta )$ for an award of $a$ MWs of FTRs in direction $\delta$. We then have the following auction maximization problem:

$$ \begin{aligned} & \mathop{\mathrm{Max}}\limits_{{a,\hat{t},\delta }}\quad \beta (a\delta ) \\ & s.t. \\ & {K^{+}}(T+a\delta )\leqslant 0, \\ & {K^{+}}(T+\hat{t}\delta +a\delta )\leqslant 0, \\ & \hat{t}\in \arg \mathop{\max}\limits_t\left\{ {tp\delta |K(T+t\delta )\leqslant 0} \right\}, \\ & \left\| \delta \right\|=1, \\ & a\geqslant 0. \\ \end{aligned} $$

(6.9)

In this model, the investor’s preference is maximized subject to the simultaneous feasibility conditions, and the best use protocol. We add a constraint on the (two-)norm²¹ of the directional vector to preclude the trivial case $\delta =0$. We want to explore if such an auction model approach can produce acceptable proxy and incremental awards. We next analyze this issue within a framework that ignores losses, and utilizes a DC-load flow approximation.

The auction model is a nonlinear optimization problem of “bi-level” nature.²² There are two optimization stages. Maximization is non-myopic since the result of the lower problem (first stage) depends on the direction chosen in the upper problem (second stage).²³ Bi-level problems may be solved by first transforming the lower problem (i.e. the allocation of proxy awards) into to a set of Kuhn-Tucker equations that are subsequently substituted in the upper problem (i.e. the maximization of the investors’ preference). The model can then be understood as a Stackelberg problem although it is not intending to optimize the same type of objective function at each stage.²⁴

The Lagrangian (L) for the lower problem is:

$$ L(\hat{t},\delta, \lambda )=\hat{t}p\delta -{\lambda^T}(K(T+\hat{t}\delta )) $$

where ${\lambda^T}$ is the Lagrange multiplier vector associated with transmission capacity on the respective transmission lines before the expansion. It is the Lagrange multiplier of the simultaneous feasibility restriction for proxy awards. The Kuhn-Tucker conditions are:

$$ \eqalign{{\frac{{\partial L\left( {\hat{t},\delta, \lambda } \right)}}{{\partial \hat{t}}}=0,}\ {\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}\ge 0,}} \displaystyle{{\lambda^T}\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}=0,\ \lambda\ge0} $$

The transformed problem is then written as:

$$ \begin{aligned} & \mathop{\mathrm{Max}}\limits_{{a,\hat{t},\delta, \lambda }}\quad \beta (a\delta ) \\ & s.t. \\ & {K^{+}}(T+a\delta )\leqslant 0,\quad \quad \quad \quad \quad (\omega ) \\ & {K^{+}}(T+\hat{t}\delta +a\delta )\leqslant 0,\quad \quad \quad (\gamma ) \\ \end{aligned} $$

$$ \begin{array}{*{20}{c}} {\frac{{\partial L\left( {\hat{t},\delta, \lambda } \right)}}{{\partial \hat{t}}}=0,} & {} & {} & {\left( \theta \right)} \\ \end{array} $$

(6.10)

$$ \begin{array}{*{20}{c}} {{\lambda^T}\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}=0,} & {} & {} & {(\zeta )} \\ \end{array} $$

(6.11)

$$ \begin{array}{*{20}{c}} {\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}\geqslant 0,} & {} & {} & {(\varepsilon )} \\ \end{array} $$

$$ \begin{array}{*{20}{c}} {\left\| \delta \right\|=1,} & {} & {} & {(\varphi )} \\ {a\geqslant 0,} & {} & {} & {(\kappa )} \\ {\lambda \geqslant 0} & {} & {} & {(\varphi )} \\ \end{array} $$

where $\omega, \gamma, \theta, \zeta, \varepsilon, \varphi, \kappa$ and $\pi$ are Lagrange multipliers associated with each constraint. More specifically, $\omega$ is the shadow price of the simultaneous feasibility restriction for existing and incremental FTRs; $\gamma$ is the shadow price of the simultaneous feasibility restriction for existing FTRs, proxy awards and incremental FTRs; $\theta,\zeta,\varepsilon$ are the shadow prices of the restriction on optimal proxy FTRs; $\varphi,\kappa$ are the shadow prices of the non-negativity constraints for a and $\lambda$, respectively; and $\pi$ is the shadow price of the unit restriction on $\delta$.

The Lagrangian of the auction problem is:

$$ \begin{aligned} & L(a,\hat{t},\delta, \lambda, \varOmega )=\beta (a\delta )-{\omega^T}({K^{+}}(T+a\delta )) \\ & -{\gamma^T}({K^{+}}(T+\hat{t}\delta +a\delta ))-{\theta^T}\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\delta \hat{t}}} \\ & -{\zeta^T}\left( {{\lambda^T}\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}} \right)+{\varepsilon^T}\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }} \\ & +{\varphi^T}\left( {1-\left\| \delta \right\|} \right)+{\kappa^T}a+{\pi^T}\lambda \\ \end{aligned} $$

(6.12)

where $\Omega =(\omega, \gamma, \theta, \zeta, \varepsilon, \varphi, \kappa, \pi )$ denotes the vector of Lagrange multipliers. Kuhn-Tucker conditions for the upper problem are:

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial a}}=\frac{{\partial \beta (a\delta )}}{{\partial a}}-{{\left[ {\frac{{\partial {K^{+}}(T+a\delta )}}{{\partial a}}} \right]}^T}\omega \\ & -{{\left[ {\frac{{\partial {K^{+}}(T+\hat{t}\delta +a\delta )}}{{\partial a}}} \right]}^T}\gamma +\kappa =0 \\ \end{aligned} $$

(6.13)

$$ \begin{gathered} \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial \hat{t}}}=-{{\left[ {\frac{{\partial {K^{+}}(T+\hat{t}\delta +a\delta )}}{{\partial \hat{t}}}} \right]}^T}\,\gamma \hfill \\ -{{\left[ {\frac{{{\partial^2}\,L(\hat{t},\delta, \lambda )}}{{\partial \hat{t}\partial \lambda }}} \right]}^T}\,\,\lambda \zeta +{{\left[ {\frac{{\partial {L^2}(\hat{t},\delta, \lambda )}}{{\partial \hat{t}\partial \lambda }}} \right]}^T}\,\,\varepsilon =0 \hfill \\ \end{gathered} $$

(6.14)

$$ \begin{aligned} & \frac{{\partial L(a,\hat{t},\delta, \lambda, \varOmega )}}{{\partial \delta }}=\frac{{\partial \beta (a\delta )}}{{\partial \delta }}-{{\left[ {\frac{{\partial {K^{+}}(T+a\delta )}}{{\partial \delta }}} \right]}^T}\,\,\omega \\ & -{{\left[ {\frac{{\partial {K^{+}}(T+\hat{t}\delta +a\delta )}}{{\partial \delta }}} \right]}^T}\gamma -{{\left[ {\frac{{{\partial^2}L(\hat{t},\delta, \lambda )}}{{\partial \delta \partial \hat{t}}}} \right]}^T}\,\,\theta \\ & -{{\left[ {\frac{{{\partial^2}L(\hat{t},\delta, \lambda )}}{{\partial \delta \partial \lambda }}} \right]}^T}\,\lambda \zeta +{{\left[ {\frac{{{\partial^2}L(\hat{t},\delta, \lambda )}}{{\partial \delta \partial \lambda }}} \right]}^T}\,\,\varepsilon -{{\left[ {\frac{{\partial \left\| \delta \right\|}}{{\partial \delta }}} \right]}^T}\varphi =0 \\ \end{aligned} $$

(6.15)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \lambda }}=-{{\left[ {\frac{{{\partial^2}L(\hat{t},\delta, \lambda )}}{{\partial \lambda \partial \hat{t}}}} \right]}^T}\theta -{{\left[ {\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}} \right]}^T}\zeta +\pi =0, $$

(6.16)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \omega }}=-{K^{+}}(T+a\delta )\ge 0, $$

(6.17)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\delta \gamma }}=-{K^{+}}(T+\hat{t}\delta +a\delta )\ge 0 $$

(6.18)

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

(6.19)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \zeta }}=-{\lambda^T}\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}=0, $$

(6.20)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \varepsilon }}=\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}\ge 0, $$

(6.21)

$$ \frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \varphi }}=1-\left\| \delta \right\|=0, $$

(6.22)

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

(6.23)

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

(6.24)

$$ {\omega^T}\frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \omega }}=0,\omega \ge 0, $$

(6.25)

$$ {\gamma^T}\frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \gamma }}=0,\gamma \ge 0, $$

(6.26)

$$ {\varepsilon^T}\frac{{\partial L(a,\hat{t},\delta, \lambda, \Omega )}}{{\partial \varepsilon }}=0,\varepsilon \ge 0, $$

(6.27)

$$ {\kappa^T}a=0,\kappa \ge 0, $$

(6.28)

$$ {\pi^T}\lambda =0,\pi \ge 0 $$

(6.29)

The constraint $\displaystyle\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}=0$ is redundant when the preset proxy preference (p) is non-zero, since it is a sub-gradient of the constraint $\displaystyle{\lambda^T}\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}=0$, and $\varepsilon$ is therefore zero when p is non-zero. We show in a later example that $\theta$ and $\varphi$ are zero because the associated constraints are redundant. The binding constraint in the lower level problem is $\displaystyle{\lambda^T}\frac{{\partial L(\hat{t},\delta, \lambda )}}{{\partial \lambda }}=0$, since some transmission constraints are fully utilized by proxy awards.

This is a nonlinear and non-convex problem, and its solution depends on the investor-preference parameters, the current partial allocation (T), and the topology of the network prior to and after the expansion.²⁵ A general solution method utilizing Kuhn-Tucker conditions would be through checking which of the constraints are binding.²⁶ One way to identify the active inequality constraints is the active set method.²⁷ In this paper we solve the problem in detail for different network topologies, including a radial line and a three-node network.

6.6 Simulations

6.6.1 Radial Line

Let us first analyze a radial transmission line that is expanded as in Fig. 6.1.

Fig. 6.1

An expanded line and its feasible expansion

The corresponding optimization problem is:

$$ \begin{aligned} & \mathop{\mathrm{Max}}\limits_{{a,\hat{t},\delta }}\quad {b_{12 }}a{\delta_{12 }} \\ & s.t. \\ & {T_{12 }}+a{\delta_{12 }}\leqslant C_{12}^{+} \\ \end{aligned} $$

$$ \begin{aligned} & {T_{12 }}+\hat{t}{\delta_{12 }}+a{\delta_{12 }}\leqslant C_{12}^{+} \\ & \hat{t}({\delta_{12 }})\in \arg \mathop{\max}\limits_t\left\{ {t{p_{12 }}{\delta_{12 }}|{T_{12 }}+t{\delta_{12 }}\leqslant {C_{12 }}} \right\} \\ & \left\| {{\delta_{12 }}} \right\|=1, \\ & a\geqslant 0 \\ \end{aligned} $$

(6.30)

where ${C_{12 }}$ is the transmission capacity of the network before the expansion, $C_{12}^{+}$ is the transmission capacity of the network after the expansion, and ${b_{12 }}$ is the investor preference. The first order conditions of the lower maximization problem can then be added as constraints to the upper problem:

$$ \begin{aligned} & \mathop{\mathrm{Max}}\limits_{{a,\hat{t},{\delta_{12 }},\lambda }}\quad {b_{12 }}a{\delta_{12 }} \\ & s.t. \\ & {T_{12 }}+a{\delta_{12 }}\leqslant C_{12}^{+} \\ & {T_{12 }}+\hat{t}{\delta_{12 }}+a{\delta_{12 }}\leqslant C_{12}^{+} \\ & {p_{12 }}{\delta_{12 }}-\lambda {\delta_{12 }}=0 \\ & \lambda ({C_{12 }}-{T_{12 }}-\hat{t}{\delta_{12 }})=0 \\ \end{aligned} $$

(6.31)

$$ \begin{aligned} & {T_{12 }}+\hat{t}{\delta_{12 }}\leqslant {C_{12 }} \\ & \delta_{12}^2=1 \\ & a,\lambda \geqslant 0 \\ \end{aligned} $$

Since the grid is being expanded, the constraint on simultaneous feasibility of incremental FTRs ${T_{12 }}+a{\delta_{12 }}\le C_{12}^{+}$ is non-binding. The solution to this problem provides the values for the decision variables, and shadow prices.²⁸ First, ${\delta_{12 }}=1$, because the network is being expanded. Additionally $\gamma ={b_{12 }}$ which implies that the higher the value of the investor-preference parameter ${b_{12 }}$ the more the investor values post-expansion transmission capacity (its marginal valuation of transmission capacity increases with the bid value).

Similarly, we get $\lambda ={p_{12 }}$ which implies that the higher the value of the preset proxy preference parameter ${p_{12 }}$ the higher marginal valuation of pre-expansion transmission capacity. Other results are $\theta =0$, $\zeta =\gamma /{p_{12 }}={b_{12 }}/{p_{12 }}$ and $\varepsilon =0$. This was expected since only one restriction for the lower problem is binding because the two other are redundant. The value of the binding Lagrange multiplier equals the ratio between the investor’s bid value and the preset proxy parameter.

It also follows that $\varphi =0$ which is to be expected because the directional vector $\delta$ is non-zero. Furthermore, $\hat{t}={C_{12 }}-{T_{12 }}$, which means that for given existing rights the higher the current capacity the larger the need for reserving some proxy FTRs for possible negative externalities generated by the expansion. Proxy awards are auctioned as a hedge against externalities generated by the expanded network.

We finally get $a=C_{12}^{+}-{T_{12 }}-\hat{t}=C_{12}^{+}-{C_{12 }}$, which shows that the optimal amount of additional MWs of FTRs in direction $\delta$ directly depends on the amount of capacity expansion. Transmission capacity is in fact fully utilized by proxy awards (in the pre-expansion network), and by incremental FTRs (in the expanded network). Likewise, the investor receives a reward equal to the MW amount of new transmission capacity that it creates.

6.6.2 Three-Node Network with Two Links

We now consider a three-node network example from Bushnell and Stoft (1997) where there is an expansion of line 1–2. The network is illustrated in Fig. 6.2 and the feasible expansion in Fig. 6.3.

Fig. 6.2

Three-node network with expansion of line 1–2

Fig. 6.3

Feasible expansion of FTRs

The network expansion problem for identical links and FTRs between buses 1–3 and 2–3 is formulated as:

$$ \begin{aligned} & \mathop{\mathrm{Max}}\limits_{{a,\hat{t},\delta }}\quad a({b_{13 }}{\delta_{13 }}+{b_{23 }}{\delta_{23 }}) \\ & s.t. \\ & \frac{2}{3}({T_{13 }}+a{\delta_{13 }})+\frac{1}{3}({T_{23 }}+a{\delta_{23 }})\leqslant {C_{13 }} \\ & \frac{2}{3}({T_{13 }}+\hat{t}{\delta_{13 }}+a{\delta_{13 }})+\frac{1}{3}({T_{23 }}+\hat{t}{\delta_{23 }}+a{\delta_{23 }})\leqslant {C_{13 }} \\ & \frac{1}{3}({T_{13 }}+a{\delta_{13 }})+\frac{2}{3}({T_{23 }}+a{\delta_{23 }})\leqslant {C_{23 }} \\ & \frac{1}{3}({T_{13 }}+\hat{t}{\delta_{13 }}+a{\delta_{13 }})+\frac{2}{3}({T_{23 }}+\hat{t}{\delta_{23 }}+a{\delta_{23 }})\leqslant {C_{23 }} \\ \end{aligned} $$

(6.32)

$$ \begin{aligned} & \frac{1}{3}({T_{13 }}+a{\delta_{13 }})-\frac{1}{3}({T_{23 }}+a{\delta_{23 }})\leqslant {C_{12 }} \\ & \frac{1}{3}({T_{13 }}+\hat{t}{\delta_{13 }}+a{\delta_{13 }})-\frac{1}{3}({T_{23 }}+\hat{t}{\delta_{23 }}+a{\delta_{23 }})\leqslant {C_{12 }} \\ \end{aligned} $$

(6.33)

$$ \begin{aligned} & -\frac{1}{3}({T_{13 }}+a{\delta_{13 }})+\frac{1}{3}({T_{23 }}+a{\delta_{23 }})\leqslant {C_{21 }} \\ & -\frac{1}{3}({T_{13 }}+\hat{t}{\delta_{13 }}+a{\delta_{13 }})+\frac{1}{3}({T_{23 }}+\hat{t}{\delta_{23 }}+a{\delta_{23 }})\leqslant {C_{21 }} \\ \end{aligned} $$

$$ \begin{aligned} & \hat{t}(\delta )\in \arg \mathop{\max}\limits_t\left\{ {t({p_{13 }}{\delta_{13 }}+{p_{23 }}{\delta_{23 }})} \right\} \\ & ({T_{13 }}+t{\delta_{13 }})\leqslant {C_{13 }} \\ & ({T_{23 }}+t{\delta_{23 }})\leqslant {C_{23 }} \\ & \left\| \delta \right\|=1 \\ & a\geqslant 0 \\ \end{aligned} $$

Appendix 2 presents the calculations to obtain the power transfer distribution factors (PTDFs) for the post expansion network. In Fig. 6.3 the pre-existing FTRs in the direction 2–3 do not use the full capacity of the pre-expansion network and become infeasible after inserting line 1–2. The preference is for FTRs in the direction 1–3 for transmission expansion. As seen from Fig. 6.3 the maximum amount of proxy and incremental FTRs in the direction 1–3 that can be obtained is 1,100, and corresponds to the point where the 1–3 and 1–2 transmission capacity constraints intersect.

In solving this problem, we get²⁹:

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

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

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

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

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

with

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

where ${\gamma_1}\;\mathrm{ and}\;{\gamma_2}$ are the Lagrange multipliers associated with transmission capacity on the lines 1–3 and 1–2, respectively, in the expanded network, and $\zeta$ is the multiplier associated with the Kuhn-Tucker condition regarding transmission capacity in the pre-expansion network for the line 1–3. This line has the Lagrange multiplier $\lambda$ associated with it before expansion. So as to characterize the solution to our model, we now calculate the Lagrange multipliers and decision variables for particular parameter values. In particular, we find the solution for the allocation presented in Fig. 6.3. We assume the following bid values, preset proxy preferences and pre-existing amount of FTRs:

$$ \begin{array}{*{20}{c}} {{b_{13 }}=40,} & {{b_{23 }}=10,} \\ {{p_{13 }}=60,} & {{p_{23 }}=10,} \\ {{T_{13 }}=100,} & {{T_{23 }}=800} \\ \end{array} $$

From these parameters we find that the marginal value of transmission capacity on line 1–3 and line 1–2 are ${\gamma_1}=39.6$ and ${\gamma_2}=33.6$, respectively. Thus the investor values transmission capacity on line 1–3 more than on line 1–2. We find that the product of the Kuhn-Tucker multiplier and the transmission capacity multiplier for the line 1–3 is $\zeta \lambda =37$ _.

Likewise, the values of the decision variables are calculated as:

$$ \begin{aligned} {\delta_{13 }}= & 0.958,\quad {\delta_{23 }}=-0.287, \\ a= & 208,\quad \hat{t}=835 \\ \end{aligned} $$

The MW amount of awarded proxy FTRs in the direction 1–3 is $\hat{t}{\delta_{13 }}=800$, and the amount of awarded incremental FTRs is $a{\delta_{13 }}=200$. The amount of incremental 1–3 FTRs corresponds to the new transmission capacity on line 1–2 that the investor has created. There is also an allocation of proxy FTRs such that the full capacity of line 1–3 is utilized. Similarly the proxy awards in direction 2–3 is $\hat{t}{\delta_{23 }}=-240$, and the amount of awarded incremental FTRs is $a{\delta_{23 }}=-60$. The amount of incremental 2–3 FTRs is minimized and corresponds to 20 % of the reduction (300) in pre-existing FTRs. The incremental 2–3 awards are mitigating FTRs, and are necessary to restore feasibility. The investor is then responsible for additional counterflows so that it pays back for the negative externalities it creates. The solution is indicated by the black arrow in Fig. 6.3 and consists of both pre-existing and incremental FTR awards amounting to ${T_{13 }}+a{\delta_{13 }}=300\;\mathrm{ and}\;{T_{23 }}+a{\delta_{23 }}=740$. The allocation of incremental 2–3 FTRs is minimized because the model takes into account that one line is expanded, and some of the pre-existing FTRs become infeasible after the expansion.

This illustrates that the amount of incremental FTRs in the preference direction must be greater than zero such that feasibility is restored. Both the proxy and incremental FTRs exhaust transmission capacity in the pre-expansion and expanded grid, respectively. The proxy FTRs help allocating incremental FTRs by preserving capacity in the pre-expansion network, which results in an allocation of incremental FTRs amounting to the new transmission capacity created in 1–2 direction.³⁰ The proxy awards are transmission congestion hedges that can be auctioned to electricity market players in the expanded network.³¹

In the example provided by Bushnell and Stoft (1997), the investor with pre-existing FTRs chooses the most profitable incremental FTR based on optimizing its final benefit. The investor is then awarded a mitigating incremental 1–2 FTR with associated power flows corresponding to the difference between the ex-ante and ex-post optimal dispatches. The pre-existing FTRs correspond to the actual dispatch of the system and become infeasible after expanding line 1–2, and therefore a mitigating 1–2 FTR³² is allocated so that feasibility is exactly restored (that is, the investor “pays back” for the negative externalities to other agents). There is no allocation of proxy awards because the pre-expansion network is fully allocated by FTRs before the expansion. The amount of incremental FTRs is minimized because they represent a negative value to the investor and decrease its revenues from the pre-existing FTRs.

6.7 The Auction Model and Welfare

Bushnell and Stoft (1997) demonstrate that the increase in social welfare will be at least as large as the ex-post value of new contracts, when the FTRs initially match dispatch in the aggregate and new FTRs are allocated according to the feasibility rule. In particular, if social welfare is decreased by transmission expansion, the investor will have to take FTRs with a negative value (If social welfare is increased there will be free riding). With only the aggregate match of FTRs and dispatch, some agents might still benefit from investments that reduce social welfare, whenever their own commercial interests improve to an extent that more than offsets the negative value of the new FTRs. Further, Bushnell and Stoft show that incentives for expansion that reduce social welfare would be removed if FTRs for each agent as a perfect hedge and match their individual net loads. In such a case, FTRs allocated under the feasibility rule ensure that no one will benefit from an expansion that reduces welfare.

Although apparently similar, our mechanism and its implications on welfare are different from those in the Bushnell and Stoft (1997) model. Bushnell and Stoft analyze the welfare implications of transmission expansion given matching of dispatch both in the aggregate and individually. In our model, we assume unallocated FTRs both before and after the expansion, so that there is no match in dispatch.³³

However, the proxy award mechanism developed in this paper implies nonnegative effects on welfare in the sense that future investments in the grid cannot reduce the welfare of aggregate use for FTR holders. The reason is that simultaneous feasibility is guaranteed before and after the enhancement project so that revenue adequacy is also guaranteed after expansion. Only those non-hedged agents in the spot market might be exposed to rent transfers. For feasible long term transactions identified ex ante, the FTRs provide perfect congestion hedges for the existing grid or for any future grid that develops under the feasibility rule. However, FTRs cannot provide perfect hedges ex post for all possible transactions. A similar property carries over to any welfare analysis under FTRs.

More formally, suppose we have a social welfare function B for dispatch in a single period. Also assume that there is no uncertainty, that all functions are known, and that agents are price takers in the electricity and FTR markets. A simple welfare model associated with transmission expansion $\Delta$ is³⁴:

$$ \begin{aligned} & \mathop{Max}\limits_{\varDelta}\,B({Y^{*}}+\Delta ) \\ & \mathrm{s}.\mathrm{t}. \\ \end{aligned} $$

(6.34)

$$ {K^{+}}({Y^{*}}+\Delta )\le 0 $$

where

$$ {Y^{*}}\in \arg \max \left\{ {\left. {B(Y)} \right|K(Y)\le 0} \right\} $$

Then ${Y^{*}}$ is the dispatch that maximizes social welfare without the expansion. Let ${\Delta^{+}}$ be the dispatch that would be provided as an increment due to transmission expansion. ${\Delta^{+}}$ solves program (6.34). Note that if ${P^{+}}=\nabla B({Y^{*}}+{\Delta^{+}})$, then under reasonable regularity conditions ${\Delta^{+}}$ is also a solution to:

$$ \begin{aligned} & \mathop{Max}\limits_{\varDelta}\,{P^{+}}\Delta \\ & \mathrm{s}.\mathrm{t}. \\ \end{aligned} $$

(6.35)

$$ {K^{+}}({Y^{*}}+\Delta )\le 0 $$

Formulation (6.35) is interpreted as the maximization of congestion rents for the incremental allocation $\Delta$.

In the context of Bushnell and Stoft assumptions,³⁵ suppose no w that the current allocation of FTRs T satisfies $T={Y^{*}}$, then (6.35) would award the maximum value of incremental FTRs. In this case, we need not know the full benefit function. We could rely on the expander to estimate ${P^{+}}$, and provide this preference ranking function as part of the bid. Then solving (6.35) would give the maximum value incremental award for expansion ${K^{+}}$, and this award would preserve the welfare maximizing property of the FTRs for the expanded grid.³⁶

Now suppose that (for some reason) $T\ne {Y^{*}}$. To preserve simultaneous feasibility the constraint ${K^{+}}(T+\Delta )\le 0$ should be imposed. A natural (second best) rule might be:

$$ \begin{aligned} & \mathop{Max}\limits_{\varDelta}\,{P^{+}}\Delta \\ & s.t. \\ \end{aligned} $$

(6.36)

$$ {K^{+}}({Y^{*}}+\Delta )\le 0 $$

$$ {K^{+}}(T+\Delta )\le 0 $$

$$ {Y^{*}}\in \arg \max \left\{ {\left. {B(Y)} \right|K(Y)\le 0} \right\} $$

Hence, the existing users of the grid could continue to do as before the expansion, and the expander receives the incremental values arising from the expansion. Then the example in Hogan (2002a, p. 12; see also Appendix 3) illustrates the case of a beneficial expansion where the only solution to (6.36) is $\Delta =0$ so that the expansion project does not occur. In fact, ${K^{+}}(T+\Delta )\le 0$ cannot be relaxed without violating the critical property of simultaneous feasibility. We illustrate the argument in the following examples.

Consider the example in Hogan (2002a, p. 12) illustrated in Fig. 6.4. Here the dispatch in the pre-existing network does not match the current allocation of FTRs. The limiting constraints for the dispatch are the 1–3 and 2–3 constraints. Likewise, the limiting constraints for the current allocation of FTRs are the 1–2 and 1–3 constraints. Assume that the incremental dispatch in the 1–3 and 2–3 directions may be caused by the increased capacity of line 1–3. The relevant constraints are ${K^{+}}({Y^{*}}+\Delta )\le 0$ for the current dispatch and ${K^{+}}(T+\Delta )\le 0$ for the current allocation of FTRs. The corresponding objective is $Max\left( {P_{13}^{+}{\Delta_{13 }}+P_{23}^{+}{\Delta_{23 }}} \right)$. Then the following constraints would apply for the specific network topology:

$$ \frac{2}{3}({T_{13 }}+{\Delta_{13 }})+\frac{1}{3}({T_{23 }}+{\Delta_{23 }})\le C_{13}^{+} $$

$$ \frac{2}{3}({Y_{13 }}+{\Delta_{13 }})+\frac{1}{3}({Y_{23 }}+{\Delta_{23 }})\le C_{13}^{+} $$

$$ \frac{1}{3}({T_{13 }}+{\Delta_{13 }})+\frac{2}{3}({T_{23 }}+{\Delta_{23 }})\le {C_{23 }} $$

$$ \frac{1}{3}({Y_{13 }}+{\Delta_{13 }})+\frac{2}{3}({Y_{23 }}+{\Delta_{23 }})\le {C_{23 }} $$

$$ \frac{1}{3}({T_{13 }}+{\Delta_{13 }})-\frac{1}{3}({T_{23 }}+{\Delta_{23 }})\le {C_{12 }} $$

$$ \frac{1}{3}({Y_{13 }}+{\Delta_{13 }})-\frac{1}{3}({Y_{23 }}+{\Delta_{23 }})\le {C_{12 }} $$

$$ -\frac{1}{3}({T_{13 }}+{\Delta_{13 }})+\frac{1}{3}({T_{23 }}+{\Delta_{23 }})\le {C_{21 }} $$

$$ -\frac{1}{3}({Y_{13 }}+{\Delta_{13 }})+\frac{1}{3}({Y_{23 }}+{\Delta_{23 }})\le {C_{21 }} $$

Fig. 6.4

Dispatch Y does not match the current allocation of FTRs

First assume that ${T_{13 }}=1100,\quad {T_{23 }}=500$ and ${Y_{13 }}=900,\quad {Y_{23 }}=900$ and that the incremental benefit of expansion is greater in the 1–3 direction than in the 2–3 direction. Also assume that $C_{13}^{+}=1000$. We notice that the mismatch between the dispatch and existing FTRs is ${Y_{13 }}-{T_{13 }}=-200\mathrm{ and}\ {Y_{23 }}-{T_{23 }}=400$. Furthermore, the marginal expansion occurs from the current dispatch to where the 1–3+ and 2–3 transmission constraints intersect. This amounts to the incremental dispatch ${\Delta_{13 }}=200\;\mathrm{ and}\;{\Delta_{23 }}=-100$. If the above numbers are substituted in the constraints we find that the transmission capacity constraint for line 1–2 and existing FTRs are violated because $\frac{1}{3}({T_{13 }}+{\Delta_{13 }})-\frac{1}{3}({T_{23 }}+{\Delta_{23 }})=\frac{1}{3}(1100+200)-\frac{1}{3}(500-100)=300>{C_{12 }}=200$. Hence the expansion does not occur. Conversely, assume that the location of the current dispatch and existing FTRs are interchanged so the mismatch between the dispatch and existing FTRs is ${Y_{13 }}-{T_{13 }}=200\mathrm{ and}\ {Y_{23 }}-{T_{23 }}=-400$ and assume that the marginal benefit of the expansion is greater in the 2–3 direction than in the 1–3 direction. Then the incremental dispatch would be ${\Delta_{13 }}=0\;\mathrm{and}\;{\Delta_{23 }}=300$. In this case the 2–3 transmission capacity constraint would be violated for the existing FTRs $\frac{1}{3}({T_{13 }}+{\Delta_{13 }})+\frac{2}{3}({T_{23 }}+{\Delta_{23 }})=\frac{1}{3}900+\frac{2}{3}(900+300)=1100>{C_{23 }}=900$.

Similar problems would arise with rules such as preserving proxy awards to allow for any possible dispatch on the existing grid, where the only expansions incented would be those that added to every constraint in the system, virtually foreclosing the possibility of investment under this rule.

Given the complicated externalities of electric grid, a first best system based on decentralized property rights is not known. Traditionally, all investment decisions relied on central decisions by regulators under certification of need. This often produced regulatory gridlock precisely because of the grid externalities considered here (not to mention the siting and environmental issues). The FTR feasibility rule always preserves the property that the incidence of any welfare reductions falls to those whose transaction were not selected ex ante to be hedged by FTRs. In dealing with the aggregate welfare effects, the second best motivation is shown in (6.35) (without going all the way to (6.36)). In the absence of the known welfare function or the possibility of allocating all the existing grid, the total award is divided between proxy awards and incremental awards for the investor. The proportional part of the resulting total award that could be achieved with the existing grid is preserved as a proxy award. The remainder is assigned to the investor. Subject to this rule, the total award is chosen to maximize the market value of the incremental award to the investor. Presumably this would reinforce the incentive for the investor to provide an accurate estimate of the market value. Given the prices, the special case of FTRs matching dispatch or $T={Y^{*}}$ (considered by Bushnell and Stoft 1996, and Bushnell and Stoft 1997) is consistent with this rule, and the welfare maximizing results apply. In the case where there is not a full allocation of the existing grid, the likely result is that there would be more scope for welfare reducing investments. The need for regulatory oversight would not be eliminated, but the intent is that the scope of the regulatory issues would be reduced. Since proxy award mechanisms are in use and more are under development, further investigation of the private incentives, welfare effects and regulatory implications would be of value.

6.8 Concluding Remarks

We proposed a merchant mechanism to expand electricity transmission. Proxy awards (or reserved FTRs) are a fundamental part of this mechanism. We defined them according to the best use of the current network along the same direction of the incremental expansion. The incremental FTR awards are allocated according to the investor preferences, and depend on the initial partial allocation of FTRs and network topology before and after expansion.

Our examples showed that the internalization of possible negative externalities caused by potential expansion is possible according to the rule proposed by Hogan (2002a): allocation of FTRs before (proxy FTRs) and after (incremental FTRs) the expansion is in the same direction and according to the feasibility rule. Under these circumstances, the investor will have the proper incentives to invest in transmission expansion in its preference direction given by its bid parameters. Likewise the larger the existing current capacity the greater the number of FTRs that must be reserved in order to deal with potential negative externalities depending on post network topology.

Our mechanism of long term FTRs is basically a way to hedge market players from long-run nodal price fluctuations by providing them with the necessary property transmission rights. The main purpose of the four basic criteria that support our model (feasibility rule, proxy awards, maximum value and symmetry) were to define property rights for increased transmission investment according to the preset proxy rule. However, the general implications on welfare, and incentives for gaming are still an open research question.

Although our model is specifically designed to deal with loop flows, and the security-constrained version of our model can take care of contingency concerns, our proposed mechanism is to be applied to small line increments in meshed transmission networks. LTFTRs are efficient under non-lumpy marginal expansions of the transmission network, and lack of market power. Regulation has then an important complementary role in fostering large and lumpy projects where investment is large relative to market size, and in mitigating market power. Since revenues from nodal prices only recover a small part of total costs, LTFTRs must be complemented with a regulated framework that allows the recovery of fixed costs. The challenge is to effectively combine merchant and regulated transmission investments or, as Hogan (2003) puts it, to establish a rule in practice for drawing a line between merchant and regulated investment.

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