Nodal, zonal, or uniform electricity pricing: how to deal with network congestion

Abstract

In this paper, the main contributions to congestion management and electricity pricing, i.e., nodal, zonal, and uniform electricity pricing, are surveyed. The key electricity market concepts are structured and a formal model framework is proposed for electricity transportation, production, and consumption in the context of limited transmission networks and competitive, welfare maximizing electricity markets. In addition, the main results of existing short-run and long-run congestion management studies are explicitly summarized. In particular, the important interconnection between short-run network management approaches and optimal long-run investments in both generation facilities and network lines are highlighted.

Keywords

nodal pricing / zonal pricing / uniform pricing / competitive electricity markets / welfare maximization / redispatch / optimization models

Introduction

Liberalized electricity markets are typically characterized by a complex interaction of production, consumption, and network transmission. In contrast to the transportation of most physical goods, electricity cannot be sent along a single, contracted path, but is divided among all the available transmission lines of a network. Given the existence of such complex transmission constraints including limited line capacities, in practice, demand and production schedules will typically be restricted by at least some of these power flow constraints. Such situations are referred to as network congestion. Depending on the market environment, congestion may yield price divergence of different submarkets with a complex spatial differentiation of the generation structure. Obviously, transmission system operators have to ensure that transmission constraints are never violated, which is the task of congestion management [–]. Efficient congestion management mechanisms and adequate electricity market designs are of high importance for efficient short-term network and market operations as well as for long-term transmission and generation investments. In particular, long-run investments will highly depend on the implemented congestion management methods and on corresponding market outcomes, including future profits of firms. In the past decades, various models have been developed in the electricity market literature to analyze and compare different ways to deal with transmission congestion. The most efficient, welfare maximizing congestion management is the implementation of node specific prices that directly account for limited transmission capacities and possible congestion between nodes—called nodal pricing []. Even though, nodal pricing avoids an overload of transmission lines and induces optimal investments in the long-run, an obvious drawback of nodal pricing in real world networks is the high number of calculated prices. As pointed out by Stoft [] and Hogan [], already a single congested transmission line may yield nodal prices that differ at every node of the network. Therefore, network nodes are sometimes partitioned into price zones that share a common price. This approach is referred to as zonal pricing (see Bjørndal and Jørnsten []). In its extreme form, all nodes belong to a single price zone and only a single system price is calculated. Obviously, the outcomes under this uniform pricing and under zonal pricing may violate transmission restrictions of the underlying network. Therefore, zonal and uniform pricing may require an expost adjustment of market outcomes that is often referred to as redispatch.

In this paper, an overview is given of main contributions to congestion management and electricity pricing, i.e., nodal, zonal, and uniform electricity pricing. The overview is restricted to comparative studies that evaluate different zonal setups relative to each other or to the nodal pricing benchmark. In addition, the case of perfect competition and/or welfare maximization is focused on, which implies that strategic behavior among producers is not considered. Finally, uncertainties are mainly abstracted and a deterministic world is assumed. Such an assumption may also be interpreted as perfect foresight.

In the first step, the findings of different research papers are explicitly summarized and set in context. The short-run congestion management and corresponding pricing measures are analyzed, where two-node and multiple-node networks are considered. In addition, long-run investments for the single-node (“copperplate”) case and for the multiple-node case are also discussed. In the single-node case, where transmission is not considered, naturally only generation investment takes place, whereas in the multiple-node case both generation and transmission investments are taken into account. These line and generation capacity investments can be seen as long-run congestion management measures that highly depend on each other. In the second step, a general model framework is provided that structures the discussed congestion management approaches and highlights their main similarities and differences. As nodal, zonal, or uniform pricing is applied in various countries around the world, this overview may directly contribute to the current discussion on real-world electricity market designs.

Does the consideration of the network matter?

Does the consideration of the network matter at all? In the present section, this question will be discussed on the basis of a simple two-node example. Even though, similar network structures have been used in previous work (see, for instance, De Vries and Hakvoort [], Green [], Inderst and Wambach [], or Grimm et al. []), a two-node network seems a good starting point to introduce and motivate the important characteristics of electricity markets as compared to markets with no electricity transmission. As network congestion will already be observed in such a simple network structure, the example also underlines the need for efficient congestion management in more complex real-world electricity networks.

As it is well known, in a perfectly competitive market, where all consumers and producers are located at a single network node (no transmission) with given generation facilities and no consideration of fixed or investment cost, variable generation cost of the marginal producer set the uniform market clearing price (see for instance Cho [] for a discussion). Such a market situation is depicted in Fig. 1 for a single time period (one hour). A single generator with a variable generation cost functionC^var(q) = q, where q denotes electricity production, is assumed. Consumption is described by an inverse demand function p(d) = 10−d, with d being the electricity consumption quantity. As can be seen in Fig. 1, for the case of no transmission restrictions the intersection point A of supply and demand characterizes both the equilibrium price of 5 €/MWh and the equilibrium consumption/production of 5 MWh. Welfare amounts to 25 €.

Now assume that demand and generation are located in two different areas (nodes) that are linked by a lossless transmission line with a capacity of 2 MWh. Generation is located at the northern node and demand is located at the southern node. Obviously, the described optimal solution with a generation of 5 MWh violates the transmission capacity. The reason for this physical infeasibility is the missing “congestion signal” of the uniform market price, i.e., the uniform price does not incentivize an optimal use of the transmission line that interconnects the two nodes. Therefore, in an optimal solution that directly accounts for the limited transmission capacity between the two nodes, more than one (uniform) market price is needed to clear the northern and the southern market. The following section will also be referred to, which discusses such an optimal nodal pricing as well as other congestion management approaches that aim at keeping a uniform price or at least uniform prices within different price zones—possibly at the cost of a welfare loss as compared to nodal pricing. Turning back to the two-node example, for the case of a limited transmission capacity, in the north the price will fall to 2 €/MWh with a corresponding optimal production of 2 MWh (point B). Analogously, in the south the price will rise to 8 €/MWh reducing southern consumption to 2 MWh (point C). The striped area in Fig. 1 describes the so-called congestion rent, transmission surplus, or merchandising surplus that a transmission system operator earns due to the price difference between the north and the south (consumers pay more than producers receive). This merchandising surplus may be used by the transmission system operator for network maintenance or extension measures. When turning to the welfare effects of the optimal network-constrained solution, as expected welfare decreases as compared to the unrestricted (single-node) case. The grey area in Fig. 1 indicates this congestion cost, which is the welfare loss of an optimal network-restricted solution as compared to the unrestricted case with a uniform market price (see for instance Hornnes et al. [], Crampes and Laffont [], or Rious et al. [] for more details on the discussed concepts). Figure 2 further illustrates congestion rents of the transmission system operator (solid line) and congestion cost (dashed line) for different line capacities. While congestion costs fall until the line capacity limit reaches the equilibrium quantity of the unrestricted case (point E), rents first increase and then decrease. For line capacities of more than 5 MWh (the unrestricted case), both congestion rents and congestion costs completely disappear. This can be seen, as the line capacity does not constrain market outcomes anymore. It further shows that non-network models can be seen as a special case of a network-constrained model, where transmission capacities are non-restricting and losses are not considered, and that binding network constraints may add substantial complexity as compared to non-network models.

It is finally noted that if transmission losses are additionally considered, even an unlimited transmission capacity of the considered network line will not yield the same outcome as compared to the single-node “copperplate” case. To illustrate this divergence, it can be assumed that in the two-node example, the transmission line linking the northern and the southern node is characterized by an infinite capacity and a loss of 25%. This transmission loss implies that consumers in the south will only be able to consume 75% of the electricity produced in the north. In direct consequence, optimal prices and corresponding optimal quantities will adjust to the additional costs that are caused by these transmission losses. In such an optimal solution, consumers in the south only buy a quantity of 3.6 MWh at a price of 6.4 €/MWh. Producers in the north supply 4.8 MWh at a corresponding price of 4.8 €/MWh. Transmission causes a loss of 1.2 MWh. As consumer payments in the south equal producer income in the north, the corresponding congestion rent is zero. However, transmission losses and the corresponding reduced electricity consumption directly yield to positive congestion cost, i.e., a reduced welfare as compared to the single-node reference case. This shows that in addition to the challenges of limited line capacities, losses may cause further complexity for efficient congestion management.

Short-run network management analysis: nodal, zonal, and uniform pricing

In this section, different short-run congestion management studies that consider nodal, zonal, and uniform pricing in the absence of investment decisions are reviewed. Appendix A can be referred to for overview tables that give model characterizations of selected research papers.

Short-run nodal pricing

Bohn et al. [] are among the first to elaborate on the effects of spatial differentiation in an electricity transmission network. The authors analyze a welfare maximization problem that is subject to generation limits, transmission capacities, and energy balance restrictions accounting for physical transmission laws. Thus, all relevant transmission restrictions are taken into account when the energy market clears, which implies that a first best welfare is realized (see also Problem (3a) to (3b) in Section 6.1). Therefore, in their model, spot prices reflect the optimal locational value of consumption, generation, and transmission. As a main result, the price difference between two nodes of the network indicates the correct optimal short-run price of transmission, which depends on marginal losses and congestion in the network. In direct consequence, prices provide an optimal incentive for efficient generation and consumption schedules and incentivize an optimal use of network facilities, i.e., limited transmission lines. Based on these findings, different mechanisms to allocate transmission capacity rights, to design transmission ownership, and to compensate the respective owners have been analyzed. In the following, two main contributions by Hogan [] and Chao and Peck [] will be described. Hogan [] studies optimal pricing in the context of contract networks and transmission capacity rights that ensure short-run efficiency. In contrast to a pure contract path model, where capacity rights correspond to individual transmission lines that were contracted, Hogan [] defines capacity rights that are assigned to node pairs. The owner of such a contract network right can either send power from one node to the other network node or receive the nodal price difference between the two nodes. Based on these results, Chao and Peck [] introduce a property rights approach that assigns network rights to contract paths together with some specified trading rule. The trading rule describes the amount of transmission rights of the relevant lines that have to be procured in order to being able to send power to a chosen network node. Again, short-run efficiency is guaranteed by this mechanism.

In addition to the described work on transmission rights, Oren et al. [] and Wu et al. [] further elaborate on nodal pricing. In particular, they show that well known properties (folk theorems) of competitive markets do not need to hold in the presence of transmission restrictions. Among other paradoxical effects, neighbouring nodes, i.e., node pairs that are linked by a transmission line, may have different nodal prices even if their capacity limit is not hurt. If two nodes are characterized by a price difference, power may be sent from the node with the higher price to the low-price node.

In Chen et al. [] nodal prices are further analyzed proposing a method to disentangle the different components of a nodal price, e.g., generation restrictions, transmission congestion, or voltage limits. This information may be used to better understand existing shortages and needed facility investments.

Even though nodal pricing yields a first best welfare and perfectly integrates generation and transmission, especially the huge number of submarkets and nodal prices is often seen as a drawback. In this context Stoft [] and Hogan [] pointed to the fact that a single congested transmission line may result in nodal prices that differ at every node of the network. Additionally, also the complex coordination of the different submarkets of a nodal pricing system and the corresponding limited number of traders in each submarket may be seen as a disadvantage. This limited number of traders may yield submarkets with a relatively low liquidity and possibly low competition. Table 1 summarizes the main advantages and disadvantages of nodal pricing. An attempt made to reduce the number of prices is the use of zonal (uniform) pricing instead of nodal pricing as discussed in the following section, where the welfare loss/inefficiencies of different variants of zonal pricing as compared to a first best nodal pricing system will be analyzed; amongst others, see Bjørndal and Jørnsten [], Bjørndal et al. [], Bjørndal and Jørnsten [], Ehrenmann and Smeers [], Green [], or Oggioni and Smeers []. In Section 4 the increasing literature that incorporates long-run investments into nodal pricing models is also elaborated on.

Short-run zonal and uniform pricing

In this section, zonal and uniform pricing models in the short-run are considered. First, the problems related to a simple two-node analysis are briefly highlighted. In a second step, general networks are focused on.

Problems of two-node models

In a two-node network without losses, only the transmission capacity between the two nodes constrains the exchange between them (see also the introductory example in Section 2). This yields an enormous simplification of power flows in general networks with loop flows, where transmission becomes much more complex. Therefore, it cannot be expected that the results of two-node networks can directly be generalized to the case of a multiple-node network. To give an example, De Vries and Hakvoort [] explicitly compare zonal pricing and uniform pricing with cost-based as well as with market-based redispatch in a simple two-node network. In their main analysis they assume inelastic demand, given generators with constant variable production cost, and an upper generation capacity. The authors show for a binding line capacity that all methods are efficient and yield an optimal dispatch of generators. In the next section, it will be seen that this equivalence may not always hold in a general setting. As shown in Grimm et al. [], market-based redispatch may result in an inefficient dispatch if more than two nodes are considered. Also in Green [] it is highlighted that for realistic problem instances power flows must be derived for meshed networks with loop flows.

Multiple-node networks

As discussed above, zonal pricing is often seen as a simplified form of nodal pricing by reducing the number of different prices. However, zonal pricing models are not necessarily a relaxation of a nodal pricing model. Instead, Bjørndal and Jørnsten [] are the first to present an ideal zonal pricing model that adds constraints to the nodal pricing model. These constraints force equal prices in the different zones (see also Problem (9a) to (9c) in Section 6.3.1). In their zonal pricing model, transmission flows are always physically feasible, which implies that no redispatch is necessary. The authors analyze different zonal configurations for given generation and transmission facilities with elastic demand. In a pseudo direct current flow model, they show that a finer node set partition, i.e., the introduction of new price zones, can only be welfare enhancing or welfare neutral. In general, zonal pricing will result in a welfare loss as compared to nodal pricing. Different zone boundaries influence total welfare and the surplus of individual agents, creating conflicts of interest between these groups. Bjørndal and Jørnsten [] also show that nodal prices may, in general, fail to indicate welfare maximizing zone allocations when the number of price zones is fixed. Instead, a nonlinear, mixed-integer program is proposed to determine an optimal zone configuration for a given number of price zones. This extends and formalizes previous studies that discussed criteria for adequate zone boundaries based on either congested lines or nodal prices. In this context, Walton and Tabors [] elaborate on within-zone and between-zone nodal price variances to determine both the number of price zones and their respective boundaries. Bjørndal and Jørnsten [] also contribute to the work of Hogan [], who has already highlighted that only in radial networks the question of an optimal definition of zone boundaries may easily be answered. In contrast, in general networks with loop flows, congestion of only a single transmission line may result in a very complex system. For such general networks, the nonlinear, mixed-integer program in Bjørndal and Jørnsten [] may provide a tool to identify optimal zone boundaries in the short-run.

Bjørndal et al. [] further elaborate on the surplus effects of zonal pricing. They show that “fake” or political capacity limits—artificial capacity limits imposed on some non-congested transmission lines—may only in special cases help in controlling overloads of the constrained transmission lines and, in general, fail as an efficient congestion management measure. The authors again highlight that zonal pricing may yield substantial different outcomes as compared to a first best nodal pricing solution. Therefore, Bjørndal et al. [] call for a detailed economic assessment before introducing zonal pricing. Based on their work, in Bjørndal and Jørnsten [] nodal and zonal pricing models are applied to the Nordic market for different zonal configurations and load scenarios. The authors confirm that an increase in the number of price zones may in general help to increase welfare, however, also few price zones may already yield a relatively high welfare level as compared to the nodal pricing reference case. This underlines that benefits of zonal pricing may highly depend on the network of consideration. Apart from that, their load scenario analysis underlines the advantages of a flexible zone system, as optimal zone configurations may vary over time.

Ehrenmann and Smeers [] analyze the effects of a second best zonal pricing approach with fixed zones, which only considers inter-zonal flow restrictions (see also Section 6.3.2). Based on Bjørndal and Jørnsten [], the authors discuss the inefficiencies of different zonal configurations in an ideal and second best zonal pricing model including the uniform pricing case. Similar to Bjørndal and Jørnsten [], Ehrenmann and Smeers [] do not explicitly present an ex post redispatch model to deal with overloads. Instead, they focus on general infeasibility issues of the uniform pricing and zonal pricing model. In addition, the problem of an adequate (aggregate) network representation of the second best zonal pricing model is elaborated on. In particular, safety standards of inter-zonal transmission lines are discussed. As in an ideal zonal pricing model the original network characteristics are always used, under second best zonal pricing the construction of adequate physical characteristics of inter-zonal lines comes on top of the already challenging decision on the optimal number of price zones and their boundaries. In this context, using a pseudo direct current network, Boucher and Smeers [] show that inter-zone transfer capacities can only be efficiently defined for the case of radial networks. In contrast, in the case of a general network with loop flows, the transfer capacity highly depends on the demand-generation pattern and not only on the physical characteristics of particular transmission lines. Observe that such ambiguities have been previously emphasized in the nodal pricing literature in the context of different load patterns (see Hogan []). This underlines that, in general, it may be very challenging to identify an optimal, welfare maximizing zonal design.

Recently, Oggioni and Smeers [] explicitly study the effects of (cost-based) redispatch in a (fixed zone) second best bilevel zonal pricing model in relation to a nodal pricing benchmark model. The authors consider the case, where both demand and generation can be redispatched as well as the case of a generation-only redispatch. It is assumed that the aggregated spot market volumes may not be changed during the redispatch. Following Rious et al. [], the authors discuss a transfer capacity regulation that is based on a worst case analysis of inter-zone line flows (“feasibility of worst case loading”). Oggioni and Smeers [] show that zonal pricing with redispatch may yield a comparatively efficient outcome if the inter-zone capacities are chosen correctly. However, a worst case transmission capacity may be too conservative, resulting in a significant welfare decrease. In addition, the authors show that both welfare losses as compared to the nodal pricing model and redispatch cost may be reduced, when consumers are involved in the redispatch.

Combining the discussed congestion management approaches, Bjørndal et al. [] analyze the case of a hybrid congestion management scheme, where a subset of nodes is operated under a nodal pricing system and the remaining network nodes under zonal pricing. The two subnetworks are interconnected and can affect the outcome of each other. In the zonal pricing subnetwork, within the different zones intra-zonal transmission restrictions are not considered. In contrast, zonal energy balance and inter-zone transmission limits are applied. The authors use the aggregated nodal pricing (optimal) power flows between two areas to construct corresponding inter-zonal capacity limits. Even though, such a procedure makes the outcomes of the different regimes comparable, Bjørndal et al. [] point to the fact that both a pure zonal pricing system and a hybrid pricing system might yield infeasible flows, as redispatch is not explicitly addressed. Indeed, their results show that in a hybrid system, the corresponding nodal prices may give wrong signals, yielding an inefficient and possibly infeasible outcome.

Only very recently, Grimm et al. [] focus on a single-zone system and explicitly compare cost-based redispatch with market-based redispatch. In their market-based redispatch model, endogenous redispatch prices are determined on a separate balancing market. In contrast to the findings in De Vries and Hakvoort [], market-based redispatch may yield an inefficient outcome as compared to cost-based redispatch. These differences may either be demand-side or supply-side driven and can only be observed in networks with at least three nodes. In a model extension, the authors analyze different degrees of demand redispatch. This extends the analysis of Oggioni and Smeers [], where consumers are either not or completely involved in the redispatch. Grimm et al. [] find that a higher availability of demand redispatch may paradoxically reduce welfare under market-based redispatch. The reason for such an effect is that under market-based redispatch the transmission system operator may have an incentive to reduce redispatch cost at the expense of welfare.

Long-run generation and transmission investments: nodal, zonal, and uniform pricing models

In this section, the discussed models will be extended to the case of long-run transmission and generation investments. The case will first be considered, where all relevant transmission, generation, consumption, and investment restrictions are taken into account (long-run nodal pricing or integrated planning models). In direct consequence of such an integrated operation and planning, these models can serve as a first best reference investment solution or as a system optimum. In a second step, zonal, uniform, and hierarchical nodal pricing models that, in general, may yield an inefficient outcome as compared to a first best solution are considered. Hierarchical nodal pricing models are market models operated under nodal pricing, where some investment decisions are made on a separate model level. In direct consequence of this separation, such models may yield a welfare loss as compared to an integrated, single-level nodal pricing model. However, from an economic perspective, such hierarchical models may better capture the interplay between different agents in real world electricity markets.

Reference long-run generation and transmission investment models: nodal pricing and integrated planning as overall system optima

Network investments

Investments in a network environment may add substantial complexity as compared to short-run models, computationally, economically, as well as physically. In this context, already in the nineties, Oren et al. [] and Wu et al. [] illustrate that a new or strengthened transmission line may paradoxically yield a network with a reduced transmission capacity.

Hirst and Kirby [] highlight economies of scale related to transmission extensions. The authors discuss the problem of new lines that are possibly overbuilt. This may, in particular, be true if network extensions take place at the beginning of the planning horizon with demand growing over time. Hirst and Kirby [] further demonstrate that congestion cost may be highly nonlinear related to transmission capacity. If there are only discrete transmission expansion options, this nonlinearity may additionally complicate optimal transmission investments. Moreover, the authors discuss that transmission and generation investments may highly influence each other, as optimal transmission investments will have a direct impact on optimal generation investments and vice versa. For more details, see the sections below.

Alguacil et al. [] further focus on network extensions in the case of discrete line investment decisions. The authors show that when losses are considered in a model with inelastic demand, the resulting network extensions may differ from the solution of a lossless model. Observe that in such models typically only a subset of lines can be extended. Otherwise, the resulting models may be too computationally challenging. To identify candidate lines that should be included in the model, a typical selection procedure will first identify the existing transmission lines, whose capacities bind. In a second step, these binding lines will be ranked according to their binding time, i.e., the aggregated time where the transmission line’s capacity is binding. Based on this ranking, a selection of candidate lines is made (see David and Wen [] for a more detailed discussion including several ranking criteria). Dating back to the early work of Garver [], linear programming may also be used to identify relevant transmission capacity shortages. However, most long-run studies mainly focus on some kind of radial network extensions, which mainly implies that existing transmission corridors can be strengthened by new candidate lines. As demonstrated by Baldick and Kahn [], the focus on such extensions may face severe problems. In particular, such radial network extensions will, in general, fail to identify optimal network configurations that may involve completely new transmission corridors. However, such simplifications may be necessary to ensure computational tractability of long-run investment models.

Generation and network investments

Besides these physical or technical challenges of long-run investments, in Arellano and Serra [] it is discussed that in the simple case of a two-node network with only two generation technologies, inelastic demand, and a single transmission line, the location of energy supply may have significant economic investment consequences, including the question of whether consumers or producers bear optimal marginal capacity cost. These effects have been analyzed for more general settings, i.e., for arbitrary networks with more than two nodes. Salerian et al. [] present a nodal pricing model with both continuous generation and transmission expansion planning. They characterize optimal investment decisions on a network and illustrate their results on a four-node test example. In particular, the authors show that in the case of binding capacities, both newly installed transmission lines and generation plants may receive revenues that exceed their construction costs (rents). Note that these results directly extend the peak-load pricing results, which consider generation investments without transmission restrictions.

Recently, Cedeño and Arora [], Jenabi et al. [], and Grimm et al. [] present clear-cut long-run nodal pricing models and compare their outcomes with the solution of different hierarchical energy markets (For more information, refer to Section 4.2.2). These long-run reference models consider both transmission and generation investments. Observe that in contrast to Jenabi et al. [] and Grimm et al. [], Cedeño and Arora [] assume inelastic demand.

Long-run generation and transmission investment models: zonal, uniform, and hierarchical nodal pricing models

Investment in generation capacity without relevant transmission constraints

This section briefly discusses generation investments at a single network node, where transmission can be neglected. Or more general, generation investments in lossless networks are considered, where transmission capacities are “large enough” and can therefore be neglected (“copperplate case”). In direct consequence, in such a model setting, there is no need to consider transmission extensions.

Steiner [] is among the first to study generation investments in such a single-node framework. He analyzes a simple two-period model with a high and a low demand, respectively. The two demand functions are decreasing, independent, and unequal. A single generator with constant variable and investment cost is considered. He shows that when the investment decision is made in order to maximize welfare, capacity investment costs are charged in the high demand period only. In the low demand period, the resulting price is equal to the variable cost of production. Crew and Kleindorfer [] extend this simple model by a consideration of two generators that are ordered such that their production cost strictly increase (investment cost strictly decrease) and satisfy a criterion under which both plants are used. Under these assumptions, in the low demand period, not only running costs are charged. Crew and Kleindorfer [] further extend these results to the case of multiple generators with several time periods. The authors characterize the technological frontier and present bounds on optimal prices. In the following years, one emerging strand of peak-load pricing literature incorporates market power or uncertainties, which are both not the focus of this paper (Amongst others, see Crew and Kleindorfer [], Kleindorfer and Fernando [], or for an overview Crew et al. []). The other stand of generation investment literature introduces limited network facilities and discusses investments in the context of networks. Such contributions are discussed in more detail below.

Transmission investments: zonal, uniform, and hierarchical nodal pricing models

Based on the transmission investment paradoxes under nodal pricing that were already discussed above, this section elaborates on investments in networks under uniform pricing, zonal pricing, or hierarchical nodal pricing. These models will, in general, yield a welfare loss as compared to a single-level nodal pricing model that simultaneously accounts for all transmission, generation, consumption, and investment constraints.

Fan et al. [] and Garcés et al. [] analyze a bilevel model, where a transmission system operator invests in the transmission grid on the first level. While in Fan et al. [] line investment cost depend on the length of a line, in Garcés et al. [] line investment has a discrete character. On the second level, energy trading takes place in a network environment (hierarchical nodal pricing). In Garcés et al. [] the possibility of load shedding is additionally included in the model. Observe that such models typically place transmission investment on the upper level, as network extensions will highly influence (anticipated) market outcomes. The authors show that their bilevel approach may yield significantly different outcomes as compared to a classical cost-minimization model.

In contrast to the above studies that focus on hierarchical nodal pricing models, Bjørndal and Jørnsten [] analyze a zonal pricing system. They discuss network modifications (the introduction of a new line) for different zone configurations with given generation facilities. As a main finding, optimal network modifications, i.e., welfare maximizing network extensions, may highly depend on the chosen zonal system. For some zonal configurations, the introduction of a new line may even yield a reduced welfare, given a decrease of the available transfer capability. This may also imply that an extension of the transmission network requires a redesign of current price zones.

Transmission and generation investments: zonal, uniform, and hierarchical nodal pricing models

In an extension of Garcés et al. [], Jenabi et al. [] propose a long-run bilevel model with continuous generation and discrete line investment. On the first level, a transmission system operator decides on optimal line expansion and anticipates the outcomes of a competitive electricity market with generation investment on the second level. Given all pseudo direct current network restrictions, firms choose optimal generation investment and production schedules on the second level on the basis of a complete network representation. Therefore, no redispatch is necessary, but congestion is directly managed through adequate prices. The authors compare the outcomes of their bilevel model with a long-run, single-level nodal pricing benchmark model. They show that in the case, where the transmission system operator maximizes its own profits in the bilevel model, investments may be less efficient as compared to the welfare maximizing bilevel or the single-level nodal pricing case. These results also extend the work of Rious et al. [], who discuss the properties of an ideal transmission system operator including the welfare-maximization perspective.

Baringo and Conejo [] further analyze investments in a market environment. The authors study a bilevel model with wind power and network facility investment decisions. On the first level, optimal investments in both wind power and new transmission lines are chosen, anticipating market clearing on the second level that accounts for all relevant transmission restrictions. The authors also analyze subsidies in wind power facilities. As a main result, they point out that optimal line and wind power investment may highly depend on each other. In addition, subsidies may help in promoting wind power investments.

The fact that different market designs (for instance nodal, zonal, or uniform pricing) may affect optimal investment decisions is further underlined by Grimm et al. [], who present a trilevel power market model that allows for generation and transmission investment and subsequent cost-based redispatch. On the first level, the transmission system operator decides on optimal line investment (and a corresponding transmission fee), while on the second level, competitive firms trade on a zonal spot market and make optimal generation investment decisions. If spot market outcomes violate transmission restrictions, on the third level, cost-based redispatch takes place. The authors show that market outcomes (prices, line and generation investments) of their trilevel model (with one or multiple zones) may differ from a nodal pricing solution. In addition, market-splitting (zonal pricing) can yield a different network expansion plan as compared to uniform pricing (and nodal pricing) and may only partially decrease the welfare loss of a uniform pricing model—independent of the chosen network fee regime. Observe that Grimm et al. [] further analyze the long-run effects of market splitting (zonal pricing) in a bilevel problem with a given transmission network.

Key model elements of electricity production, transportation, and consumption

General overview

In this section, key model elements of electricity production, transportation, and consumption that are used in the different research papers discussed above are summarized. Assuming a discretized time horizonT with time periods or load blocks t of equal length (for instance one minute, hour, day, week, or year), the case of an inter-temporal problem that involves operational and investment decisions is considered. This multiple-period framework includes the special case of a static model, where only a single time period is considered (for instance the peak-load period).

Assume further a connected graph G=(N,L) with a node set (or bus set) N and a set L that describes the different transmission lines. If neither a consumer nor a generator is located at a given node n∈N, this node will be denoted as an intermediate network node. Otherwise, node n is a final node. Transportation from producers to consumers takes place by using the different lines that interconnect the nodes, where the respective production and consumption is located at. Figure 3 depicts the main relation between the three functions of electricity production, transportation, and consumption that will be described in more detail in this section. In addition, Figure 4 shows the three corresponding planning and scheduling processes of exchange capacity allocation, market outcome determination, and grid status analysis, which will be the focus of Section 6. As will be seen below, the interrelation between economic and technical forces drives to a significant degree optimal decisions of the relevant agents.

Electricity production

In the electricity market literature, different generators G are assumed, which are described by various technical and economic characteristics. Each generatorg∈G is located at one network node. By {\vartheta }_n^{\mathrm{gen}}\left(G\right) the subset of generators of G that are located at node n is denoted. Let q_g,t describe the power production of generator g in time period t∈T . Power production must be nonnegative and must not violate its upper generation capacity

\mathrm{0}\le q_{g,t}\le {\overline{Q}}_{g,t}({\overline{q}}_g^{\mathrm{ex}},{\overline{q}}_g^{\mathrm{new}})\mathrm{ for all }t\in T, g\in G,

where the upper generation bound {\overline{Q}}_{g,t}is a function of the existing capacity {\overline{q}}_g^{\mathrm{ex}} and the newly invested capacity {\overline{q}}_g^{\mathrm{n}\mathrm{e}\mathrm{w}}\ge 0. By assumption, {\overline{q}}_g^{\mathrm{new}}\mathrm{is set to }\mathrm{0} if investment cannot take place. As in Stoft [], Bjørndal and Jørnsten [], Bjørndal et al. [], or Ehrenmann and Smeers [], in its simplest form{\overline{Q}}_{g,t} just equals {\overline{q}}_g^{\mathrm{ex}}, which may also be infinite. Bohn et al. [], Salerian et al. [], Green [], Jenabi et al. [], or Grimm et al. [] also use equivalent availability factors of existing and new generation capacity in the upper capacity bound{\overline{Q}}_{g,t}. Such availability factors may reflect (known) outages that are caused for instance by repair and maintenance work or by different weather conditions (for example summer vs. winter period).

Both electricity production q_g,t and generation investment {\overline{q}}_g^{\mathrm{new}} involve costs. It is further assumed that each generator is characterized by a variable production cost function{\mathrm{C}}_g^{\mathrm{var}}\left(q_{g,t}\right) and an investment cost function {\mathrm{C}}_g^{\mathrm{i}\mathrm{n}\mathrm{v}}\left({\overline{q}}_g^{\mathrm{n}\mathrm{e}\mathrm{w}}\right). Tables 2 and 3 show examples of such cost functions, where for simplicity, the indices t and g are omitted. While most of the considered studies use constant generation investment cost functions (see Table 3), variable production cost functions cover a large variety of different function types including constant, linear, affine, or stepwise linear functions (see Table 2).

Note that in a general context there may be further technical or inter-temporal production constraints that can be included in the model. For instance interdependencies with the heating market, start-up costs, minimal downtimes, minimum runtimes, or technology-specific constraints may be added (see for instance Leuthold et al. [] or Neuhoff et al. []).

Denote by

q={{\displaystyle \left({({{\displaystyle q}}_{\mathrm{1},\mathrm{1}},{{\displaystyle q}}_{\mathrm{2},\mathrm{1}},\dots ,{{\displaystyle q}}_{\left|G\right|,\mathrm{1}})}^{\mathrm{T}},\dots ,{({{\displaystyle q}}_{\mathrm{1},\left|T\right|},{{\displaystyle q}}_{\mathrm{2},\left|T\right|},\dots ,{{\displaystyle q}}_{\left|G\right|,\left|T\right|})}^{\mathrm{T}},{{\displaystyle \left(\stackrel{-\mathrm{n}\mathrm{e}\mathrm{w}}{{{\displaystyle q}}_{\mathrm{1}}},\stackrel{-\mathrm{n}\mathrm{e}\mathrm{w}}{{{\displaystyle q}}_{\mathrm{2}}},\dots ,\stackrel{-\mathrm{n}\mathrm{e}\mathrm{w}}{{{\displaystyle q}}_{\left|G\right|}}\right)}}^{\mathrm{T}}\right)}}^{\mathrm{T}},

the vector of decision variables of all generators, where for instance q_i,j is the amount of production of generator i in time period j. Then, Ω^gen describes the set of q that satisfies all generation restrictions given in this section.

Electricity consumption/demand

Let a set of consumers C be given. Typically, a consumer refers to a household, a firm, or a public entity. By {\vartheta }_n^{\mathrm{d}\mathrm{e}\mathrm{m}}\left(C\right) the subset of consumers C is denoted that are located at node n. The demand d_{c,t}\ge \mathrm{0} of each consumer c∈C can either be completely inelastic or vary with the electricity price. Inelastic demand is often justified by a comparatively low short-term demand elasticity (see Ackermann []). For the case, where elastic demand is used, decreasing inverse demand functionsp_c,t(d_c,t) that describe the relation between prices and demand are typically assumed. Table 4 shows examples of different demand functions including constant, linear, or quadratic functions. For the sake of clear exposition, the indices c and t are omitted.

Observe that instead of using explicit (inverse) demand functions, some authors assume (increasing) concave benefit functions (CBF) for the individual consumers. As in Bohn et al. [], Hogan [], Chao and Peck [], Wu et al. [], or Green [], such functions give the value added for each consumerc, when consuming a given quantity d_c,t in period t. Mathematically, such benefit functions can be expressed as the integral of the underlying inverse demand function up to the electricity consumed.

It should further be noted that similar to Ding and Fuller [] or Garcés et al. [], bounds on electricity demand may additionally be used to limit demand, e.g., a maximum demand for each consumer may be specified.

As above, by

d={\left({\left({{\displaystyle d}}_{\mathrm{1},\mathrm{1}},{{\displaystyle d}}_{\mathrm{2},\mathrm{1}},...,{{\displaystyle d}}_{\left|C\right|,\mathrm{1}}\right)}^{\mathrm{T}},...,{\left({{\displaystyle d}}_{\mathrm{1},\left|T\right|},{{\displaystyle d}}_{\mathrm{2},\left|T\right|},...,{{\displaystyle d}}_{\left|C\right|,\left|T\right|}\right)}^{\mathrm{T}}\right)}^{\mathrm{T}},

the vector of consumption decisions of all consumers is described, where for instance d_i,j denotes demand of consumer i in time period j. Ω^demis the set of d that satisfies all relevant restrictions imposed on demand.

Electricity transportation

Each transmission line l∈L is described by different physical characteristics v(l). Amongst others, these characteristics comprise the transmission capacity {\overline{P}}_l, the conductance g_l, the succeptance b_l, or the resistance r_l. By L^{\mathrm{new}}\subset L the set of candidate transmission lines is denoted that can be invested in.y_l is the corresponding investment variable. Alguacil et al. [], Garcés et al. [], Baringo and Conejo [], Jenabi et al. [], and Grimm et al. [] assume that y_l is a binary variable. In contrast, for instance, Salerian et al. [] or Arellano and Serra [] use continuous line investment variables.

C_l^{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}\left(y_l\right) describes the cost of building a line l, which for the case of binary decision variables is naturally a fixed amount if a given candidate line is built. In the case of continuous line investment, like in Arellano and Serra [] affine functions may be used.

Energy transportation requires an adequate modeling of the respective transmission flows. In this section energy models with a pseudo direct current (DC) and a transmission-capacity constrained (CC) flow model—sometimes also referred to as some kind of a transshipment model are mainly considered. Both models can be derived from an alternating current (AC) flow model. Observe that in the extreme case, only a single node is considered with no transmission (see Fig. 5).

The most precise way to model power transmission is the nonlinear AC flow model with complex power. As can be seen in Table 5, a complete AC model involves for each network noden and time period t, real power P_n,t, reactive power Q_n,t, voltage magnitude V_n,t, and voltage angles δ_n,t. Power flows on a given line l are described by real power flow P_l,t and reactive power flow Q_l,t. These power flows can be modeled by a set of nonlinear constraints that involve all of these variables. Amongst others, the flow constraints include Kirchhoff’s laws on loop flows and nodal flow balance (the so-called Kirchhoff’s first and second laws). In addition, capacity constraints are typically imposed on the flow variables that account for the transmission capacities of the electricity network. Such capacity constraints are intended to prevent the network from any physical damage or from system blackouts. As a complete derivation of all AC power flow equations is out of the scope of this paper, only the two main power flow equations are presented for an arbitrary linel= (n,m) in an arbitrary time period t∈T and the interested reader is referred for further information to Bjørndal and Jørnsten [], Frank et al. [], or Schweppe et al. []:

$$\begin{gathered} P_{l,t}=V_{n,t}^{\mathrm{2}}{g}_l-V_{n,t}{V}_{m,t}{g}_l\cos ({\delta }_{n,t}-{\delta }_{m,t}) \\ -V_{n,t}{V}_{m,t}{b}_l\sin ({\delta }_{n,t}-{\delta }_{m,t}), \end{gathered}$$

Q_{l,t}=V_{n,t}{V}_{m,t}{b}_l\cos ({\delta }_{n,t}-{\delta }_{m,t})-V_{n,t}{V}_{m,t}{g}_l\sin ({\delta }_{n,t}-{\delta }_{m,t})

-V_{n,t}^2\left(g_l+{\tilde{g}}_l\right).

In the DC model, some simplifying assumptions regarding phase angle differences and voltage magnitudes are made (Fig. 5). In addition, in DC flow models, reactive power (flows)Q_n,t (Q_l,t) is (are) ignored and only real power that provides energy is considered. Note, however, that these models are not direct current models but an approximation of the nonlinear AC model. Similar to the AC case, the main (real) power flow equation is briefly stated as

P_{l,t}=b_l({\delta }_{n,t}-{\delta }_{m,t}) \mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}\mathrm{a}\mathrm{l}\mathrm{l}\hspace{0.17em}l=\left(n,m\right)\in L,t\in T.

CC flow models further simplify the DC model by ignoring phase angle restrictions and by only accounting for the transmission capacity constraints between network nodes and nodal flow balance. For a better overview, Table 6 gives examples of papers that use the different flow models including the single-node case, where transmission constraints do not play any role.

Independent of the chosen flow model (AC, DC, or CC), in addition to the mentioned flow restrictions that typically only involve physical variables, reactive powerP_n,t is explicitly linked to consumption and production at each node n\in N:

P_{n,t}=\sum _{g\in {\vartheta }_n^{\mathrm{gen}}\left(G\right)}{q}_{g,t}-\sum _{c\in {\vartheta }_n^{\mathrm{dem}}\left(C\right)}{d}_{c,t}

(1)\mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}\mathrm{a}\mathrm{l}\mathrm{l}\hspace{0.17em}n\in N, t\in T.

In settings, where investment can take place, Constraint (1) does not only interconnect generation, demand, and power, but indirectly imposes restrictions on optimal investment decisions.

Like in Salerian et al. [], Garcés et al. [], or Jenabi et al. [] further constraints may be added that affect electricity transportation, for instance in form of an upper bound on the number of new transmission lines or on the total budget that is available for transmission investments.

Let f be the vector of all flow (and decision) variables. Given one of the above flow models (AC, DC, or CC), Ω^flow describes the set of all f that satisfy all relevant flow constraints except from Eq. (1). By Ω^gdf, the set of all demand, generation, and flow variables (d, q, f) that satisfy Eq. (1) is denoted.

Welfare formulation

As in Bjørndal and Jørnsten [] or Ehrenmann and Smeers [], energy market studies typically focus on the efficiency of a congestion management approach, i.e., on the welfare effects of a considered congestion management. In the case of integrable (elastic) demand and cost functions, welfare is expressed as the aggregated difference of consumer surplus—the area under the demand function (or the value of the used benefit function)—and all production or investment costs:

(2)\begin{array}{l}w:={\displaystyle \sum }_{t\in T}{\displaystyle \sum }_{c\in C}\hspace{0.17em}{\displaystyle \int \begin{array}{l}{{\displaystyle d}}_{c,t}\hfill \\ \mathrm{0}\hfill \end{array}{{\displaystyle p}}_{c,t}\left(h\right)\mathrm{d}h}-{{\displaystyle \sum }}_{g\in G}\hspace{0.17em}{\displaystyle \int \begin{array}{l}{{\displaystyle q}}_{g,t}\hfill \\ \mathrm{0}\hfill \end{array}{{\displaystyle C}}_g^{\mathrm{var}}\left(h\right)\mathrm{d}h}\hfill \\ -{\displaystyle \sum }_{g\in G}{\displaystyle \int \begin{array}{l}{\overline{q}}_g^{\mathrm{n}\mathrm{e}\mathrm{w}}\hfill \\ \mathrm{0}\hfill \end{array}{{\displaystyle C}}_g^{\mathrm{i}\mathrm{n}\mathrm{v}}\left(h\right)\mathrm{d}h-{\displaystyle \sum }_{l\in L}\hspace{0.17em}{{\displaystyle C}}_l^{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}\left(y_l\right)y_l-C^{\mathrm{a}\mathrm{d}\mathrm{d}}\left(d,q,f\right).}\hfill \end{array}

ƒThe last term C^add describes additional cost components that may affect welfare. For instance, Oggioni and Smeers [] include average redispatch cost in their spot market welfare function to model a grid access charge.

Observe that in De Vries and Hakvoort [], Alguacil et al. [], Arellano and Serra [], Baringo and Conejo [], or Trepper et al. [] welfare maximization reduces to a minimization of the cost that arise in order to satisfy inelastic demand.

Congestion management: basic models

In this section, a general model framework is presented for the discussed congestion management approaches based on the model elements introduced in Section 5. For an illustration of this model framework, we refer to our three-node example that is described in Appendix B.

Nodal pricing model

As discussed in Oren et al. [], Wu et al. [], or Schweppe et al. [], a nodal pricing model fully integrates energy trading and transmission, i.e., it takes all transmission restrictions of the chosen flow model into account when clearing the market. Following Bohn et al. [], nodal pricing uses the common welfare maximization objective and accounts for all demand, production, and transmission restrictions. In consequence, the resulting nodal prices directly reflect scarcity of transmission facilities and ensure physical feasibility. In a general form, the nodal pricing model can be formulated as

(3a)\max w\left(q,d,f\right)

(3b)\mathrm{s}.\mathrm{t}. q\in {\Omega }^{\mathrm{g}\mathrm{e}\mathrm{n}},d\in {\Omega }^{\mathrm{d}\mathrm{e}\mathrm{m}},f\in {\Omega }^{\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}},(d,q,f)\in {\Omega }^{\mathrm{g}\mathrm{d}\mathrm{f}},

where the welfare function w accounts for all relevant cost components.

As in the nodal pricing model welfare is maximized, like in Bjørndal and Jørnsten [] or Jenabi et al. [] nodal pricing is often used as a reference, when different congestion management approaches are evaluated.

It should be further noted that both zonal and uniform pricing models either relax some of the nodal pricing constraints or add new constraints. As this changes the feasible region and may yield physical infeasibilty, an expost adjustment of equilibrium quantities may be needed in order to restore a physically feasible solution (see also the sections below).

Uniform pricing model

In contrast to nodal pricing, uniform pricing models neglect all transmission constraints and completely separate energy trading from transmission. As already shown in the two-node example in Section 2, in direct consequence, the corresponding uniform market price will, in general, fail to give adequate signals for an efficient use of transmission capacities.

Observe that uniform pricing models contain an explicit market clearing constraint to ensure that demand and generation is balanced. Note that this market clearing condition results from Kirchhoff’s law on nodal flow balance together with Eq. (1) for the case of a single network node (see also Fig. 5).

(4){\displaystyle \sum _{c\in C}{d}_{c,t}}={\displaystyle \sum _{g\in G}{q}_{g,t}} \mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}\mathrm{a}\mathrm{l}\mathrm{l}\hspace{0.17em}t\in T.

The uniform pricing model reads as

(5a)\max w\left(q,d,f\right),

(5b)\mathrm{s}.\mathrm{t}. q\in {\Omega }^{\mathrm{g}\mathrm{e}\mathrm{n}},d\in {\Omega }^{\mathrm{d}\mathrm{e}\mathrm{m}},

(5c){\displaystyle \sum _{c\in C}{d}_{c,t}}={\displaystyle \sum _{g\in G}{q}_{g,t}} \mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{\hspace{0.17em}}t\in T.

Despite special cases, where networks have infinite or at least “enough” transmission capacities, outcomes under uniform pricing will typically violate network restrictions. Therefore, in general, uniform pricing will be followed by redispatch—a corrective measure operated outside the spot market (see Section 6.3.2).

Zonal pricing models

This section presents zonal pricing models that differ in the extent to which they account for energy transmission restrictions, when clearing the market. Different models will be explicitly related to nodal and uniform pricing.

Throughout this paper, the partition of the node set N into different price zones Z_k will be denoted by Z with

(6)\varnothing \in Z,

(7)\underset{k\in \left\{\mathrm{1},\mathrm{...},\left|Z\right|\right\}}{\cup }{{\displaystyle z}}_k=N,

(8)Z_j\in Z,Z_k\in Z\hspace{0.17em}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\hspace{0.17em}j\ne k\Rightarrow Z_j\cap Z_k=\varnothing .

Ideal zonal pricing model

The ideal zonal pricing model of Bjørndal and Jørnsten [] or Bjørndal and Jørnsten [] takes all demand, production, and transmission constraints into account. In addition to these restrictions that are also used in the nodal pricing model, an ideal zonal pricing model forces equal prices in each of the given price zones. Observe that no ex post adjustment of market quantities is needed, as ideal zonal pricing always yields a physically feasible flow.

In a general form, the ideal zonal pricing model can be stated as

(9a)\max w\left(q,d,f\right)

(9b)\mathrm{s}.\mathrm{t}. q\in {\Omega }^{\mathrm{g}\mathrm{e}\mathrm{n}},d\in {\Omega }^{\mathrm{d}\mathrm{e}\mathrm{m}},f\in {\Omega }^{\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}},\left(d,q,f\right)\in {\Omega }^{\mathrm{g}\mathrm{d}\mathrm{f}},

{{\displaystyle p}}_{c,t}={{\displaystyle p}}_{\overline{c},t} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} k\in \left\{\mathrm{1},...\left|Z\right|\right\},

(9c)c,\tilde{c}\in {{\displaystyle \vartheta }}_{{{\displaystyle Z}}_k}^{\mathrm{d}\mathrm{e}\mathrm{m}}\left(C\right),t\in T,

where with slightly abuse of notation the price equality in Eq. (9c) also accounts for the equilibrium incentive compatibility of the respective generators. As above, the welfare objectivew considers all relevant costs.

Second best zonal pricing

In contrast to ideal zonal pricing that uses all physical transmission constraints of the chosen flow model, in the second best zonal pricing model, the original network is replaced by an aggregated network as in Ehrenmann and Smeers [] or Oggioni and Smeers []. Four main steps are identified that are typically applied to construct such an aggregated network representation:

•Step1:‚Use of a prespecified mapping M of the original network G to an aggregated network \widetilde{G^{\prime }}

•Step 2:‚Deletion of all self-loops of \widetilde{G^{\prime }} yielding \tilde{G}

•Step3:‚Decision on the new flow model of the aggregated network \tilde{G} (for instance a DC or CC flow model)

•Step‚4:‚Calculation of relevant network parameters \tilde{v} of \tilde{G} (for example transmission capacities, susceptances, ...)

Observe that M maps each original node to a price zone that is represented by a new node in the aggregated network. Analogously, each transmission line is mapped to a line in the new network\tilde{G}. Figure 6 depicts a four-node example and the derivation of the respective aggregated two-node network.

In general, there may be different mappings M in Step 1 of the above scheme that, for instance, depend on the number of price zones and the price zone configuration. Therefore, different rules exist on how to choose a reasonableM (see Stoft [] for a critical discussion). It should be further noted that depending on whether the original DC/AC flow model or a simplified CC model is used for the aggregated network\tilde{G} in Step 3, the terms flow-based market coupling (splitting) or transfer-capacity-based market coupling (splitting) are often used (see also Ehrenmann and Smeers [] or Grimm et al. []). Moreover, the problem of a calculation of adequate network parameters\tilde{v} of \tilde{G} in Step 4 should be pointed out. In this context, especially the choice of inter-zonal transfer capacities is very challenging. As in Grimm et al. [] or Grimm et al. [], the sum of line capacities of inter-zonal transmission lines may be used as some kind of “naive” inter-zonal transfer capacities. Also, a worst case analysis of inter-zone line flows may indicate reasonable transfer capacity values. However, as pointed out by Oggioni and Smeers [], transfer capacities that base on such rules may not always be efficient. It should be further noted that only recently, Tong et al. [], Luna et al. [], Luo et al. [], or Bucksteeg et al. [] additionally analyze the impact of (uncertain) wind generation on transfer capacities, which, however, is not the focus of this paper.

Observe that the incomplete consideration of network constraints in \tilde{G} may yield flows that are not physically feasible for the original network G. Overload may, in particular, be observed if no or too low flow reliability margins are used. In the following, both a flexible and a fixed zone system that differ in the way, in which they deal with infeasible network flows, will be considered.

1)‚Flexible zone systems

As proposed in Bjørndal and Jørnsten [], Bjørndal et al. [], or Bjørndal and Jørnsten [], in a flexible zone system, starting with a uniform pricing model (|Z|=1), new zones are iteratively added and a nodal pricing model is solved on a corresponding extended network until a solution is obtained that yields a physically feasible solution inG. The main functioning of such a flexible zone system is expressed in Algorithm 1, where it is assumed that in the aggregated network\tilde{G} and in the original network G the same flow model is used (for instance a DC model).

Algorithm 1: Functioning of a flexible zone system

Input: network, demand, and generation parameters for the zonal pricing model.

Output: solution to the zonal pricing model.

Step 1:‚Solve the uniform pricing model (5a) to (5c).

Step2:‚While the corresponding network flows f are ƒƒƒƒinfeasible for the underlying network Gdo

ƒƒƒIntroduce one or several new price zones Z^new ƒƒƒƒaccording to a specified criteria.

ƒƒƒConstruct a corresponding network \tilde{G}.

ƒƒƒSolve the nodal pricing model (3a) to (3b) on ƒƒƒƒthe network \tilde{G}.

Step 3:‚Return zonal pricing solution.

Note that in the literature there are different rules on how to choose the new zones added in Step 2 of Algorithm 1. For instance, congested network links can be used as discussed in Bjørndal and Jørnsten [].

2)‚Fixed zone systems with subsequent redispatch

In a fixed zone system, a zonal partition into price zones Z_k ∈ Z is a priori given, with the extreme form again being a single-zone or uniform pricing system. Obviously, if |Z| = |N| and the original flow model is chosen for the aggregated network \tilde{G}, the nodal pricing model is reached. Therefore, in the following, 1<|Z|<|N| will be assumed.

In a first step, a nodal pricing model is solved for the aggregated network \tilde{G} with the |Z| price zones and the chosen flow model. As only an aggregated and simplified network is used, consumption and generation quantities will, in general, cause physical infeasibility in the original networkG. Therefore, in a second step, demand and generation is redispatched by a transmission system operator in order to restore the physical feasibility inG avoiding any damage to the network (rescheduling of consumption and/or generation). These redispatch measures are ex post operated outside the spot market. Denote byDd and Dq the demand and generation redispatch adjustment, i.e., the amount by which demand and generation is decreased or increased.d +Dd and q +Dq will describe final demand and generation, respectively. If demand or generation is regulated down (up), as in Ding and Fuller [] or Blijswijk and De Vries [] the term constrained-off-regulation (constrained-on-regulation) is often used.

The redispatch costs that result due to the market interventions can either be cost-or market-based and will be denoted by C (Dd, Dq). Typically, redispatch measures restore physical feasibility while minimizing the involving redispatch costs (For an overview of possible redispatch cost functions, see Table 7, where both cost-based and market-based functions are presented). Being a corrective method, in some model specifications, redispatch may arrive at the same outcome as a nodal pricing model, however, possibly involving high redispatch cost.

The general redispatch problem will be stated as

(10a)\min C\left(\mathrm{\Delta }d,\mathrm{\Delta }q\right)

(10b)\mathrm{s}.\mathrm{t}.\left(q+\mathrm{\Delta }q\right){\in \Omega }^{\mathrm{g}\mathrm{e}\mathrm{n}},\left(d+\mathrm{\Delta }d\right)\in {\Omega }^{\mathrm{d}\mathrm{e}\mathrm{m}},f\in {\Omega }^{\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}},

(10c)\left(d+\mathrm{\Delta }d,q+\mathrm{\Delta }q,f\right)\in {\Omega }^{\mathrm{g}\mathrm{d}\mathrm{f}},

(10d)\left(d+\mathrm{\Delta }d,q+\mathrm{\Delta }q,f\right)\in {\Omega }^{\mathrm{a}\mathrm{d}\mathrm{d}},

where the feasible flow sets correspond to the original network G and f describes the respective final flow variables. W^add indicates additional constraints that may be imposed on redispatch market variables. Such additional constraints include the above relation between the final demand and generation variablesd +Dd and q +Dq, respectively. To give other examples, Oggioni and Smeers [] additionally require the aggregated spot market volume not to be changed by the redispatch measure and in Green [] the total transmission surplus must be equal to the merchandising surplus of a nodal pricing system. It should also be noted that, for instance, in De Vries and Hakvoort [], Oggioni and Smeers [], or Trepper et al. [] only one market side—mainly the generation side—can be redispatched. In the case where demand cannot be redispatched, final demand is fixed to spot market demand. It should again be noted that if, on the opposite, only (peak-load) demand is adjusted, the so-called load-shedding is often applied with shed load being compensated according to a fixed cost parameter (see also Garcés et al. [].)

(Further) Multilevel electricity market models

As optimal decisions of economic agents highly depend on optimal decisions of other agents, in recent years (further) multilevel models have been developed (see Table 8 for examples). In these models, different problem levelsL₁, L₂, ... are considered. Each level consists of an objective O₁, O₂, ... that can, for example, be welfare w or redispatch cost C. The decision variables at the different levels of L₁, L₂, ... are denoted by (d₁, q₁, f₁), (d₂, q₂, f₂), ... and are restricted to corresponding feasible sets W₁,W₂, ..., with

{\Omega }_i\subset \{\left(q_i,d_i,f_i\right):q_i\in {\Omega }^{\mathrm{gen}},d_i\in {\Omega }^{\mathrm{d}\mathrm{e}\mathrm{m}},f_i\in {\Omega }^{\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}},

(d_i,q_i,f_i)\in {\Omega }^{\mathrm{g}\mathrm{d}\mathrm{f}}\},

for all levels i. As optimal decisions at the different levels depend on each other, a general multilevel problem is represented as the following hierarchical problem (see also Dempe []).

(11a)\max {{\displaystyle O}}_{\mathrm{1}}\mathrm{s}.\mathrm{t}. \left({{\displaystyle d}}_{\mathrm{1}},{{\displaystyle q}}_{\mathrm{1}},{{\displaystyle f}}_{\mathrm{1}}\right)\in {{\displaystyle \mathrm{\Omega }}}_{\mathrm{1}},

(11b)\mathrm{s}.\mathrm{t}.\max {{\displaystyle O}}_{\mathrm{2}}\mathrm{s}.\mathrm{t}. \left({{\displaystyle d}}_{\mathrm{2}},{{\displaystyle q}}_{\mathrm{2}},{{\displaystyle f}}_{\mathrm{2}}\right)\in {{\displaystyle \mathrm{\Omega }}}_{\mathrm{2}},

(11c)...

Observe that the second best market splitting model with redispatch is such a bilevel model with O₁ = w and O₂ = −C. Accordingly, both the nodal pricing and the ideal zonal pricing model can be seen as a multilevel problem consisting of only a single level (withO₁ = w).

Future research challenges

In this section, key challenges and open questions for future research are briefly discussed.

Further hierarchical long-run models

When analyzing congestion management methods under different market environments, long-run effects of both generation and transmission investments are obviously highly relevant. As pointed out by Grimm et al. [], market models that account for different economic players (like firms or transmission system operators) and their interdependent decisions may yield investments that differ from a first-best solution of an integrated planning benchmark. In this context, especially new multilevel energy market models that account for the expectations of different players would help to capture highly relevant long-run effects for real world energy markets. Open research questions may, for instance, relate to the long-run effects of different types of redispatch including cost-based and market-based redispatch; capacity markets; technology-specific subsidy programs; or different degrees of planning cooperation among the relevant players in a multilevel market environment.

Hybrid congestion management models

As shown in Bjørndal et al. [], hybrid congestion management schemes, where subsets of network nodes are operated under different congestion management schemes, will shed light on how different congestion management mechanisms affect each other in an interconnected network structure. Such interconnected systems are of high relevance, for instance, in Europe, where different congestion management measures including market coupling or redispatch are currently applied. It is obvious that model extensions that consider various congestion management schemes and account for both short-run operational and long-run investment effects may shed new light on the functioning of interconnected congestion management systems. Such models could also be used to quantify and assess deviations from single congestion management models that have mainly been investigated in the past.

Optimal price zone configurations and efficient transfer capacities

With regard to zonal systems, the question of an optimal number of price zones and their corresponding boundaries still needs additional research, as different zonal systems will typically induce significant welfare effects. In particular, models that endogenously determine optimal zone configurations would highly contribute to a better understanding of market splitting/coupling. In this context, based on the short-run work of, for instance, Bjørndal and Jørnsten [], new models that account for the aforementioned long-run or hybrid-market effects may be of high relevance. As market splitting or coupling is applied in different countries around the world (for instance, in Australia or Europe), new insights would also be relevant from a practical point of view. In addition, further work on efficient rules to construct general inter-zone transfer capacities is needed, as current approaches are either too conservative or may yield physical infeasibility.

Conclusions

In this paper, different approaches to congestion management were surveyed. In particular, nodal, zonal, and uniform electricity pricing for the case of perfect competition and/or welfare maximization were focused on. Strategic interactions among producers as well as uncertainties were mainly abstracted from.

In a first step, the short-run congestion management literature for the cases of two-node and multiple-node networks was focused on. In a second step, long-run investments were separately discussed for networks consisting of a single node (“copperplate”) and of multiple nodes. Naturally, only generation investments were analyzed in the single-node case, where transmission was not considered. Both generation and transmission investments were discussed in the case of networks consisting of several nodes. In the latter case, The interconnection between transmission and generation investments in a network environment was also highlighted. In a third step, key concepts of electricity production, transportation, and consumption together with general model formulations for the different congestion management approaches were summarized. The model framework structured the discussed congestion management methods and highlighted their interconnection. Since nodal, zonal, or uniform pricing are applied in various countries around the world, this work may also contribute to the current discussion on real-world electricity market designs.

As a main conclusion, nodal pricing yields a first best outcome both in the short- and in the long-run, since congestion is directly reflected in optimal spot prices. However, given several disadvantages of nodal pricing like the high number of nodal prices that have to be calculated or the possibly complex coordination of the corresponding submarkets, some countries refrain from choosing a nodal pricing system in practice. Instead, some form of zonal or uniform pricing is implemented, for instance, in several European countries. In general, such uniform and zonal pricing will be accompanied by a welfare loss as compared to optimal nodal pricing. However, a detailed apriori analysis should quantitatively elaborate on the welfare effects associated with the market design under consideration of specific network structures in order to assess corresponding advantages and disadvantages. In such quantitative studies, typically a trade-off between the depiction of technical aspects (including minimum downtimes, modelling of combined heat and power plants, or minimum runtimes) and the mathematical solvability of the model will occur. However, corresponding model simplifications will in particular be necessary in the case, where long-run effects are taken into account.

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