6.3.1 Joint Balancing with flexibility
The settings within the Joint Balancing with Flexibility cluster deal with the question of how prosumers can make use of their flexibility to optimize a common objective. The settings and approaches within this cluster are all rather similar to each other, although some settings highlight particular aspects, such as the preference between different ’types’ of electricity, such as, e.g., locally produced or green electricity. Due to the decentralized structure of computation, individual data and parameters of the participants, such as the flexibility or the valuation of electricity, do not need to be shared. Therefore, most settings take data privacy into account. As mentioned in Sect.
3, all of the settings in the Joint Balancing with Flexibility cluster do take flexibility into account. In some settings, this flexibility directly stems from devices such as batteries, EVs, or heat pumps, but there are also some problem definitions, in which flexibility only stems from the flexible part of the load. These settings often reduce the problem to a bare minimum and do not model any devices. Nevertheless, the models are able to represent the key problems that may occur in (local) energy trading. Regarding the time horizon, some settings directly formulate models for multiple time slots, while quite a few of the considered settings within this cluster only formulate single time slot models. Nevertheless, it is often noted that for the sake of simplicity and notation only a single time slot is modeled, but the presented approach can easily be adapted for multiple time slots. However, even though larger time horizons can be modeled at once, uncertainty is not taken into account in any of the settings. Grid constraints are also not considered by the majority of settings, although a few use approximations of power flow to create price signals for overloaded lines within the grid. In all settings, decentralized optimization techniques, such as ADMM, relaxed C+I, or decentralized primal-dual algorithms are used.
The approaches in this cluster are Baroche et al. (
2019), Guo et al. (
2021), Jiang et al. (
2021), Khorasany et al. (
2018,
2020), Le Cadre (
2019), Moret et al. (
2018,
2020), Morstyn and McCulloch (
2019), Sorin et al. (
2019) and Sousa et al. (
2019). In Baroche et al. (
2019), Guo et al. (
2021), Le Cadre (
2019), Moret et al. (
2018), Moret et al. (
2020), Sorin et al. (
2019) and Sousa et al. (
2019), the approaches are based on simplified and reduced models in which no devices are directly modeled. Nevertheless, different types of local trading, such as direct peer-to-peer trading, communal trading, or a hybrid version are formulated and solved either via ADMM or relaxed C+I, see Sect.
5 for a short introduction to these techniques. Due to the structure of the simplified models, there is no difference in the objective value of the proposed decentralized and centralized optimization algorithms. In addition to an optimal solution, the relaxed C+I in Sorin et al. (
2019) also computes prices for each individual trade, which are based on the economic concept of shadow prices, that are the dual variables of the trade constraints. In Khorasany et al. (
2018) and Khorasany et al. (
2020), the same simplified model is extended by grid constraints in the form of distribution load flow. Based on the load flow, the power transfer distribution factor (PTDF), which computes the contribution of each trade between prosumers in the power flow, is computed for each line in the grid and is used as a price signal for the prosumers. Before solving this model with an adapted decentralized primal-dual gradient method, Lagrangian multipliers move global constraints into the objective function. Instead of introducing grid constraints to the simplified models, in Jiang et al. (
2021), Lee et al. (
2019) and Morstyn and McCulloch (
2019) different devices, such as batteries are directly modeled. While Morstyn and McCulloch (
2019) introduce prosumer preferences over different classes of electricity, such as green or local electricity, Jiang et al. (
2021) focus on a payment scheme in a second stage, which is based on a Nash Bargaining game. All three models are again solved using ADMM. In Moret et al. (
2020) risk levels for prosumers are introduced to model different human behavior in the presence of uncertainty. The model is again solved using ADMM.
6.3.2 Equilibrium balancing with flexibility
Another approach to make use of flexibility is offered by the settings in the Equilibrium Balancing with Flexibility cluster. In contrast to the Joint Balancing with Flexibility cluster, the participants behave more selfishly and do not simply act as distributed computing units for the goal of the whole microgrid. Individual objectives and goals are more important and techniques that lead to stable solutions in which no participant can improve anymore have to be used. Beyond the considered characteristics, for most aspects, there are large similarities between the settings, although there are some exceptions. Comparable with the previous cluster, due to the decentralized structure of computations, in most cases, sensitive data, such as flexibility or utility functions, can remain private for each participant. Apart from Shilov et al. (
2021), no other setting considers grid constraints in its approach. Devices are mostly explicitly modeled, although there are a few settings, in which there are either no devices modeled, or PV generation is indirectly included via the load profiles. Regarding the considered time horizon, the settings are evenly split up between considering only a single time slot and multiple time slots at once. Furthermore, unlike the previous cluster, settings covering only one time slot can not always easily be upgraded to multiple time slot models. This is mainly a consequence of the absence of one central model which can be split up into subproblems for each participant. Adapting all individual models while ensuring that the used techniques still converge to an equilibrium is more challenging. Regarding uncertainty, only some settings take that into account, even if a larger time horizon is modeled. Hence, no exact pattern between time horizon and uncertainty can be recognized, as there are settings with only one time slot, but also settings covering multiple time slots, which consider uncertainty. Due to the focus on individual objectives, techniques in this cluster have to be able to represent this selfish behavior, while ensuring that a stable solution is found. Game theory offers the right tools for such problems, and in most settings, a Stackelberg game is used to model the relation between the different participants. Other settings ignore the leader-follower dynamic of Stackelberg games and focus on general non-cooperative games. In some settings, either the non-cooperative or Stackelberg games are complemented by other techniques, such as auctions.
The first group consists of Stackelberg games in which prosumers are leaders and followers. The notion of prosumer is generalized beyond the definition in Sect.
4, as also companies that either buy or sell electricity are included. This setting is considered in Anoh et al. (
2020), El Rahi et al. (
2019), Lee et al. (
2015), Liu et al. (
2018), Liu et al. (
2017) and Paudel et al. (
2019). In Anoh et al. (
2020), Lee et al. (
2015) and Paudel et al. (
2019), the set of prosumers is divided into a set of sellers and a set of buyers. The sellers act as the leaders in a multi-leader multi-follower Stackelberg game, while the buyers are the followers. The strategies of sellers and buyers can differ from one approach to the other. In Anoh et al. (
2020) and Lee et al. (
2015), the sellers start by announcing the amount of electricity they are willing to sell, and the buyers react with the prices they are able to pay. Based on these prices, the sellers update the amount of electricity and the game continues until convergence to a Stackelberg equilibrium. In Paudel et al. (
2019), the strategies are quite different. The sellers announce their prices and the amount of electricity they are able to sell first, and then the buyers react with a selection of the sellers. This selection is a probability distribution for each buyer over the complete set of sellers and should indicate the probability of a buyer choosing a specific seller. The buyers compute this selection using an evolutionary game. Based on this selection, the sellers update their prices using a non-cooperative game. Note that hereby the amount of electricity to sell is a fixed parameter in this setting. Again, it is shown that the iterative Stackelberg game converges to a Stackelberg equilibrium. In El Rahi et al. (
2019), Liu et al. (
2018) and Liu et al. (
2017) on the other hand, the prosumers are not in advance divided into buyers or sellers. In all these settings, the leader is a single entity that can buy and sell electricity and the followers are the set of prosumers. In Liu et al. (
2018) and Liu et al. (
2017), the leader is a storage system within the microgrid, which can buy excess electricity or sell electricity to prosumers with a demand. Its goal is to maximize its profit, while the objectives of the prosumers are to maximize their own utility. The leader starts by announcing internal prices for the prosumers. Based on these prices, the prosumers can each solve their own (bounded) optimization problem to maximize their utility. They then announce their optimal amount of electricity to buy or sell, and the leader reacts to this by adjusting its prices. While the convergence of this iterative approach to a Stackelberg game is shown in Liu et al. (
2018), in Liu et al. (
2017) the model is based on a bi-level optimization problem, and no guarantees for convergence are made. In El Rahi et al. (
2019), instead of a storage system, a power company is the leader of the Stackelberg game. It first announces a price, based on which the prosumers play a non-cooperative game among themselves to determine how much electricity to buy or sell. Two different ways to achieve a Stackelberg game are proposed, with the first one being an iterative one leading to an
\(\epsilon\)-Stackelberg equilibrium, while in the second, the leader solves a non-linear optimization problem to directly find the Stackelberg equilibrium.
The second group of settings uses Stackelberg games to model the relation between the prosumers and their MGO or DSO. This setting is considered in Askeland et al. (
2021), Aussel et al. (
2020), Cui et al. (
2018), Le Cadre (
2019), Le Cadre et al. (
2019), Liu et al. (
2017), Rajasekhar et al. (
2019), Tushar et al. (
2014) and Zugno et al. (
2013). In Cui et al. (
2018), Le Cadre et al. (
2019) and Liu et al. (
2017), the leader of the game is the MGO, while the prosumers are the followers. The goal of the leader is to maximize its profit and it starts by submitting initial internal buying and selling prices to the prosumers. The prosumers use these prices as input to their utility maximization problems and optimize them on their own. The prosumers then announce the amount of electricity to buy or sell and the leader updates its prices. The existence of a Stackelberg equilibrium is shown. In Askeland et al. (
2021), Le Cadre (
2019), Tushar et al. (
2014) and Zugno et al. (
2013), the leader is either a central power station, which wants to buy surplus electricity from the prosumers, the DSO, which wants to minimize the grid cost of the microgrid or retailers, who want to maximize their profit of selling electricity to the prosumers. The followers are once again the prosumers, who want to maximize their utility, or local MGOs, who want to maximize the social welfare of their set of prosumers. The leader announces initial prices or grid tariffs and based on this, the prosumers optimize their utility. In contrast to Cui et al. (
2018) and Liu et al. (
2017), the prosumers either solve a generalized Nash equilibrium (GNE) game to decide how much to sell to the central power station, or they need to solve a complementary problem to compute an equilibrium. In both cases, the prosumers then announce their electricity consumption, either on an individual base (Tushar et al.
2014), or on an aggregated level (Askeland et al.
2021). Based on the reaction of the prosumers, the leader updates its prices and this iterative scheme continues until some convergence criterion is met. In Zugno et al. (
2013) on the other hand, the bilevel problem is reformulated into a single-level MILP, which can easily be solved. A similar approach is taken in Aussel et al. (
2020), where the trilevel problem is reformulated twice to obtain a tractable formulation. In a first step, an explicit formulation is derived for the prosumers, which removes the bottom layer. The remaining two layers, with the supplier being the leader and the MGOs being followers, are then reformulated using the KKT conditions of the followers in the leader’s problem. In Rajasekhar et al. (
2019), the MGO also acts as the leader of the Stackelberg game, but instead of using price signals as a strategy, it uses demand profiles. In the beginning, the MGO collects the load profiles of all prosumers and optimally schedules its own battery usage. It then broadcasts the aggregated load profiles minus the prosumer’s load profile to each prosumer. In addition, also boundaries for the aggregated load profile and penalty prices are announced. The prosumers then optimize their utility function, which is a weighted sum of electricity costs, the comfort level, and the minimization of interruption to increase the life span of appliances. The prosumers announce their resulting load profiles to the MGO, which updates its battery schedule and possibly also the penalty prices. This process continues until the difference in the objective function of the MGO is reasonably small. It is shown that the iterative process converges to a Stackelberg equilibrium.
The third group of approaches uses general non-cooperative games to model the interactions between prosumers and other participants. The corresponding approaches are Devine and Bertsch (
2018), Dvorkin et al. (
2019), Grübel et al. (
2020), Kim et al. (
2013), Le Cadre et al. (
2020), Li (
2021), Shilov et al. (
2021), Tushar et al. (
2017) and Zhang et al. (
2019). In Devine and Bertsch (
2018), Grübel et al. (
2020), Kim et al. (
2013) and Le Cadre et al. (
2020), a non-cooperative game is played among all prosumers. The utility functions of the players consist of the cost and the satisfaction of electricity consumption, while the strategies of the prosumers are their load profiles. In Grübel et al. (
2020), the prosumer may be equipped with storage devices and the market equilibrium problem is reformulated into a mixed complementarity problem using the KKT conditions. The resulting formulation is then solved via ADMM. A similar solution approach is presented in Devine and Bertsch (
2018), where a non-cooperative game between prosumers, consumers, and generator units with possible failure times is modeled. The formulation results in a stochastic mixed complementarity problem, which solves the optimization problems of each prosumer and results in an equilibrium solution. In Kim et al. (
2013), a tailor-made billing scheme penalizes heavy electricity user and it is shown that an iterative gradient-based algorithm converges to the NE of the game. In Le Cadre et al. (
2020) on the other hand, the coupling constraints between the prosumers lead to a GNE. A detailed analysis provides insights into the efficiency of the GNE compared to a central solution. In Shilov et al. (
2021), a GNE game is played between the DSO and the prosumers. The strategy of the prosumers is based on the amount of flexibility that they are willing to offer, while the strategy of the DSO is based on the fraction of the prosumers’ flexibility that it wants to use, as well as a congestion price. Due to a coupling constraint between the prosumers as well as the DSO, a GNE is computed. In Tushar et al. (
2017), a similar setting is considered, in which a power company is interested in buying surplus electricity from the prosumers, given a fixed budget. A cake-cutting game is proposed and a variational equilibrium is found using a decentralized algorithm. In Dvorkin et al. (
2019), a non-cooperative game between a market operator, producers and consumers is modeled. It is shown that a unique equilibrium always exists and a distributed algorithm is presented, in which producers and consumers react to the market operator’s prices by adapting their production and consumption. In Zhang et al. (
2019), two non-cooperative games between prosumers, the MGO, and suppliers are played. The MGO acts as a local aggregator between the prosumers on the one side and the suppliers on the other side. For the non-cooperative game between MGO and suppliers, the suppliers offer bids to the MGO. The MGO then uses these bids and the net demand of the prosumers to compute external trading prices with the suppliers. The utility function of the suppliers represents the profit they make by selling electricity to the MGO. The second non-cooperative game in Zhang et al. (
2019) is played among the prosumers, who decide on their load profiles, given some predefined buying and selling prices for the given time interval. The utility functions of the prosumers consist of the cost of buying or the profit of selling electricity locally as well as the utility of electricity consumption. For both non-cooperative games, it is shown that a unique NE exists and an iterative algorithm is given, which converges to the NE. Both non-cooperative games are then connected via the MGO, which updates the external and internal prices after a change in either bids from the suppliers or the electricity consumption from the prosumers. In Li (
2021) on the other hand, a trilevel problem between the DSO, MGOs, and prosumers is modeled. Two different solution approaches, one cooperative and one non-cooperative are proposed. In both cases, the trilevel model is first reduced to a bilevel model by deriving an analytical solution to the non-cooperative game between MGOs and their respective prosumers. The remaining bilevel problem is then solved either in a cooperative or non-cooperative way using price and demand as signals.
The last group combines Stackelberg or general non-cooperative games with auctions (see Doan et al.
2021; He and Zhang
2021; Saad et al.
2011; Tsaousoglou et al.
2021; Tushar et al.
2016; Wang et al.
2014). In the considered literature, there are two main ways to combine these approaches with each other. In Saad et al. (
2011) and Wang et al. (
2014), a non-cooperative game is played among a set of prosumers with a surplus of electricity. The strategies of the sellers are specified by the amount of electricity they are willing to sell, while the utility is the profit they gain by selling electricity to the buyers. The prices are computed using a standard double auction between buyers and sellers, as is also often seen in the Strategic Matching without Flexibility cluster. After initializing the amounts to sell, the double auction is run, and based on the new clearing price, each prosumer one after the other finds best responses by communicating with the MGO, which acts as the auctioneer. It is shown that this iterative algorithm converges to a NE. In He and Zhang (
2021), a non-cooperative game is played among the prosumers of a microgrid. Each prosumer first solves a simple optimization problem to determine how much electricity to offer or ask for in the auction. Following a double auction, the winners participate in a non-cooperative game, in which each participant finds an optimal deviation from its original bid. This deviation maximizes a utility function, which consists of profit and the reluctance to deviate from the original bid. Afterward, the clearing price of the double auction is updated and the non-cooperative game continues, with each participant finding its best response to the new clearing price until a stopping criterion is met. Similarly to the above approaches, in Tsaousoglou et al. (
2021), a modified version of a combinatorial auction is run. Within each iteration, players are added to the set of winners of the auction, based on the outcome of a non-cooperative game. In Doan et al. (
2021) and Tushar et al. (
2016) on the other hand, a double auction is run first to determine the set of winners of the auction, as well as the clearing price limits. Then, a Stackelberg game is played, with the MGO being the leader and the followers are either the winning buyers or sellers of the double auction. Using the range of possible clearing prices, the objective of the MGO is to maximize the average social welfare of the remaining set of winners. The strategy of the followers is to adapt the amount of electricity they are willing to sell or buy. This iterative process continues until the Stackelberg equilibrium is found.