Visual Market equilibrium model for power seller multi-microgrid based on a graphic game

Microgrids (MGs) have characteristics of flexibility and intelligence. These grids provide potential for integrating renewable energy sources. Hybrid Microgrids make possible to manage complementary between sources and storage schemes, enabling exploration of stronger commercial impact of these energy producing stations. This paper proposes a new market model of equilibrium operation for MGs in local energy markets. MGs remotely located in the grid are virtually associated to compete with other MGs better located. The model meets the interests of the MGs by remuneration of the delivered power and the other stakeholders via loss reduction. A deterministic method based on reduction of the constraint set size and application of Karush-Kuhn-Tucker (KKT) conditions process visually the Nash equilibrium (NE) and the Pareto efficiency (PE). The non-cooperative static game has the ability to encourage the participation of small agents in the grid.


INTRODUCTION
Increased energy consumption, the urbanization, the dismembered the energy supply chain, the pursuit of efficiency and sustainability, and the loss of traditional economies of scale through centralized generation (LASSETER; PAIGI, 2004), resulted in opportunities for consumers to participate more actively in the network. This participation can occur by demand-side management, demand response, and energy production through distributed energy resources (DERs).
In market, DERs participate directly or through intermediary entities, either individually or by aggregation as microgrids (MGs). MGs are a subsystem within defined area acting as a single entity. MGs can integrate renewable energy sources (RESs), conventional generators, energy storage systems (ESSs), main grid and loads (LASSETER;PAIGI, 2004;MENG et al., 2016). When the MG has power surplus, it can supply the grid as an independent energy producer (DOU et al., 2019). Therefore, MGs may be contracted for power supply and/or ancillary services as voltage control, backup supply, and network stability.
Configurations with more than one type of source are called hybrid MGs. In general, hybrid systems composed of renewable resources take advantage of the complementarity of sources, in this way, can contribute to the regularity of the system's supply during its operation.
From an economic point of view, Hanna et al. (2017) find that the changes from the pure distribution service (electric utilities) to the diversified system (electric utilities and MGs) are advantageous.
From the investor's perspective, the reason of MGs existence is to serve its own load at a lower cost than the utility companies. The next generation of retail electricity market will offer new business opportunities for prosumers (CHEN et al., 2018).
Although the spread of prosumers could create an environment competitive on sector, there is evidence of market power in the segment (PEREZ et al., 2016). High financial investment for the MGs implementation and their remote location in the grid are suggestive examples of inequality on competition between them. DERs located at more remote points in the grid (i.e., farthest from the substation) have greater operational importance than less remote DERs, because the firsts improve power quality in less at_tractive portions for the utility companies to invest. However, DERs in more remote portions of the network suggests a smaller number of neighboring consumers to be served because the DER is farther away from the highest loads, therefore, smaller market share than DERs centrally located in the grid that have the potential to customers on several directly connected extensions.
The selfish profile of energy producers makes difficult the cooperation among them (TANG, et. al., 2017). The interaction among MGs involves the conflict of the market share, therefore game theory is a suitable methodology to analyze strategically the behavior of these agents. The objective of this analysis is to find the Nash equilibrium (NE), which is the desired behavior for all participants. The strength of the NE concept imposes a computational stalemate: not all games have pure equilibrium and those that do can give rise to search of high computational complexity Admitting the remuneration of the power sold by MGs based on bilateral contracts, therefore, there are favorable prices for investor remuneration. This paper aims to propose a market model competition among MGs active in local energy market. MGs remotely installed in the grid participate of a static game on associated way. This virtual association seeks justice for market competition due to the importance of these MGs to power quality of the grid. These associate MGs share the local market with another locally better installed MG. The graphical game model is proposed to competition between remotely installed MGs and MG more centralized in the grid. Through the KKT conditions, the game is solved and NE and PE points are obtained in polynomial time.
The model aims to remunerate the main agents of electric grid: MGs, utility companies, and consumers to reduce the potential disagreements among stakeholders.
In addition to the development of a graphical platform to allow visualization of NE and PE points and contribute to the reform of the electricity market other contributions are listed as follows: • Boost investment in DERs, especially RESs, given that autonomous MGs participate intensively in the energy market.
• Encourage MGs, particularly those remotely installed in the network to identify energy businesses opportunities.
• Promote fairness and equality on market share by strengthening the remote MGs in the network.
• Propose a competitive benefit model of multiple stakeholders: utility companies, consumers, and MGs.

LITERATURE REVIEW
MG can be defined as a set of DERs, ESSs, loads and coordinated agents that can operate connected or not to the grid, in low or medium voltage.
Due to the spread of DERs, utility companies may innovate in their business models. They can support MGs with professional knowledge and ensure energy equilibrium. It is from this perspective that the Distribution System Operator (DSO) has the role of managing the energy flows and offsets involving MGs in the retail market as in Fig. 1. The DSO can negotiate preferentially with suppliers that use renewable resources in their portfolio. Unlike Boloukat e Foroud (2018), where the DSO is considered a protagonist with decision-making power over the MGs, in this paper, the DSO is a referee of contracts with energy and monetary settlement skills.  Boloukat and Foroud (2018) In order to guarantee the reliability of the electrical grid, the hybrid MR needs a robust power management system. In this sense, the Multi Agent System (MAS) based on control mode becomes efficient to facilitate the decision-making process within MGs (KHAN et al., 2019). The MAS structure is a composition of multiple intelligent agents (machines) that interact with each other, offering robustness and flexibility to the managed system. These agents work autonomously and are employed to strengthen the system as they solve problems that could not be solved by a single agent (LASSETER; PAIGI, 2004). Fig. 2 shows the internal macro design of interaction among intelligent agents on decentralized control arrangement performed by the Hybrid Microgrid Operator. Once the agents are in EN do not have incentives to deviate, an approximate Nash means a low incentive to do. For a continuous game, Zarei e Salami (2016) discretize the space of strategies and resort for modified solving matrix game to decrease the complexity of an auction game. Although discrete strategies contribute to reduce search time, there are a risk of losing some equilibria. The Nikaido-Isoda function and the relaxation algorithm are used in the retail market for sale power surplus in Marzband et al. (2016).
In this case, small prosumers come together to become big enough to participate in the market.
The bi-level optimization model also called hierarchical optimization, which one of the players considered commander (leader) decides, and then the other agents (followers) decide in sequence, is also known as sequential play. This hierarchy process is used sometimes for reasons inherent to the processes, sometimes for computational convenience, and even if the second option is a consequence of the first, if necessary. The follower only optimizes its strategy as a result of the choice of leader, and thus reduces the algorithm's search space (HINCAPIE; GALLEGO; MANTOVANI, 2019). Table 1 shows the positioning of players in bi-level games involving MRs and others electrical system agents.

CHARACTERIZATION OF THE PROBLEM
In multi-microgrids systems of several owners, it will be very unlikely to carry out cooperative management between them, therefore it is proposed a non-cooperative model considering the competitive relationship among MGs. It is assumed that prosumers have a strictly growing desire to remunerate themselves.
Considering the economic motivation for dispatch, it is proposed a multi-objective optimization problem (1)  is filled with the smallest own load between the MGs and its size is equal to the number of them: G + 1. This measure guarantees an approximation to the left of the point in the Fig. 3, which begins the evaluation through of Algorithm 1. Until ∀ inequality restrictions of (1) be violated The following section presents a graphical model that requires two goods, therefore it is necessary to replace loss for loss reduction ( ). To each feasible , the loss reduction can be calculated by (2). Likewise, it can be obtained − .

GRAPHIC TOOL
The Edgeworth box in Fig. 4 is a graphical visualization, which represents all possible allocations of goods (EDGEWORTH, 1881). After solving (1)   In this model, loss reduction is considered the remuneration of the utility companies and consumers. The utility companies gain in quality, image, and improvement operational problems that avoid sanctions imposed by the regulator. As the cost of loss is reduced, consumers benefit from reduced energy bills.
As MGs earn by selling energy, therefore they prefer to operate in the direction shown in Fig. 4 with > > . The utility companies and consumers prefer high loss reduction, such that > > as shown in the Fig. 4. It is assumed that the convex curve in relation to each axis provides the utility (U) of each player. The curves B and C provide utility for the i-axis and curve A provides utility for the -i-axis. Therefore, midpoints of the curve are preferable to extremes. From the Fig. 4, it is observed that for i-axis ( , ) > ( , ) and ( , ) > ( , ) because the curve C is more distant from the origin of the i-axis than B, therefore C provides a greater utility than B. Thereby, the point E corresponds to the maximum possible utility. The same analysis is admitted for the player -i-axis.
According to eigenvalues of , the conic is classified as an ellipse.
Whether the players are rational, they have a convex objective function and share a convex constraint, thereby a convex problem can be given by (

RESULTS AND DISCUSSION
A modified IEEE 13-bus distribution network is shown in Fig. 6 that is used as a case study (IEEE, 2020  We assume that MGs are hybrid, therefore, these agents are able to sustain the selling contracts backed through the mix of renewable and non-renewable resources. In Edgeworth box, a possible set of operation curve for the problem (1) and the convex set for the system under study is illustrated in Fig. 7. The ordinate axes relate the total loss reduction according to (2). The convex set in the figure is formed by the most distant concave curve: blue continues for the i-axis and red discontinues for the -i-axis. By nonlinear fit, these curves are given by (13) and (14). The initial conic has the normalized matrix (15).
The NE can also be obtained by fixed point theorems, therefore it can be expected that all the NE points are non-degenerate (Rosen, 1965). The point E in Fig. 4 is obtained for all curves, this result can be observed in Fig. 8 for modes 1 and 2 of operation . These are the contract curves that contain all NE and PE points. The Fig. 8  The association of remotely installed MGs on the model favors their participation in the local market. This participation promotes justice for small prosumers located in remote portions of the network, which have greater operational importance than central prosumers.
The present simulation was performed on a processor computer Intel Core i5 with 3.1 Ghz of processing speed and 16 GB of RAM, with average approximate time of 580s for solution of multiobjective processes: (1) and (3).
Once operating on the contract curve, no agent can increase its utility unless decrease other. PE and NE are a strong concept, because they mean that mutual trade gains are exhausted. The PE frontier contains the non-dominated values of the objective functions and it is illustrated in Fig. 9, which axes represent the utility. The i-axis corresponds to the associated MGs, the another (-i-axis) represents the large MG together with the utility companies and the consumers. In another way, the -i-axis represents only the large MG and the i-axis aggregates the utilities of associated MGs, utility companies, and consumers according to modes of operation. As utility of i-axis involves the participation of the associated MGs, these MGs are also privileged by the model for obtaining greater utility gain. For this example, the gain of utility involving associated MGs is 88% greater than that involving large MG.

CONCLUSIONS
The energy systems of the future are planned to cover several autonomous entities that interact each other with the network and other consumers. The reduction in costs involved in hybrid systems due to the rapid expansion of global markets should expand investments for generation focused on energy trade.
The proposed methodology aims to address the market share of MGs according to their inequalities. The Edgeworth box allows obtain visually NE and PE results. The virtual association among MGs remotely installed in the network aims a market justice for the participation of these agents.
To infinite games, the strategy search space is usually large, therefore special techniques are needed to search. The concave/convex condition of the operation/utility function associated a graphical representation provides computationally tractable mechanisms for obtaining an analytical NE and PE results.
We analyzed a macro perspective for the game where MGs compete each other.
For future work, the economic dispatch of hybrid MG acting as a firm will be analyzed as a cooperative game.

This work was supported by Coordination of Improvement of Higher Education
Personnel of Brazil (Capes) in partnership with Research Support Foundations of Goias state (Fapeg). The authors thank these institute for their support.