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# Trustless coordination mechanism for smart grid energy markets, a game theoretic approach

Increasing the amount of  installed renewable energy sources such as solar and wind is an essential step towards the decarbonization of the energy sector.

From a technical point of view, however, the stochastic nature of distributed energy resources (DER) causes operational challenges. Among them, unbalance between production and consumption, overvoltage and overload of grid components are the most common ones.

As DER penetration increases, it is becoming clear that incentive strategies such as Net Energy Metering (NEM) are threatening utilities, since NEM doesn’t reward prosumers to synchronize their energy production and demand.

In order to reduce congestions, distributed system operators (DSOs) currently use a simple indirect method, consisting of a bi-level energy tariff, i.e. the price of buying energy from the grid is higher than the price of selling energy to the grid. This encourages individual prosumers to increase their self-consumption. However, this is inefficient in regulating the aggregated power profile of all prosumers.

Utilities and governments  think that a better grid management can be achieved by making the distribution grid ‘smarter’, and they are currently deploying massive amount of investments to enforce this vision.

As I explained in my previous post on the need of decentralized architectures for new energy markets,  the common view of the scientific community is that a smarter grid requires an increase in the amount of communication between generators and consumers, adopting near real-time markets and dynamic prices, which can steer users’ consumption during periods in which DER energy production is higher, or increase their production during high demand. For example, in California a modification of NEM that allows prosumers to export energy from their batteries during evening peak of demand has been recently proposed.

But as flexibility will be offered at different levels and will provide a number of services, from voltage control for the DSOs to control energy for the transmission system operators (TSOs), it is important to make sure that these services will not interfere with each other. So far, a comprehensive approach towards the actuation of flexibility as a system-wide leitmotiv, taking into account the effect of DR at all grid levels, is lacking.

In order to optimally exploit prosumers’ flexibility, new communication protocols are needed, which coupled with a sensing infrastructure (smart meters), can be used to safely steer aggregated demand in the distribution grid, up to the transmission grid.

The problem of coordinating dispatchable generators is well known by system operators and has been studied extensively in the literature. When not taking into account grid constraints, this is known under the name of economic dispatch, and consists in minimizing the generation cost of a group of power plants . When operational constraints are considered, the problem increases in complexity, due to the power flow equations governing currents and voltages in the electric grid. Nevertheless, several approaches are known for solving this problem, a.k.a. optimal power flow (OPF), using approximations and convex formulations of the underlying physics. OPF is usually solved in a centralized way by an independent system operator (ISO). Anyway, when the number of generators increases, as in the case of DERs, the overall problem increases in complexity but can be still effectively solved by decomposing it among generators.

The decomposition has other two main advantages over a centralized solution, apart from allowing faster computation. The first is that generators do not have to disclose all their private information in order for the problem to be solved correctly, allowing competition among the different generators. The second one is that the computation has no single point of failure.

In this direction, we have recently proposed a multilevel hierarchical control which can be used to coordinate large groups of prosumers located at different voltage levels of the distribution grid, taking into account grid constraints. The difference between power generators and prosumers is that the latter do not control the time of generated power, but can operate deferrable loads such as heat pumps, electric vehicles, boilers and batteries.

The idea is that prosumers in the distribution grid can be coordinated only by means of a price signal sent by their parent node in the hierarchical structure, an aggregator.  This allows the algorithm to be solved using a forward-backward communication protocol. In the forward passage each aggregator receives a reference price from its parent node and sends it downwards, along to its reference price, to its children nodes (prosumers or aggregators), located in a lower hierarchy level. This mechanism is propagated along all the nodes, until the terminal nodes (or leafs).  Prosumers in leaf nodes solve their optimization problems as soon as they are reached by the overall price signal. In the backward passage, prosumers send their solutions to their parents, which collect them and send the aggregated solution upward.

Apart from this intuitive coordination protocol, the proposed algorithm has other favorable properties. One of them is that prosumers only need to share information on their energy production and consumption with one aggregator, while keeping all other parameters and information private. This is possible thanks to the decomposition of the control problem. The second property is that the algorithm exploits parallel computation of the prosumer specific problems, ensuring minimum overhead communication.

However, being able to coordinate prosumers is not enough.

The main difference between the OPF and DR problem, is that the latter involves the participation of self-serving agents, which cannot be a-priori trusted by an independent system operator (ISO). This implies that if an agent find it profitable (in terms of its own economic utility), he will compute a different optimization problem from the one provided by the ISO. For this reason, some aspects of DR formulations are better described through a game theoretic framework.

Furthermore, several studies have focused on the case in which grid constraints are enforced by DSOs, directly modifying voltage angles at buses. Although this is a reasonable solution concept, the current shift of generation from the high voltage network to the low voltage network lets us think that in the future prosumers and not DSOs could be in charge of regulating voltages and mitigating power peaks.

With this in mind, we focused on analyzing the decomposed OPF using game theory and mechanism design, which study the behavior and outcomes of a set of agents trying to maximize their own utilities $u(x_i,x_{-i})$, which depend on their own actions $x_i$ and on the action of the other agents $x_{-i}$, under a given ‘mechanism’. The whole field of mechanism design tries to escape from the Gibbard–Satterthwaite theorem, which can be perhaps better understood by means of its corollary:

If a strict voting rule has at least 3 possible outcomes, it is non-manipulable if and only if it is dictatorial.

It turns out, that the only way to escape from this impossibility result, is adopting money transfer. As such, our mechanism must define both  an allocation rule and a taxation (or reward) rule. In this way, the overall value seen by the agents is equal to their own utility augmented by the taxation/remuneration imposed by the mechanism:

$v_i (x_i,x_{-i})= u_i(x_i,x_{-i}) + c_i(x_i,x_{-i})$

Anyway, monetary transfers are as powerful as perilous. When designing taxes and incentives, one should always keep in mind two things:

• Designing wrong incentives could result in spectacular failures, as we learned from the case of a very anecdotal misuse of incentives from British colonial history, known as the cobra effect
• If there is a way to fool the mechanism, self-serving prosumers will almost surely find it out.  Know that some people will do everything they can to game the system, finding ways to win that you never could have imaginedSteven D. Levitt

A largely adopted solution concept, used to rule out most of the strategic behaviors from agents (but not the same as strategyproof mechanism), is the one of ex-post Nash Equilibrium (NE), or simply equilibrium,  which is reached when the following set of problems are jointly minimized:

\begin{aligned} \min_{x_i \in \mathcal{X}_i} & \quad v(x_i, x_{-i}) \quad \forall i \in \{N\} \\ s.t. & \quad Ax\leq b \end{aligned}

where $x_i \in \mathcal{X}_i$ means that the agents’ actions are constrained to be in the set $\mathcal{X}_i$, which could include for example the prosumer’s battery maximum capacity or the maximum power at which the prosumer can draw energy from the grid.  The linear equation in the second row represents the grid constraints, which is a function of the actions of all the prosumers, $x = [x_i]_{i=1}^N$, where N is the number of prosumers we are considering.

Rational agents will always try to reach a NE, since in this situation they cannot improve their values given that the other prosumers do not change their actions.

Using basic optimization notions, the above set of problems can be reformulated using KKT conditions, which under some mild assumptions ensure that the prosumers’ problems are optimally solved. Briefly, we can augment the prosumers objective function using a first order approximation, through a Lagrangian multiplier $\lambda_i$, of the coupling constraints and using the indicator function to encode their own constraints:

$\tilde{v}_i (x_i,x_{-i}) = v_i (x_i,x_{-i}) + \lambda_i (Ax-b) + \mathcal{I}_{\mathcal{X}_i}$

\begin{aligned} 0& \in \partial_{x_i} v_i(x_i,\mathrm{x}_{-i}) + \mathrm{N}_{\mathcal{X}_i} + A_i^T\lambda \\ 0 & \leq \lambda \perp -(Ax-b) \geq 0 \end{aligned}

where $\mathrm{N}_{\mathcal{X}_i}$ is the normal cone operator, which is the sub-differential of the indicator function.

Loosely speaking, Nash equilibrium is not always a reasonable solution concept, due to the fact that multiple equilibria usually exists. For this reasons equilibrium refinement concepts are usually applied, in which most of the equilibria are discarded a-priori. Variational NE (VNE) is one of such refinement. In VNE, the price of the shared constraints paid by each agent is the same. This has the nice economic interpretation that all the agents pay the same price for the common good (the grid). Note that we have already considered all the Lagrangian multiplier as equal $\lambda_i = \lambda \quad \forall i \in \{N\}$ in writing the KKT condition.

One of the nice properties of the VNE is that for well behaving problems, this equilibrium is unique. Being unique, and with a reasonable economic outcome (price fairness),  rational prosumers will agree to converge to it, since at the equilibrium no one is better off changing his own actions while the other prosumers’ actions are fixed. It turns out that a trivial modification of the parallelized strategy we adopted to solve the multilevel hierarchical OPF can be used to reach the VNE.

On top of all this, new economic business models must be actuated in order to reward prosumers for their flexibility. In fact, rational agents would not participate in the market if the energy price they pay is higher than what they pay to their current energy retailer. One of such business models is the aforementioned Californian proposal to enable NEM with the energy injected by electrical batteries.

Another possible use case is the creation of an self-consumption community, in which a group of prosumers in the same LV grid, pays only at the point of common coupling with the grid of the DSO (which e.g. could be the LV/MV transformer in figure 1). In this way, if the group of prosumers is heterogeneous (someone is producing energy while someone else is consuming), the overall cost that they pay as a community will be always less than what they would have paid as single prosumers, at the loss of the DSO. But if this economic surplus drives the prosumers to take care of power quality in the LV/MV, the DSO could benefit from this business model, delegating part of its grid regulating duties to them.

How does blockchain fits in? Synchronizing thousands of entities connected to different grid levels is a technically-hard task. Blockchain technology can be used as a thrust-less distributed database for creating and managing energy communities of prosumers willing to participate to flexibility markets. On top of the blockchain, off-chain payment channels can be used to keep track of the energy consumed and produced by prosumers and to disburse payments in a secure and seamless way.

Different business models are possible, and technical solutions as well. But we think that in the distribution grid, the economic value lies in shifting the power production and consumption of the prosumers, enabling a really smarter grid.

At Hive Power we are enabling the creation of energy sharing communities where all participants are guaranteed to benefit from the participation, reaching at the same time a technical and financial optimum for the whole community.