# Automated Linear Function Submission-Based Double Auction as Bottom-up Real-Time Pricing in a Regional Prosumers’ Electricity Network

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- We propose an LFS-DA mechanism for a regional prosumers’ electricity network. The mechanism is able to achieve an exact balance between electricity demand and supply at each moment during iteration.
- We prove that the changes in the price profile obtained by the LFS-DA become the same as those obtained by the RTP derived from the dual decomposition framework, except for a constant factor.

## 2. Background and Related Work

#### 2.1. Penetration of Distributed Energy Resources and Smart Grids

#### 2.2. Demand-Side Management

#### 2.3. Multi-Agent System and Double Auction

## 3. Problem Definition

#### 3.1. Basic Assumptions

^{+}and ·

^{−}represent the direction from the viewpoint of a smart meter, which is at the center of electric energy flows in an agent’s house. Specifically, ·

^{+}and ·

^{−}represent outflow and inflow, respectively. We assume that the energy flow through the smart meter adheres to the law of the conservation of energy for each time slot t as follows.

_{i}∈ [0,1] must be taken into consideration. If η

_{i}= 1, the charged electricity can be fully charged without any loss. The storage profile ${s}_{i}^{t}$ represents the state of charge (SOC) of the i-th agent’s storage device at time t and is expressed by the following equation.

_{i}can be different depending on each agent. Each constant of the system, i.e., γ, η

_{i}, ${p}_{t}^{G\pm}$ and ${s}_{i}^{max}$, is fixed when a day-ahead market is held. In the long term, e.g., over several years, they can vary over time.

#### 3.2. Social Welfare

^{2}, and define the utility for consuming electric energy ${l}_{i}^{t-}$ for the i-th agent by ${D}_{i}^{t}:\mathbb{R}\to \mathbb{R}$, where ${D}_{i}^{t}$ is a concave function of class C

^{2}. The welfare W

_{i}: ℝ

^{8}

^{T}× ℝ

^{T}→ ℝ of the i-th agent is defined as follows.

_{t}is a buyer’s price profile in the regional electricity market at time $t\in \mathcal{J}$ and ${\varphi}_{i}^{t}({x}_{t})$ is a utility function for the i-th agent at time t, including payments to an outside grid. Because of the electricity losses in transmission, the seller’s price and buyer’s price become different. To balance the amount of money paid by sellers and buyers, the seller’s price becomes γp

_{t}. The i-th agents behavior at time t is represented by ${x}_{i}^{t}$, concisely. We call ${x}_{i}^{t}$ a state vector for the i-th agent at time t. Next, we define the social welfare of the network by W(x, p).

#### 3.3. Real-Time Pricing Algorithm

**Problem 1**(Primal problem).

k ← 0 |

Initialize the price profile ${p}^{(k)}={\left({p}_{t}^{(k)}\right)}_{t\in \mathcal{J}}$ |

repeat |

// Each agent solves its sub-problem (A7) and obtains its solution. |

Update ${x}_{i}^{(k+1)}\leftarrow {x}_{i}^{*}({p}^{(k)})$. |

// A central utility solves the master problem (A9) under the condition that ${x}_{i}={x}_{i}^{(k+1)}$. |

Update ${p}_{t}^{(k+1)}\leftarrow {p}_{t}^{(k)}-{\theta}_{k}^{t}{\xi}_{t}({p}^{(k)})$, for each t. |

k ← k + 1 |

until a predefined stopping criterion is satisfied. |

return Transact
${\left\{{x}_{i}^{(k)}\right\}}_{i\in \mathcal{N}}$ with p^{(}^{k}^{)} as the price profile. |

_{i}and p provides the numerical solution of the dual problem (see Appendix A). This solution is also a solution of the primal problem Equations (13)–(16). Therefore, the social welfare of the network is expected to be maximized. Further details are provided in Appendix A. From the viewpoint of the pricing mechanism, solving the dual problem equates to the determination of a price profile by a utility, so as to maximize the social welfare. Therefore, we refer to this algorithm as an RTP algorithm.

^{t}. One of the simplest methods is that the utility compensates this difference when it is not zero by selling or buying δm

^{t}to or from the outside grid at the price of ${p}_{t}^{G+}$ or ${p}_{t}^{G-}$, respectively. In this work, we assume that the i-Rene gateway employs this heuristic procedure if i-Rene adopts the RTP algorithm.

^{t}cannot be ignored. In this context, a mechanism that would be able to achieve an exact balance between the demand and supply is desirable. The LFS-DA is such a mechanism, as it has the ability to balance demand and supply exactly for each time slot and every iteration step and is guaranteed to solve the problem in the same way as the dual decomposition-based RTP algorithm.

## 4. The Linear Function Submission-Based Double Auction

_{t}is determined exactly by calculating the point at which the aggregate demand and supply functions intersect. After several iterations between an auctioneer and the prosumers, all prosumers transact their electricity using the determined clearing price. In this section, we formulate the LFS-DA mechanism and disclose its theoretical properties.

#### 4.1. Transaction with LFS-DA

_{t}is determined that would suffice to clear the market by searching for the point at which the aggregate demand and supply functions intersect. The constraint for balancing demand and supply Equation (2) becomes:

_{t}divides the agents into two groups. If for the i-th agent $\frac{{\alpha}_{i}^{t}}{{\beta}_{i}^{t}}\le {p}_{t}$, the agent becomes a seller, i.e., ${\beta}_{i}^{t}{p}_{t}-{\alpha}_{i}^{t}={m}_{i}^{t+}>0$ and ${m}_{i}^{t-}=0$. In contrast, if for the i-th agent $\frac{{\alpha}_{i}^{t}}{{\beta}_{i}^{t}}>{p}_{t}$, the agent becomes a buyer, i.e., ${m}_{i}^{t+}=0$ and $-{\beta}_{i}^{t}{p}_{t}+{\alpha}_{i}^{t}={m}_{i}^{t-}>0$. We define the set of suppliers at time t by ${I}_{t}^{+}({p}_{t}):=\{i|\frac{{\alpha}_{i}^{t}}{{\beta}_{i}^{t}}\le {p}_{t},i\in \mathcal{N}\}$ and the set of consumers at time t by ${I}_{t}^{-}({p}_{t}):=\{i|\frac{{\alpha}_{i}^{t}}{{\beta}_{i}^{t}}>{p}_{t},i\in \mathcal{N}\}$. In this case, on the basis of the constraint Equation (22), an appropriate clearing price has to satisfy the following equation.

_{t}, we obtain:

_{t}satisfying Equation (24) exactly balances demand and supply and fulfills the constraint in Equation (22).

**Lemma 1**(The clearing price). The clearing price p

_{t}that satisfies Equation (24) uniquely exists and is calculated exactly.

**Proof**. $\gamma \lfloor {\beta}_{i}^{t}{p}_{t}-{\alpha}_{i}^{t}\rfloor -\lfloor -{\beta}_{i}^{t}{p}_{t}+{\alpha}_{i}^{t}\rfloor $ is a piecewise linear and monotonically increasing function of p

_{t}, and its value at p

_{t}= 0 is negative. Therefore, $\gamma {\displaystyle {\sum}_{i\in \mathcal{N}}\lfloor {\beta}_{i}^{t}{p}_{t}-{\alpha}_{i}^{t}\rfloor -}{\displaystyle {\sum}_{i\in \mathcal{N}}\lfloor -{\beta}_{i}^{t}{p}_{t}-{\alpha}_{i}^{t}\rfloor}$ is also a piecewise linear and monotonically increasing function of p

_{t}, and its value at p

_{t}= 0 is negative. Therefore, the solution of Equation (22) uniquely exists and can be calculated exactly.

k ← 0 |

Initialize the price profile ${p}^{(k)}={\left({p}_{t}^{(k)}\right)}_{t\in \mathcal{J}}$ |

repeat |

// Each agent solves its sub-problem (A7) and obtains its solution. |

Update ${x}_{i}^{*}={\left({x}_{i}^{*}\right)}_{t\in \mathcal{J}}={x}_{i}^{*}({p}^{(k)})$. |

// Each agent submits to $\left({\alpha}_{i}^{*t},{\beta}_{i}^{t}\right)$ the market.. |

Update p^{(}^{k} ^{+ 1)} ← market_clearing${\left({\alpha}_{i}^{*t},{\beta}_{i}^{t}\right)}_{i\in \mathcal{N},t\in \mathcal{J}}$, |

Update $\left({m}_{i}^{(k+1)t+},{m}_{i}^{(k+1)t-}\right)\leftarrow \left({\mu}_{i}^{t+}\left({p}^{(k+1)}\right),{\mu}_{i}^{t-}\left({p}^{(k+1)}\right)\right)$ |

// Reconfiguration by each agent |

Update ${x}_{i}^{(k+1)}\leftarrow \mathbf{pro}{\mathbf{j}}_{{\mathfrak{X}}_{i}}\left({x}_{i}^{*}|{m}_{i}^{t+}={m}_{i}^{(k+1)t+},{m}_{i}^{t+}={m}_{i}^{(k+1)t-}\right)$, for each i |

k ← k + 1 |

until a predefined stopping criterion is satisfied. |

return Transact
${\left\{{x}_{i}^{(k)}\right\}}_{i\in \mathcal{N}}$ with p^{(}^{k}^{)} as the price profile. |

#### 4.2. LFS-DA as a bottom-up RTP Method

_{t}are fixed. Therefore, the optimal solution ${\alpha}_{i}^{*t}\left({p}_{t}^{(k)}\right)$ can be obtained using the following procedure.

**market_clearing**can be obtained by solving Equation (24).

^{(}

^{k}

^{+1)}, the amount of electricity for transacting through the regional electricity market $\left({m}_{i}^{t+},{m}_{i}^{t-}\right)$ is updated as follows.

**market_clearing**can be regarded as an update formula as follows.

**Lemma 2**(Change of price profile in LFS-DA). The price profile of the LFS-DA is updated as follows if${I}_{t}^{\pm}({p}_{t}^{(k+1)})={I}_{t}^{\pm}({p}_{t}^{(k)})$.

**Proof**. For all $t\in \mathcal{J}$,

**Theorem 1.**This update Formula (30) of the LFS-DA is the same as that of the dual decomposition-based RTP algorithm Equation (17), except for the constant factor${\overline{\theta}}_{k}^{t}$

^{k}, if${I}_{t}^{\pm}({p}_{t}^{(k+1)})={I}_{t}^{\pm}({p}_{t}^{(k)})$.

## 5. Experiment

#### 5.1. Experimental Conditions

_{i}= 0.7 for all $i\in \mathcal{N}$. We set the efficiency of transmitting electricity within i-Rene to γ = 0.8, and the parameters of the sub-gradient method to θ

_{k}= 0.1. We set the maximum values to ${m}_{i}^{+,\mathrm{max}}=5$ and ${m}_{i}^{-,\mathrm{max}}=5$ for all $i\in \mathcal{N}$, $t\in \mathcal{J}$. We set the prices of the outside grid to ${p}_{t}^{G-}=20$ and ${p}_{t}^{G+}=0$ for all $t\in \mathcal{J}$.

#### 5.2. Result

## 6. Conclusions

## Acknowledgments

## Appendix

## A. Dual Decomposition and Real-Time Pricing

^{t}

^{1}, …, h

^{t}

^{14}are constraints for the domains of the state vector ${x}_{i}^{t}$. The two constraints for the battery profiles, h

^{t}

^{15}and h

^{t}

^{16}, represent the storage capacity constraints. The constraint h

^{t}

^{17}represents the balance of the electricity flow measured by the i-th agent’s smart meter.

**Problem 2**(Dual problem).

_{t}[18]. A comparison between Equations (7) and (A6) provides us with a clear understanding of this relationship. In this problem, λ

_{t}represents a buyer’s price of electricity traded in the local electricity market at time t. The simultaneous solution of the sub-problems and the master problem results in the solution of the dual problem, and this is achieved in practice by adopting an iterative optimization technique.

**Problem 3**(Sub-problems).

_{i}

^{∗}(λ) to the problem Equation (A7) can be obtained as follows.

^{T}. For RTP formulation purposes, we assume that there is a central utility determining the price profile in the regional electricity market. On the basis of each agent’s optimal strategy ${\left({x}_{i}^{*}\left(\lambda \right)\right)}_{t\in \mathcal{N}}$, the central utility has to solve the following problem.

**Problem 4**(Master problem).

_{k}> 0 is the learning rate of the sub-gradient method. By updating x

_{i}and λ iteratively, the numerical solution of the dual problem can be obtained. The solution of the dual problem is also a solution of the primal problem. Therefore, the social welfare of the network is expected to be maximized.

## Author Contributions

## Conflicts of Interest

## Nomenclature

${l}_{i}^{t+}\in \left[{l}_{i}^{t+,\mathrm{max}},\infty \right)$ | Electric energy consumption profile |

${l}_{i}^{t-}\in \left[0,{l}_{i}^{t-,\mathrm{max}}\right)$ | Electric energy generation profile |

${b}_{i}^{t+}\in \left[0,{b}_{i}^{t+,\mathrm{max}}\right]$ | Battery charge profile |

${b}_{i}^{t-}\in \left[0,{b}_{i}^{t-,\mathrm{max}}\right]$ | Battery discharge profile |

${m}_{i}^{t+}\in \left[0,{m}_{i}^{t+,\mathrm{max}}\right]$ | Profile of electric energy sold to the local electricity market |

${m}_{i}^{t-}\in \left[0,{m}_{i}^{t-,\mathrm{max}}\right]$ | Profile of electric energy bought from the local electricity market |

${g}_{i}^{t}\in \left[0,\infty \right)$ | Profile of electric energy sold to the outside grid |

${g}_{i}^{t-}\in \left[0,{g}_{i}^{t-,\mathrm{max}}\right]$ | Profile of electric energy bought from the outside grid |

${x}_{i}^{t}$ | Profile of state vector |

${s}_{i}^{t}\in \left[0,{s}_{i}^{\mathrm{max}}\right]$ | Profile of the state of charge (SOC) of the battery |

η_{i} ∈ [0, 1] | Storage efficiency |

γ ∈ [0, 1] | Electricity transmission efficiency |

${C}_{i}^{t}$ | Cost function for generating electric energy |

${D}_{i}^{t}$ | Utility function for consuming electric energy |

${\varphi}_{i}^{t}$ | Individual utility function |

${W}_{i}^{t}$ | Individual welfare function |

p_{t} | Price profile |

${p}_{t}^{G+}$ | Price of electricity sold to the outside grid |

${p}_{t}^{G-}$ | Price of electricity bought from the outside grid |

${\alpha}_{i}^{t}$ | Constant term of parameters of the bidding function |

${\beta}_{i}^{t}$ | Primary coefficient term of parameters of the bidding function |

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**Figure 2.**(

**Left**) Real-time pricing in a conventional power grid; and (

**right**) double auction pricing mechanism in a regional prosumers’ electricity network.

**Figure 5.**Social welfare after iterations with ${\left({\displaystyle {\sum}_{i\in \mathcal{N}}{\beta}_{i}^{t}}\right)}^{-1}={\theta}_{k}=0.1$.

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## Share and Cite

**MDPI and ACS Style**

Taniguchi, T.; Kawasaki, K.; Fukui, Y.; Takata, T.; Yano, S.
Automated Linear Function Submission-Based Double Auction as Bottom-up Real-Time Pricing in a Regional Prosumers’ Electricity Network. *Energies* **2015**, *8*, 7381-7406.
https://doi.org/10.3390/en8077381

**AMA Style**

Taniguchi T, Kawasaki K, Fukui Y, Takata T, Yano S.
Automated Linear Function Submission-Based Double Auction as Bottom-up Real-Time Pricing in a Regional Prosumers’ Electricity Network. *Energies*. 2015; 8(7):7381-7406.
https://doi.org/10.3390/en8077381

**Chicago/Turabian Style**

Taniguchi, Tadahiro, Koki Kawasaki, Yoshiro Fukui, Tomohiro Takata, and Shiro Yano.
2015. "Automated Linear Function Submission-Based Double Auction as Bottom-up Real-Time Pricing in a Regional Prosumers’ Electricity Network" *Energies* 8, no. 7: 7381-7406.
https://doi.org/10.3390/en8077381