An Integrated Two‐Stage Hybrid P2P‐DR Transactive Energy Trading Platform Using Iterative Distributed‐Mixed Integer Linear Optimisation

Abstract

ABSTRACT

Transactive energy frameworks, such as demand response (DR) and peer‐to‐peer (P2P) trading, can enhance the welfare of electricity market participants by fully utilising distributed energy resources. This research proposes an iterative two‐stage DR‐P2P framework using a combined alternating direction method of multipliers (ADMM) and mixed‐integer linear programming (MILP) approach to capture the inter‐stage dependence of the individual DR and P2P frameworks. MILP solves the DR schedule of the market participants based on the input of the P2P platform, whereas the trading behaviour of the customers is optimised using the ADMM approach during the P2P stage. An iterative two‐stage approach is designed to find the combined optimal solution for both P2P and DR stages. In addition to the DR and P2P constraints, the power transfer distribution factor‐based method is suggested to formulate the trading losses and network utilisation fee models for the combined framework trading over the physical distribution network. Additionally, the voltage variations in the network are determined using the voltage sensitivity coefficients. The combined P2P‐DR platform is tested for IEEE‐13 bus distribution system for different test scenarios. It is shown that the suggested P2P‐DR network increases the savings of the market participants by 7.8% compared to the individual P2P and DR stages. Additionally, using the combined P2P‐DR network, the welfare of the market participants is increased by approximately 14.32%.

Introduction

With the emergence of smart grids, transactive energy trading platforms are gaining popularity to enhance the social-economic welfare of the market customers [1, 2]. Transactive energy structures devise coordinated and intelligent control techniques to improve the energy utilisation in a market framework. This coordinated energy sharing between entities utilising the power grid can overcome the drawbacks of vertically aligned and deregulated centrally operated systems. In traditional electric energy markets, the uni-direction flow of energy from generation to distribution system results in less flexibility for market participants [3]. With a transactive market, the customers can opt for different generation resources based on their cost/comfort preference. This not only increases flexibility, but also reduces the reliance on vertical grids and central resources transferred over the electric transmission network. Among such intelligent transactive platforms, peer-to-peer (P2P) [4] and demand response (DR) [5] are commonly used to increase the market savings for the participants.

P2P trading refers to a local market network in which residential customers having distributed energy resources (DERs) such as photovoltaic (PV) systems can sell their surplus to neighbouring consumers [4]. Such market participants having rooftop PV energy resources or other generating (or storage) resources are termed as prosumers. By engaging in P2P trading, both prosumers and neighbouring consumers can be offered a competitive price compared to the central market value, which increases their monetary advantages. The advantages of P2P networks are flexible rates, reduced risk, better competition, and improved social welfare for the trading participants [6].

DR is a cost-minimisation framework that finds an optimal schedule of the participating customers by shifting or shedding device load during intervals from high prices (or system peaks) to lower prices [7]. By optimising the device schedule of the customers in response to real-time prices (RTP), the total grid cost and the peak demand on the system can be reduced. The advantages of the DR program are: (i) increase in the market welfare of the participants by providing the monetary advantages; and (ii) enhanced grid reliability by reducing the peak demand on the system [8]. The U.S. Federal Energy Regulatory Commission (FERC) issued order numbers FERC 745 and FERC 2222 [9] to regulate such transactive networks by allowing the participation of DERs in the bulk electric energy market through grid operators.

The P2P and DR platforms can be combined into a P2P-DR framework to further increase the market savings, reduce the grid cost, and enhance grid stability. However, the integrated DR-P2P platform imposes the challenge of inter-stage dependence and network sensitivity. DR scheduling is sensitive to the market values of the P2P network (such as P2P price and shared power) [10]; similarly, the P2P network utilises the load schedule of the market participants (DR schedule) to determine the optimal set of market quantities. The perturbation in one of the networks can change the output parameters of the counterpart stage. Therefore, it is imperative to devise an integrated network for combining both P2P and DR platforms to include the inter-stage sensitivity and network parameters [11]. This research proposes a combined P2P-DR transactive energy market framework by using an iterative two-stage process to optimally compute the convergence point for both P2P and DR platforms while considering stage-dependence and system constraints.

Related Work

P2P and DR networks have been explored extensively in literature using various optimisation techniques and network constraints. The formulation of the P2P network depends on the type of bidding/matching mechanism used and the physical parameters of the network. A continuous double auction (CDA) market has been presented in ref. [12] to model the P2P platform while comparing the zero-intelligent (ZI) and eyes on best price (EOB) trading strategies. A privacy-based CDA market for urban microgrids is presented in ref. [13] to enhance the social-welfare of P2P participants in a decentralised configuration. K-continuous market for P2P platforms has been suggested in ref. [14] to dynamically alter the clearing mechanism based on market conditions. A dynamic equilibrium matching principle is suggested in ref. [15] to find the optimal trade-off between market price and traded quantity. A game-theory (GT) approach has been developed in ref. [16] to model a block-chain based P2P market while using the auction mechanism. A decentralised GT-based P2P market is formulated in ref. [17] to optimise the P2P welfare of microgrid participants. GT approach while considering the priority of the market customers has been suggested in ref. [18] using the cooperative mechanism. An alternating direction method of multipliers (ADMM) based approach has been presented in refs. [19–21] to find the optimal welfare of the P2P market in a distributed manner.

Briefly, the P2P markets have been formulated primarily in literature using auction mechanisms [12–15], preference-based GT method [16–18], and distributed optimisation [19–21]. The second part of the literature on the P2P platform deals with the physical constraints of the system. Network-constraints P2P market has been suggested in refs. [22, 23] by developing the trading losses and utilisation fee model for the P2P transactions. Distribution locational marginal pricing (DLMPs) based P2P trading platforms have been formulated in refs. [24, 25] to model the market price based on the system configuration and congestion. Network constraints for P2P-based low-voltage (LV) networks have been suggested in refs. [26, 27] to model the constrained trading platforms. Authors in refs. [28, 29] have suggested the implementation of the distributed optimisation-based framework for the P2P markets using the ADMM. However, the integration of the DR platform and the hybrid transactive networks are not incorporated in the presented framework. In this research, the authors utilise the ADMM to solve the P2P stage in a distributed manner while using the power-transfer distribution factors to model the network parameters of the distribution system.

For DR networks, optimisation techniques such as heuristic methods have been presented in refs. [30–32] to find the near-optimal solution of the appliance schedule while reducing the implementation complexity of the DR platform. Contrarily, linear-programming for the DR networks has been suggested in refs. [33–35] while considering DERs and energy storage systems. Similarly, different constraints such as the comfort factor [36–38] and peak-to-average ratio (PAR) for the energy consumption [39, 40] have been suggested in literature to develop the constrained DR-markets. For solving the DR stage, the authors suggest an extended DR-platform utilising mixed-integer linear programming (MILP) while considering the comfort factor constraint of the market participants.

Acceptance of the new power system technologies, such as the renewable energy-based power grids depend on three main sectors including socio-political, community and market welfare [41]. In ref. [42], the authors justify that in a household environment, the primary drivers for P2P energy trading include financial benefits and behaviour control. The presented P2P-DR platform is formulated considering the mentioned factors to enhance the social-economic welfare of the market participants while incorporating consumer comfort preferences into their trading strategies.

Research Gap and Contributions

The combined DR-P2P platforms have not been extensively studied in the literature. Authors in refs. [43–45] have suggested DR platforms with P2P trading, however, the inter-stage dependence of the DR and P2P networks have not been considered. The system parameters of each stage are sensitive to the changes in the counterpart network. This inter-dependence of the networks requires a sophisticated two-stage optimisation process to optimally compute both DR and P2P stages. However, this aspect of the integrated P2P-DR networks has not been extensively explored in the literature to the best knowledge of the authors. Based on the mentioned research gap, the following are the contributions of this work:

• Design of a novel iterative two-stage optimisation process for the P2P-DR platform using combined distributed-mixed integer optimisation to model the stage sensitivity of the two market frameworks;

• Incorporation of trading losses and network utilisation fee model using power transfer distribution factors and voltage sensitivity coefficients;

• Validation of the proposed P2P-DR framework for IEEE-13 and IEEE-33 bus distribution networks while considering the physical constraints of the system and uncertainty in the dataset.

The rest of the paper is organised as follows: Section 2 presents the system model for the integrated P2P-DR framework. In Section 2.1, the individual DR and P2P platform formulations are derived. The combined iterative two-stage P2P-DR network is developed in Section 3. Section 4 presents the network constraints for the suggested P2P-DR network. The simulation parameters and results are discussed in Section 5 and Section 6 concludes the findings of this research.

P2P-DR System Model

The suggested framework consists of two major stages interconnected with each other. First stage represents the DR network where the participants adjust their load schedule based on the market price and P2P power. Contrarily, P2P stage optimises the power set points and price based on the rescheduled demand pattern of the DR participants. MILP is used to optimise the DR stage, whereas, distributed approach using ADMM computes the optimal set points for P2P network. An iterative MILP-ADMM-based algorithm is suggested to find the combined optimal for both P2P and DR stages while considering the inter-stage dependence of the two networks. The presented network consists of $${N}_{\mathrm{p}}$$ market participants. Each participant has $${D}_{n}$$ number of devices which they can reschedule based on the price signals and individual preference. During DR optimisation, $${N}_{\mathrm{p}}$$ participants reschedule their load based on the RTP and P2P price. The second stage formulates the P2P trading platform which has $$P$$ prosumers having PV generation and $$C$$ customers such that $$P\cap C=\varnothing $$ and $$P\cup C={N}_{\mathrm{p}}$$. The total market has been divided into $${T}_{n}$$ time intervals of equal duration $${\Delta }T$$. P2P trading takes place over the solar hours $$\left({S}_{\mathrm{s}},{S}_{\mathrm{e}}\right)$$ where the prosumers $$P$$ sell their excess PV power to the customers $$C$$ using the distributed optimisation. The entire process using an iterative ADMM and MILP framework is shown in Figure 1. The subsequent sections provide the brief mathematical formulation for the individual DR and P2P platforms. In the current market framework, the exchange of the data between the two stages and overlooking the market operations can be done by the DSO/aggregator. This can be formulated as the profit maximisation problem for DSO/aggregator for providing the market services integrated with the P2P-DR network. However, in the current formulation the validation of the hybrid P2P-DR transactive market is presented without considering the for-profit optimisation for DSO. The presented framework utilises the DER-assets of the prosumers to be utilised in the P2P-market and formulate the hybrid P2P-DR community while scheduling the load demand of the consumers during the DR stage.

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Formulation of DR and P2P Platforms

For device scheduling, MILP formulation is suggested, whereas the distribution optimisation (ADMM) is used to find the optimal solution for the P2P stage. The details of each are given as follows:

DR Formulation

DR stage optimises the device scheduling of each customer by shifting the load to the intervals of lower RTP. Mathematically, DR optimisation problem for a particular market participant $$i\in {N}_{\mathrm{p}}$$ can be written as follows: 1 $$\min \quad \sum\limits _{n=1}^{{D}_{n}}\sum\limits _{t=1}^{{T}_{n}}{x}_{n}^{t}{P}_{n}{R}^{t},$$ 2 $$\mathrm{s}.\mathrm{t}.\quad \sum\limits _{t=1}^{{T}_{n}}{x}_{n}^{t}={d}_{n}\hspace*{.5em}\hspace*{.5em}\forall n\in \left\{1,2,3\text{\ldots },{D}_{n}\right\},$$ 3 $${x}_{n}^{t}\le 0\hspace*{.5em}\hspace*{.5em}\forall n\in \left\{1,2,..,{D}_{n}\right\},\forall \hspace*{.5em}t< {t}_{\text{start}},\hspace*{.5em}t > {t}_{\text{end}},$$ 4 $${x}_{n}^{t}\le 1-{s}_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}n\in {N}_{\mathrm{nint}},t,$$ 5 $${x}_{n}^{t-1}-{x}_{n}^{t}\le {s}_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}n\in {N}_{\mathrm{nint}},\forall \hspace*{.5em}t\in \left\{2,3,..,{T}_{n}\right\},$$ 6 $${s}_{n}^{t-1}\le {s}_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}n\in {N}_{\mathrm{nint}},\forall \hspace*{.5em}t\in \left\{2,3,\text{\ldots },{T}_{n}\right\},$$ where $${x}_{n}^{t}\in \left\{0,1\right\}$$ corresponds to binary decision for the $${n}^{\mathrm{t}\mathrm{h}}$$ device and $$t$$ time slot. The cost function in Equation (1) minimises the total cost of the $${D}_{n}$$ number of devices based on the RTP tariff $${R}_{\mathrm{t}}$$ for $${T}_{n}$$ time intervals. $${D}_{n}={N}_{\text{int}}+{N}_{\text{nint}}$$, where, $${N}_{\text{int}}$$ and $${N}_{\text{nint}}$$ represent the number of interruptible and uninterruptible devices. Equation (2) ensures that a particular device $$n$$ operates for its total duration $${d}_{n}$$. Equation (3) ensures that the device $$n$$ will be scheduled in its availability window $$\left({t}_{\text{start}},{t}_{\text{end}}\right)$$. Equations (4–6) ensures the uninterruptible operation for $${N}_{\text{nint}}$$ by introducing the auxiliary variable $${s}_{n}^{t}\in \left\{0,1\right\}$$ [46].

P2P Trading Platform Formulation

P2P stage determines the amount of power $${p}_{i,j}$$ traded by the $${i}^{\mathrm{t}\mathrm{h}}$$ prosumer with its neighbouring customer $$j$$ while determining the P2P price $${\pi }_{\mathrm{p}2\mathrm{p}}^{t}$$ shared with the customers. The authors have adopted the F-ADMM formulation presented in ref. [19] for determining the required quantities. Compared to the noncooperative game solutions (such as Stackelrberg Game) which follows leader-follower configuration, ADMM optimises the cost function for each market agent based on the individual cost coefficients and power constraints. This section provides the formulation for a single time interval $$t$$. The same methodology has been extended for all the solar intervals. The centralised optimisation problem for P2P trading can be defined as the cost minimisation for the market participants given as follows: 7 $$\underset{{\bar{p}}_{i}}{\min }\quad \sum\limits _{i\in {N}_{\mathrm{p}}}\left({\alpha }_{i}{{\bar{p}}_{i}}^{2}+{\beta }_{i}{\bar{p}}_{i}+{\gamma }_{i}\right),$$ 8 $$\mathrm{s}.\mathrm{t}.\quad {{\bar{p}}_{i}}^{\text{min}}\le {\bar{p}}_{i}\le {{\bar{p}}_{i}}^{\text{max}}\hspace*{.5em}\forall i\in {N}_{\mathrm{p}},$$ 9 $${p}_{i,j}+{p}_{j,i}=0\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in {N}_{\mathrm{p}},\forall \hspace*{.5em}j\in {N}_{\mathrm{p}}{\backslash}i,$$ 10 $${p}_{i,j}\ge 0\hspace*{.5em}\hspace*{.5em}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in P,\forall \hspace*{.5em}j\in {N}_{\mathrm{p}}{\backslash}i,$$ 11 $${p}_{i,j}\le 0\hspace*{.5em}\hspace*{.5em}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in C,j\in {N}_{\mathrm{p}}{\backslash}i,$$ where the cost function minimises the power set points $${\bar{p}}_{i}={\sum }_{j\in {N}_{\mathrm{p}}{\backslash}i}{p}_{i,j}$$ (the total amount of traded power) of the market participants. $${N}_{\mathrm{p}}{\backslash}i$$ excludes the set $$i$$ from $${N}_{\mathrm{p}}$$. Constraint in Equation (8) limits the traded power within the minimum $${{\bar{p}}_{i}}^{\text{min}}$$ and maximum $${{\bar{p}}_{i}}^{\text{max}}$$ power limits. Constraint Equations (9–11) ensures that the traded power is positive if the participant $$i$$ is selling the power to participant $$j$$ and negative if $$i$$ buys from $$j$$. Therefore, $${p}_{i,j}$$ and $${p}_{j,i}$$ are equal and opposite in polarity as given in the constraint Equation (9). The F-ADMM formulation can be developed by reformulating the constraint in Equation (9) by introducing the auxiliary variable $${\Delta }$$ as follows: 12 $${{\Delta }}_{i,j}={p}_{i,j}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in {N}_{\mathrm{p}},\forall \hspace*{.5em}j\in {N}_{\mathrm{p}}{\backslash}i,$$ 13 $${{\Delta }}_{i,j}+{{\Delta }}_{j,i}=0\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in {N}_{\mathrm{p}},\forall \hspace*{.5em}j\in {N}_{\mathrm{p}}{\backslash}i,$$ 14 $${{\Delta }}_{j,i}={p}_{j,i}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in {N}_{\mathrm{p}},\forall \hspace*{.5em}j\in {N}_{\mathrm{p}}{\backslash}i.$$

In the above set of equations, $${\Delta }$$ represents the amount of P2P power traded by customers/prosumers as an auxiliary variable. Equation (13) shows that the P2P power traded by the prosumers and customers are equal and opposite in convention. By defining the dual variable $${\pi }_{i,j}$$ associated with Equation (12), the Lagrangian for the centralised problem defined previously can be written as follows: 15 $\begin{align*}\hfill L\left({\bar{p}}_{i},{\pi }_{i,j},{{\Delta }}_{i,j}\right)& =\sum\limits _{i\in {N}_{\mathrm{p}}}[{c}_{i}\left({\bar{p}}_{i}\right)+\sum\limits _{j\in {N}_{\mathrm{p}}{\backslash}i}\left\{{\pi }_{i,j}\left({{\Delta }}_{i,j}-{p}_{i,j}\right)\,\right.\hfill \\ \hfill & +\left.\frac{{k}_{\mathrm{p}}}{2}{\left({{\Delta }}_{i,j}-{p}_{i,j}\right)}^{2}\right\}],\hfill \end{align*}$ where $${c}_{i}\left({\bar{p}}_{i}\right)={\alpha }_{i}{{\bar{p}}_{i}}^{2}+{\beta }_{i}{\bar{p}}_{i}+{\gamma }_{i}$$ is the cost function for the $${i}^{\mathrm{t}\mathrm{h}}$$ participant. Using the F-ADMM, the first problem $$\mathrm{S}{\mathrm{P}}_{1}$$ solves the local problem to find $${p}_{i,j}$$ based on the current values of $${{\Delta }}_{i,j}$$ and $${\pi }_{i,j}$$ given as follows: 16 $$\underset{{p}_{i,j}}{\min }\quad {c}_{i}\left({\bar{p}}_{i}\right)+\sum\limits _{j\in {N}_{\mathrm{p}}{\backslash}i}\left\{{\pi }_{i,j}^{t}\left({{\Delta }}_{i,j}^{t}-{p}_{i,j}\right)+\frac{{k}_{\mathrm{p}}}{2}{\left({{\Delta }}_{i,j}^{t}-{p}_{i,j}\right)}^{2}\right\},$$ 17 $$\mathrm{s}.\mathrm{t}.\quad {{\bar{p}}_{i}}^{\text{min}}\le {\bar{p}}_{i}\le {{\bar{p}}_{i}}^{\text{max}}\hspace*{.5em}\forall i\in {N}_{\mathrm{p}},$$ 18 $${p}_{i,j}\ge 0\hspace*{.5em}\hspace*{.5em}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in P,\forall \hspace*{.5em}j\in {N}_{\mathrm{p}}{\backslash}i,$$ 19 $${p}_{i,j}\le 0\hspace*{.5em}\hspace*{.5em}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in C,j\in {N}_{\mathrm{p}}{\backslash}i,$$ where $$t$$ is the current iteration index. Note in this formulation, the iteration index $$t$$ is different from the market interval $$t$$ used in the subsequent sections. The second problem $$\mathrm{S}{\mathrm{P}}_{2}$$ involves optimising for the auxiliary variables $${{\Delta }}_{i,j}$$ and $${{\Delta }}_{j,i}$$ as follows: 20 $\begin{align*}\hfill \underset{{{\Delta }}_{i,j},{{\Delta }}_{j,i}}{\min }\quad & \sum\limits _{j\in {N}_{\mathrm{p}}{\backslash}i}\left\{{\pi }_{i,j}\left({{\Delta }}_{i,j}-{p}_{i,j}^{(t+1)}\right)+\frac{{k}_{\mathrm{p}}}{2}{\left({{\Delta }}_{i,j}-{p}_{i,j}^{(t+1)}\right)}^{2}\right\}\hfill \\ \hfill & \,+\left\{{\pi }_{j,i}\left({{\Delta }}_{j,i}-{p}_{j,i}^{(t+1)}\right)+\frac{{k}_{\mathrm{p}}}{2}{\left({{\Delta }}_{j,i}-{p}_{i,j}^{(t+1)}\right)}^{2}\right\},\hfill \end{align*}$ s.t. Equation (13).

Based on the solution of $$\mathrm{S}{\mathrm{P}}_{1}$$ and $$\mathrm{S}{\mathrm{P}}_{2}$$, the dual variable $${\pi }_{i,j}$$ can be defined as follows: 21 $${\pi }_{i,j}^{(t+1)}={\pi }_{i,j}^{t}+{k}_{\mathrm{p}}\left({{\Delta }}_{i,j}^{(t+1)}-{\pi }_{i,j}^{(t+1)}\right).$$

Using the KKT conditions, the closed form solution for the problem $$\mathrm{S}{\mathrm{P}}_{1}$$ can be derived using the analysis provided in ref. [19]. 22 $${p}_{i,j}^{(t+1)}=\frac{{\pi }_{i,j}^{t}-{{\Theta }}_{i}^{t}+{\zeta }_{i}^{t}-{\beta }_{i}+{k}_{\mathrm{p}}{{\Delta }}_{i,j}^{t}}{2{\alpha }_{i}+{k}_{\mathrm{p}}},$$ where $${{\Theta }}_{i}$$ and $${\zeta }_{i}$$ represent the local Lagrangian variables associated with the constraint in Equation (17). Based on the constraints in Equations (18) and (19), we can define $${p}_{i,j}$$ as follows: 23 $${p}_{i,j}^{(t+1)}=\left\{\begin{array}{@{}ll@{}}\mathrm{m}\mathrm{a}\mathrm{x}\left(0,{p}_{i,j}^{(t+1)}\right)\quad \hfill & \forall \hspace*{.5em}i\in P,\forall \hspace*{.5em}j\in {N}_{\mathrm{p}}{\backslash}i\hfill \\ \mathrm{m}\mathrm{i}\mathrm{n}\left(0,{p}_{i,j}^{(t+1)}\right)\quad \hfill & \forall \hspace*{.5em}i\in C,j\in {N}_{\mathrm{p}}{\backslash}i\hfill \end{array}\right.,$$ the next step involves updating the local Lagrangian variables $${{\Theta }}_{i}$$ and $${\zeta }_{i}$$ as follows: 24 $${{\Theta }}_{i}^{(t+1)}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{0,\hspace*{.5em}{{\Theta }}_{i}^{t}+{k}_{i}\left({{\bar{p}}_{i}}^{(t+1)}-{{\bar{p}}_{i}}^{\text{max}}\right)\right\},$$ 25 $${\zeta }_{i}^{(t+1)}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{0,\hspace*{.5em}{\zeta }_{i}^{t}+{k}_{i}\left({{\bar{p}}_{i}}^{\text{min}}-{{\bar{p}}_{i}}^{(t+1)}\right)\right\},$$ where $${k}_{i}$$ is a positive tuning parameter usually in the range of [0, 1]. Similarly, for $$\mathrm{S}{\mathrm{P}}_{2}$$, the closed form solution can be written using the KKT conditions as follows: 26 $${{\Delta }}_{i,j}^{(t+1)}=\frac{{k}_{\mathrm{p}}\left({p}_{i,j}^{(t+1)}-{p}_{j,i}^{(t+1)}\right)-\left({\pi }_{i,j}^{t}-{\pi }_{j,i}^{t}\right)}{2{k}_{\mathrm{p}}},$$

Based on the solution of $$\mathrm{S}{\mathrm{P}}_{1}$$ and $$\mathrm{S}{\mathrm{P}}_{2}$$, dual variable $${\pi }_{i,j}$$ can then be updated using Equation (21). For further mathematical details and analysis, the readers are encouraged to go through the F-ADMM discussion provided in ref. [19].

Integrated P2P-DR Platform

This section provides the integrated P2P-DR platform by combining both P2P and DR stages. DR stage takes the input of P2P price $${\pi }_{\mathrm{p}2\mathrm{p}}$$ for solar hours based on the dual variable $$\pi $$ and the power shared with each customer $${\bar{p}}_{i}$$ from the ADMM stage. Therefore, the updated DR scheduling problem is updated using the received input parameters from P2P stage to consider P2P price in the scheduling problem.

Reformulating the DR Platform

The updated DR problem is reformulated based on the P2P price $${\pi }_{\mathrm{p}2\mathrm{p}}$$ and power shared with the customers $${\bar{p}}_{i}$$. Let us introduce the additional binary variables $$\gamma \in \left\{0,1\right\}$$ and $${\gamma }^{\prime }\in \left\{0,1\right\}$$ which dictates whether the customer will opt for the grid cost or P2P value. The updated optimisation problem for a particular customer $$i\in C$$ can then be written as follows: 27 $$\min \quad \sum\limits _{t=1}^{{T}_{n}}\sum\limits _{n=1}^{{D}_{n}}\left\{{\gamma }_{n}^{t}{x}_{n}^{t}{P}_{n}{R}^{t}+{\gamma }_{\prime n}^{t}{x}_{n}^{t}{P}_{n}{\pi }_{\mathrm{p}2\mathrm{p}}^{t}\right\},$$ 28 $$\mathrm{s}.\mathrm{t}.\quad {\gamma }_{n}^{t}+{\gamma }_{\prime n}^{t}=1\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}\hspace*{.5em}n,t,$$ 29 $${\gamma }_{\prime n}^{t}\le 0\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}\hspace*{.5em}t< {S}_{\mathrm{s}},t > {S}_{\mathrm{e}},$$ 30 $${\gamma }_{\prime n}^{t}\le 0\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}{{\bar{p}}_{i}}^{t}\le 0,i\in C,$$ 31 $$\sum\limits _{n\in {D}_{n}}{\gamma }_{\prime n}^{t}{x}_{n}^{t}{P}_{n}\le {{\bar{p}}_{i}}^{t}\hspace*{.5em}\hspace*{.5em}\forall \hspace*{.5em}i\in C,t,$$ 32 $$(2)-(6).$$

Constraint in Equation (28) ensures that for a particular time interval $$t$$, the device $$n$$ will be scheduled at either retail or P2P price. Constraint Equation (29) shows that the device $$n$$ will not be scheduled on P2P value outside the solar hours $$\left({S}_{\mathrm{s}},{S}_{\mathrm{e}}\right)$$. Constraint Equation (30) ensures that the device $$n$$ for a particular time interval $$t$$ will only be scheduled at P2P price if the $${i}^{\mathrm{t}\mathrm{h}}$$ customer has some available P2P power. Constraint Equation (31) ensures that the customer $$i$$ can not schedule more power at P2P value for a particular time interval $$t$$ than the amount available from the P2P market $${{\bar{p}}_{i}}^{t}$$. Each customer solves the DR problem based on the input price parameters which dictates whether to schedule the load at the grid price or P2P value. $${R}^{t}$$ represents the real time tariff whereas the $${\pi }_{\mathrm{p}2\mathrm{p}}^{t}$$ shows the P2P price determined from the P2P stage.

In the above optimisation problem, we have the product of two binary decision variables $${\gamma }_{n}^{t}{x}_{n}^{t}$$ and $${\gamma }_{\prime n}^{t}{x}_{n}^{t}$$. To linearise this product, let us introduce additional auxiliary binary variables $$w{g}_{n}^{t}\in \left\{0,1\right\}$$ and $$w{p}_{n}^{t}\in \left\{0,1\right\}$$ such that the above optimisation problem can be written as follows: 33 $$\min \quad \sum\limits _{t=1}^{{T}_{n}}\sum\limits _{n=1}^{{D}_{n}}\left\{w{g}_{n}^{t}{P}_{n}{R}^{t}+w{p}_{n}^{t}{P}_{n}{\pi }_{\mathrm{p}2\mathrm{p}}^{t}\right\},$$ 34 $$\mathrm{s}.\mathrm{t}.\quad w{g}_{n}^{t}\le {\gamma }_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 35 $$w{g}_{n}^{t}\le {x}_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 36 $$w{g}_{n}^{t}\ge {\gamma }_{n}^{t}+{x}_{n}^{t}-1\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 37 $$w{p}_{n}^{t}\le {\gamma }_{\prime n}^{t}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 38 $$w{p}_{n}^{t}\le {x}_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 39 $$w{p}_{n}^{t}\ge {\gamma }_{\prime n}^{t}+{x}_{n}^{t}-1\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 40 $$\sum\limits _{n\in {D}_{n}}w{p}_{n}^{t}{P}_{n}\le {{\bar{p}}_{i}}^{t}\hspace*{.5em}\hspace*{.5em}\forall n,t,i\in C,$$ 41 $$(28)-(32),$$ where $$w{g}_{n}^{t}={\gamma }_{n}^{t}{x}_{n}^{t}$$ and $$w{p}_{n}^{t}={\gamma }_{\prime n}^{t}{x}_{n}^{t}$$. The additional constraints added to ensure that the auxiliary binary variable $$w{g}_{n}^{t}\left(w{p}_{n}^{t}\right)$$ is 0 if either $${\gamma }_{n}^{t}\left({\gamma }_{\prime n}^{t}\right)$$ or $${x}_{n}^{t}$$ is zero given from the constraints Equations (34) and (35). On the other hand, auxiliary variable $$w{g}_{n}^{t}\left(w{p}_{n}^{t}\right)$$ is 1 only if both $${\gamma }_{n}^{t}\left({\gamma }_{\prime n}^{t}\right)$$ and $${x}_{n}^{t}$$ are 1 given from the constraint Equation (36).

Comfort Level Based DR Scheduling for Customers

The next step in formulating the DR problem is to include the comfort factor constraint for the customers/prosumers. We define the comfort factor of the participants using the following criterion: 42 $${c}_{n}^{\mathrm{resc}}\le {\eta }_{i}{c}_{n}^{\mathrm{org}},$$ where $${\eta }_{i}$$ given in the range [0, 1] defines the comfort level for the $${i}^{\mathrm{t}\mathrm{h}}$$ customer. The added constraint ensures that each customer has its own preference while changing its original load schedule based on the monetary advantages received from participating in the P2P-DR market. $${\eta }_{i}$$ is sensitive to each customer and models the behaviour aspect of the market participants. A particular customer $$i\in C$$ will reschedule its device only if the market savings satisfy the constraint in Equation (42). The rescheduled cost $${c}_{n}^{\mathrm{resc}}$$ and the original cost $${c}_{n}^{\mathrm{org}}$$ for the $${n}^{\mathrm{t}\mathrm{h}}$$ device for a particular customer $$i\in C$$ can be determined as follows: 43 $${c}_{n}^{\mathrm{resc}}=\sum\limits _{t=1}^{{T}_{n}}\left\{w{g}_{n}^{t}{P}_{n}{R}^{t}+w{p}_{n}^{t}{P}_{n}{\pi }_{\mathrm{p}2\mathrm{p}}^{t}\right\},$$ 44 $${c}_{n}^{\mathrm{org}}=\sum\limits _{t=1}^{{T}_{n}}{x}_{n,\mathrm{o}\mathrm{r}\mathrm{g}}^{t}{P}_{n}{R}^{t},$$ where $${x}_{n,\mathrm{o}\mathrm{r}\mathrm{g}}^{t}$$ corresponds to the binary variable for the original device schedule. Note that $${x}_{n,\mathrm{o}\mathrm{r}\mathrm{g}}^{t}$$ is not a decision variable, rather it is pre-determined based on the original load profile of the participants. The $${i}^{\mathrm{t}\mathrm{h}}$$ customer will schedule device $$n$$ only if the inequality in Equation (42) holds. For a particular customer $$i$$, the following constraints are added to the MILP framework using the Big-M method: 45 $$M{\delta }_{n}-\left({\eta }_{i}{c}_{n}^{\mathrm{org}}-{c}_{n}^{\mathrm{resc}}\right)\ge 0,$$ 46 $$M\left(1-{\delta }_{n}\right)-\left({c}_{n}^{\mathrm{resc}}-{\eta }_{i}{c}_{n}^{\mathrm{org}}\right)\ge 0,$$ where $$M$$ is a big number usually given as $$1{0}^{4}$$. The selection of $$M$$ plays an important role in the convergence of algorithm. It should be large enough to avoid restricting the number of the feasible solutions for the problem. Contrarily, a very large value of $$M$$ can result in numerical instability [47]. $${\delta }_{n}\in \left\{0,1\right\}$$ is a binary variable which decides whether to reschedule a particular device $$n$$ or not. The updated objective function for the DR framework can be written as follows: 47 $\begin{align*}\hfill \begin{array}{rl}\hfill \min \quad & \sum\limits _{t=1}^{{T}_{n}}\sum\limits _{n=1}^{{D}_{n}}\left\{{\delta }_{n}w{g}_{n}^{t}{P}_{n}{R}^{t}+{\delta }_{n}w{p}_{n}^{t}{P}_{n}{\pi }_{\mathrm{p}2\mathrm{p}}^{t}\right\}\hfill \\ \hfill & \,+\sum\limits _{t=1}^{{T}_{n}}\sum\limits _{n=1}^{{D}_{n}}\left(1-{\delta }_{n}\right){x}_{n,\mathrm{o}\mathrm{r}\mathrm{g}}^{t}{P}_{n}{R}^{t},\hfill \end{array}\end{align*}$ 48 $$\mathrm{s}.\mathrm{t}.\quad (34)-(41)\hspace*{.5em}\hspace*{.5em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace*{.5em}\hspace*{.5em}(45)-(46),$$ where if $${\delta }_{n}=0$$, the cost of the device $$n$$ will be constant equal to the original cost of $${c}_{n}^{\mathrm{org}}$$. On linearising the above expression problem by introducing the additional binary auxiliary variables, the updated optimisation problem can be written as follows: 49 $\begin{align*}\hfill \begin{array}{rl}\hfill \min \quad & \sum\limits _{t=1}^{{T}_{n}}\sum\limits _{n=1}^{{D}_{n}}\left\{W{g}_{n}^{t}{P}_{n}{R}^{t}+W{p}_{n}^{t}{P}_{n}{\pi }_{\mathrm{p}2\mathrm{p}}^{t}\right\}\hfill \\ \hfill & +\sum\limits _{t=1}^{{T}_{n}}\sum\limits _{n=1}^{{D}_{n}}\left(1-{\delta }_{n}\right){x}_{n,\mathrm{o}\mathrm{r}\mathrm{g}}^{t}{P}_{n}{R}^{t},\hfill \end{array}\end{align*}$ 50 $$\mathrm{s}.\mathrm{t}.\quad W{g}_{n}^{t}\le {\delta }_{n}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 51 $$W{g}_{n}^{t}\le w{g}_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 52 $$W{g}_{n}^{t}\ge {\delta }_{n}+w{g}_{n}^{t}-1\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 53 $$W{p}_{n}^{t}\le {\delta }_{n}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 54 $$W{p}_{n}^{t}\le w{p}_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 55 $$W{p}_{n}^{t}\ge {\delta }_{n}+w{p}_{n}^{t}-1\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 56 $$(48),$$ where $$W{g}_{n}^{t}\in \left\{0,1\right\}$$ and $$W{p}_{n}^{t}\in \left\{0,1\right\}$$ are the additional auxiliary binary variables such that $$W{g}_{n}^{t}={\delta }_{n}w{g}_{n}^{t}$$ and $$W{p}_{n}^{t}={\delta }_{n}w{p}_{n}^{t}$$. This completes the formulation of the DR problem for $$i\in C$$.

Comfort Level Based DR Scheduling for Prosumers

For prosumers $$i\in P$$, we assume that the prosumers will shift their load based on the $${R}^{t}$$ while considering the comfort level constraint. The DR optimisation problem for the prosumers $$i\in P$$ can be written as follows: 57 $\begin{align*}\hfill \min \quad \sum\limits _{t=1}^{{T}_{n}}\sum\limits _{n=1}^{{D}_{n}}Wp{g}_{n}^{t}{P}_{n}{R}^{t}+\sum\limits _{t=1}^{{T}_{n}}\sum\limits _{n=1}^{{D}_{n}}\left(1-{\delta }_{n}\right){x}_{n,\mathrm{o}\mathrm{r}\mathrm{g}}^{t}{P}_{n}{R}^{t},\end{align*}$ 58 $$\mathrm{s}.\mathrm{t}.\quad Wp{g}_{n}^{t}\le {\delta }_{n}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 59 $$Wp{g}_{n}^{t}\le {x}_{n}^{t}\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 60 $$Wp{g}_{n}^{t}\ge {\delta }_{n}+{x}_{n}^{t}-1\hspace*{.5em}\hspace*{.5em}\forall n,t,$$ 61 $$(2)-(6)\hspace*{.5em}\hspace*{.5em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace*{.5em}\hspace*{.5em}(45)-(46),$$ where $$Wp{g}_{n}^{t}\in \left\{0,1\right\}$$ is an auxiliary binary variable such that $$Wp{g}_{n}^{t}={\delta }_{n}{x}_{n}^{t}$$. Note that for the prosumers $$i\in P$$, the rescheduled cost $${c}_{n}^{\mathrm{resc}}$$ is given as $${c}_{n}^{\mathrm{resc}}={\sum }_{n=1}^{{D}_{n}}{\sum }_{t=1}^{{T}_{n}}{x}_{n}^{t}{P}_{n}{R}^{t}\hspace*{.5em}\hspace*{.5em}(\forall i\in P)$$. The above cost equation reschedules the demand of the prosumers based on the RTP price while considering the comfort factor constraint. Note that the prosumers will use the PV power locally first before going into the P2P market for trading the excess PV generation. This completes the reformulation of the DR stage based on the input parameters of the P2P platform. The next part of the section reformulates the P2P stage based on the inputs of the DR platform.

1

ALGORITHM

Iterative P2P-DR platform.

1: Input: DR Platform: $${D}_{n},{T}_{n},{R}^{t},{N}_{\mathrm{int}},{N}_{\mathrm{nint}},{\eta }_{i},M$$ 2: P2P Platform: $${\alpha }_{i},{\beta }_{i},{\gamma }_{i},{k}_{\mathrm{p}},{k}_{i},{{\Theta }}^{0},{\zeta }^{0},{{\Delta }}^{0},{\pi }^{0}$$ 3: Initialisation: $${N}_{\mathrm{p}},C,P,{\pi }_{\mathrm{p}2\mathrm{p}}^{0}={R}^{t}\hspace*{.5em}\hspace*{.5em}\forall t\in \left({S}_{\mathrm{s}},{S}_{\mathrm{e}}\right),\hspace*{.5em}{{\bar{p}}_{i}}^{t}=0\hspace*{.5em}\hspace*{.5em}\forall i\in C,\hspace*{.5em}\hspace*{.5em}\forall t\in \left({S}_{\mathrm{s}},{S}_{\mathrm{e}}\right),\hspace*{.5em}T$$ iterations. 4: Input the original load profile for all market participants $${N}_{\mathrm{p}}=P\cup C$$ . 5: Input the PV curves for the prosumers $$\forall i\in P$$. 6: for $$m=1$$ to $$T$$ do 7:  Perform the DR stage for customers based on the current values of $${\pi }_{\mathrm{p}2\mathrm{p}}^{0}$$ and $${{\bar{p}}_{i}}^{0}$$ and find the $$d{r}_{i}^{m},\hspace*{.5em}\forall i\in C$$ Equations (49–56). 8:  Use PV power locally and perform the DR optimisation for prosumers to find $$e{g}_{i}^{m}\hspace*{.5em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace*{.5em}d{r}_{i}^{m},\hspace*{.5em}\forall i\in P$$ Equations (57–61). 9:  Use the updated parameters $$e{g}_{i}^{m},\hspace*{.5em}\forall i\in P\hspace*{.5em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace*{.5em}d{r}_{i}^{m},\hspace*{.5em}\forall i\in C$$ for the ADMM stage. 10:  Perform the ADMM stage and find the updated values for $${\pi }_{\mathrm{p}2\mathrm{p}}^{m}$$ and $${{\bar{p}}_{i}}^{m}$$ Equations (7–26). 11:  Update the parameters for the first stage $${\pi }_{\mathrm{p}2\mathrm{p}}^{0}={\pi }_{\mathrm{p}2\mathrm{p}}^{m}$$ and $${{\bar{p}}_{i}}^{0}={{\bar{p}}_{i}}^{m}$$. 12: end for 13: return $${\pi }_{\mathrm{p}2\mathrm{p}},{\bar{p}}_{i},e{g}_{i},d{r}_{i}$$ 14: Based on $${\pi }_{\mathrm{p}2\mathrm{p}},{\bar{p}}_{i},e{g}_{i},d{r}_{i}$$ compute the total savings of the market participants and rescheduled load.

Reformulating the P2P Platform and Iterative Process

P2P network takes the rescheduled load profile of the customers along with the excess PV power for the prosumers. The centralised problem presented in Equation (7) can be reformulated by adjusting the power limit constraint of Equation (8) as follows: 62 $${=}\left\{\begin{array}{@{}l@{}}0\le {\bar{p}}_{i}\le {e{\bar{g}}_{i}}^{t}\hspace*{.5em}\hspace*{.5em}\forall i\in P\quad \hfill \\ -d{r}_{i}^{t}\le {\bar{p}}_{i}\le 0\hspace*{.5em}\hspace*{.5em}\forall i\in C\quad \hfill \end{array}\right.,$$ where $${e{\bar{g}}_{i}}^{t}$$ is the excess PV power for $$i\in P$$ for a particular time interval $$t$$. $$d{r}_{i}^{t}$$ is the rescheduled demand of the customers $$\forall i\in C$$ for the time interval $$t$$. The negative sign is due to the convention introduced for the ADMM method in Section 2.1. Note that the binary variables are only optimised in the first stage (DR optimisation) using MILP, whereas the ADMM algorithm minimises the cost function as described in Section 2.1.2 which does not include the binary variables. The overall iterative process for the combined P2P-DR problem has been shown in Algorithm 1. The combined P2P-DR platform has the following major steps.

• Declaring input parameters: The first step is to declare system parameters for the suggested platform.

• Load and PV profile of participants: Determine the load profile for each market participant and PV profile for all prosumers $$\forall i\in P$$.

• Initialisation of DR-P2P platform: For the initial iteration, the participants will perform the DR scheduling based on the market values. For customers $${\pi }_{\mathrm{p}2\mathrm{p}}$$ will be equal to $${R}^{t}$$ and the power shared $${\bar{p}}_{i}$$ by the prosumers $$\forall i\in P$$ will be zero for the initial DR scheduling.

• Iterative process: After performing the initial DR stage, compute the updated quantities from the P2P network. The updated quantities $${\pi }_{\mathrm{p}2\mathrm{p}}$$ and $${\bar{p}}_{i}$$ will be used for the DR network for the next iteration. Similarly, the DR stage based on the updated parameters, compute the $$e{g}_{i}^{m},\hspace*{.5em}\forall i\in P\hspace*{.5em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace*{.5em}d{r}_{i}^{m},\hspace*{.5em}\forall i\in C$$ for the ADMM stage. This process continues till the termination criterion is met. We define the termination criterion as the convergence of the total market savings of the participants.

Network Constraints for P2P-DR Platform

In distribution networks, the transactions for P2P platform take place over the branches which are prone to trading losses. Additionally, the market participants use the services of the third party entity such as market operator due to which they need to pay the transaction fee. In this research, the authors propose the Power Transfer Distribution Factor (PTDF)-based method to compute the trading losses and network utilisation fee for the market participants. The details of each model are given in the subsequent subsections.

P2P Trading Losses

In the physical network, the transfer of P2P energy takes place over the distribution lines, which can result in trading losses for each transaction. Therefore, it is imperative to consider the network losses while formulating the P2P network. This research adopts the formulation of PTDF matrix as suggested in ref. [48] to compute the trading losses for the P2P stage. PTDF can be defined as the linear approximation to the first order sensitivity of the active power flow. It shows the sensitivity of the flow over a particular branch caused by the change in the generation or load in the power system. The following relation is used to compute the PTDF $${{\Lambda }}_{mn}^{L}$$ for a particular branch $$L$$ of the distribution network through which the power is transferred by the prosumer (seller) $$m\in P$$ to the customer (buyer) $$n\in C$$ and is given as follows: 63 $${{\Lambda }}_{mn}^{L}={{\Psi }}_{m}^{L}-{{\Psi }}_{n}^{L},$$ where $${{\Psi }}_{m}^{L}$$ and $${{\Psi }}_{n}^{L}$$ are the injection shift factor (ISF) for the nodes $$m$$ and $$n$$ respectively. ISF matrix $${\Psi }\in {\mathbb{R}}^{{N}_{\mathrm{L}}\times {N}_{\mathrm{B}}}$$, where $${N}_{\mathrm{L}}$$ and $${N}_{\mathrm{B}}$$ represent the number of line segments and nodes in the system, can be determined as follows: 64 $${\Psi }\triangleq BA{C}^{-1}\in {\mathbb{R}}^{{N}_{\mathrm{L}}\times {N}_{\mathrm{B}}},$$ 65 $$B\triangleq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left[{B}_{1},{B}_{2},{B}_{3},\text{\ldots }.,{B}_{\mathrm{L}}\right]\in {\mathbb{R}}^{{N}_{\mathrm{L}}\times {N}_{\mathrm{L}}},$$ 66 $$A\triangleq {\left[{A}_{1},{A}_{2},{A}_{3},\text{\ldots }.,{A}_{\mathrm{L}}\right]}^{T}\in {\mathbb{R}}^{{N}_{\mathrm{L}}\times {N}_{\mathrm{B}}},$$ 67 $${A}_{L}^{T}\triangleq [0\text{\ldots }0\hspace*{.5em}\hspace*{.5em}1(n)\hspace*{.5em}\hspace*{.5em}0\text{\ldots }0\hspace*{.5em}\hspace*{.5em}-1(m)\hspace*{.5em}\hspace*{.5em}0\text{\ldots }0],$$ 68 $$C\triangleq {A}^{T}BA\in {\mathbb{R}}^{{N}_{\mathrm{B}}\times {N}_{\mathrm{B}}},$$ where $$A\in {\mathbb{R}}^{{N}_{\mathrm{L}}\times {N}_{\mathrm{B}}}$$ represents the branch to node incidence matrix, $$B\in {\mathbb{R}}^{{N}_{\mathrm{L}}\times {N}_{\mathrm{L}}}$$ shows the diagonal branch susceptance matrix, and $$C\in {\mathbb{R}}^{{N}_{\mathrm{B}}\times {N}_{\mathrm{B}}}$$ represents the reduced nodal susceptance matrix. Based on the PTDF, the trading losses can be computed as follows [49]: 69 $${P}_{mn}^{\mathrm{loss}}=\sum\limits _{l\in L}\left(\frac{{r}^{l}}{{V}_{b}^{2}}\right){\left({{\Lambda }}_{mn}^{l}{p}_{m,n}\right)}^{2},$$ where $${P}_{mn}^{\mathrm{loss}}$$ represents the trading loss between the prosumer $$m\in P$$ and customer $$n\in C$$, $${r}^{l}$$ and $${V}_{\mathrm{B}}$$ represent the line resistance nominal grid voltage respectively, and $${p}_{m,n}$$ shows the P2P traded energy between the prosumer $$m$$ and customer $$n$$. This completes the formulation of the trading losses for the P2P network. The next part formulates the network utilisation fee model for the market participants.

Network Utilisation Fee

P2P transactions take place over the distribution network lines which are owned by the market operator (such as distribution system operator). Therefore, market participants need to pay certain amount to operator for engaging in the P2P market. Additionally, network fee is required for overlooking the operation and maintenance of the P2P market. This research uses the electrical-distance based approach for formulating the network utilisation fee model using the PTDF [49]. The mathematical relation for computing the network fee for the market participants can be written as follows: 70 $${D}_{mn}=\sum\limits _{l\in L}\vert {{\Lambda }}_{mn}^{l}\vert ,$$ 71 $$\mathrm{T}\mathrm{F}\left({p}_{m,n}\right)=\varrho {D}_{mn}{p}_{m,n},$$ where $${D}_{mn}$$ represents the electrical distance between the prosumer $$m$$ and customer $$n$$. $$\mathrm{T}\mathrm{F}\left({p}_{m,n}\right)$$ represents the transaction fee paid to the market operator and $$\varrho $$ shows the charge rate coefficient as determined by the operator.

Voltage Sensitivity Coefficients

The suggested P2P-DR framework can result in the change in the voltage profile of the network due to the power injections at different nodes. These voltage variations in the network due to the power injections can be determined using the voltage sensitivity coefficients (VSC's). The conventional way of computing these sensitivity coefficients is through the Jacobian matrix given [27] as follows: 72 $$J=\left[\begin{array}{@{}cc@{}}\hfill \frac{\partial p}{\partial \vert v\vert }\hfill & \hfill \frac{\partial p}{\partial {\theta }_{v}}\hfill \\ \hfill \frac{\partial q}{\partial \vert v\vert }\hfill & \hfill \frac{\partial q}{\partial {\theta }_{v}}\hfill \end{array}\right],$$ where $$p,q,v$$, and $${\theta }_{v}$$ represent the real power injection, reactive power injection, voltage magnitude, and voltage angle vectors respectively. The change in the voltage $${\Delta }v$$ due to the power injections $$({\Delta }p,{\Delta }q)$$ can be based on the Jacobian inverse and is given as follows: 73 $${\Delta }{v}_{i}=\left(\frac{\partial {v}_{i}}{\partial {p}_{i}}\right){\Delta }{p}_{i}+\left(\frac{\partial {v}_{i}}{\partial {q}_{i}}\right){\Delta }{q}_{i}.$$

One of the alternative ways of computing the VSC's without determining the full load flow is through the analytical derivation presented in refs. [27, 50]. The analytical derivation presented in refs. [27, 50] solves the following system of equations given as follows: 74 $${\mathbf{1}}_{\left\{i=k\right\}}=\frac{\partial {v}_{i}^{\ast }}{\partial {p}_{k}}\sum\limits _{j\in {N}_{\mathrm{B}}}{y}_{ij}{v}_{j}+{v}_{i}^{\ast }\sum\limits _{j\in {N}_{\mathrm{B}}/0}{y}_{ij}\frac{\partial {v}_{j}}{\partial {p}_{k}},$$ where the above set of equations is linear with respect to $$\frac{\partial {v}_{i}}{\partial {p}_{k}}$$ and $$\frac{\partial {v}_{i}^{\ast }}{\partial {p}_{k}}$$. Based on the solution of the above set of equations, the partial derivative of the voltage magnitudes can be determined as follows: 75 $$\frac{\partial \vert {v}_{i}\vert }{\partial {p}_{k}}=\frac{1}{\vert {v}_{i}\vert }\mathrm{R}\mathrm{e}\left({v}_{i}^{\ast }\frac{\partial {v}_{i}}{\partial {p}_{k}}\right),$$ 76 $${\Delta }\vert {v}_{i}\vert =\frac{{\Delta }{p}_{k}}{\vert {v}_{i}\vert }\mathrm{R}\mathrm{e}\left({v}_{i}^{\ast }\frac{\partial {v}_{i}}{\partial {p}_{k}}\right).$$

Voltage variations can then be determined based on the power changes in the specific buses of the network.

Case Study and Results

The suggested integrated P2P-DR platform is investigated using the modified IEEE 13 bus distribution system suggested in ref. [49] and is shown in Figure 2. The parameters of the distribution system are taken from [49]. The original schedule and the device data of the market participants are obtained using the queueing load model explained in refs. [51, 52]. Load queueing models the start time of each device randomly using the stochastic process to capture the impact of seasonal variations and individual behaviour on the load profile of the market participants. PV profile for the prosumers is determined using the fractional integral model explained in refs. [53, 54]. To focus on the suggested P2P-DR platform, the authors skip the explanation of these modelling components and encourage the readers to go through the details provided in the mentioned references. A total of 10 customers and 5 prosumers are considered for the network and are placed at different nodes of the network as shown in Figure 2. The cost coefficients for each market participant for the ADMM stage are taken from ref. [19]. The values of $${k}_{\mathrm{p}}$$ and $${k}_{i}$$ are taken as 0.5 and 0.25 respectively and adopted from ref. [19].

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The charge rate coefficient $$\varrho $$ is taken as 0.005 and adopted from ref. [23]. The comfort level factor of the market participants $${\eta }_{i}$$ is randomly initialised using the coefficient of variation method described in ref. [32]. The total time intervals $${T}_{n}$$ are taken to be 24 divided in equal duration of 1 h $${\Delta }t=1$$. The value of M for Big-M method is taken to be $$1{0}^{4}$$ to ensure the feasibility of the problem while also considering the computational issues. The market price data are taken from the ComEd website and is adopted from ref. [23]. The following three cases are formulated given as follows:

• Case I: DR scheduling without PV resources.

• Case II: DR scheduling with PV and without P2P trading.

• Case III: DR scheduling with P2P trading.

For each case, the total rescheduled cost and savings of the market participants are compared. Additionally, the P2P-DR network results are tested for the IEEE-13 bus distribution network in OpenDSS for the voltage profile and sensitivity coefficients. The P2P-DR algorithm is implemented on IEEE 33-bus distribution network to validate the scalability of the presented framework. The suggested platform is also tested for monthly data to compute the metrics for the market participants.

Case I: DR Scheduling Without PV Resources

The first case solves the DR platform for all the market participants (prosumers and customers) without considering the participation of the PV resources. The comfort factor level constraint is considered in addition to the conventional DR optimisation problem presented in Section 2.1. Figure 3 shows the results of the DR scheduling for Case I.

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In Figure 3, the participants schedule their demand based on the RTP signals while considering the DR constraints. The peak demand on the system has been reduced by optimising the load schedule of the market participants. Note that in Figure 3, the original load schedule (unscheduled load) is obtained using the device data obtained from the queueing theory as mentioned in the case study. Compared to the original cost $15.583, the participants managed to lower their electricity cost to $15.401 for Case I. The original cost is computed based on the original load schedule where the market participants buy their demand from the utility at RTP without considering the DR and PV resources.

Case II: DR Scheduling With Utilising PV Resources Locally

In second case, the prosumers utilise their PV generation locally and sell any excess PV power at the FIT to the utility without engaging in the P2P market. DR scheduling is performed for all the market participants sensitive to the RTP value. Figure 4a–c shows the results of the Case II. As shown in Figure 4c, the prosumers utilise their PV power locally and sell any remaining PV energy at the FIT to the utility. Note that in Figure 4b, the total scheduled load includes both the power scheduled at grid value and the locally used PV power. However, the scheduled grid power has been reduced while performing the DR optimisation. For Case II, the participants were able to reduce the total electricity cost to the value of $13.665. For Case II, since there is no P2P trading, the prosumers have to sell their excess PV power at the fixed FIT to the utility which can lower the overall savings of the market participants.

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Case III: DR Scheduling With P2P Trading

The last case solves the integrated P2P-DR platform for the suggested IEEE 13 bus distribution network. In this case, the participants perform the DR optimisation while also engaging in P2P market as suggested in Section 3. Figure 5 shows the results of the integrated P2P-DR platform for the suggested network. In this particular case, the customers perform the DR optimisation based on both RTP and P2P price signals. During the solar hours, the customers are offered a competitive P2P value compared to the RTP price. Additionally, the amount of the PV power sold to the utility has been reduced since the prosumers are offered with a P2P price which is competitive to the FIT value as shown in Figure 5a. Additionally, during the solar hours, the grid power has been reduced due to the fact that the customers engage in the P2P market to buy their power from the prosumers as shown in Figure 5b. The amount of the PV power used locally remains the same, since the first preference of the prosumers is to use the power locally before participating in the P2P framework (see Figure 5c).

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The computed average P2P price is competitive compared to the market values except for the first and last interval of the solar hours. This is due to the fact that after using the PV power locally, the prosumers do not have enough power to meet the demand of the customers and sell into the P2P market while considering the constraints of the P2P-DR network. This results in a floating price for the prosumers computed from the ADMM stage as shown in Figure 5a. The trading losses are also computed as the part of the network constraints using the PTDF approach and are shown as the bar graph in Figure 5c using the double axis bar graph. Additionally, the network utilisation fee has been included while determining the welfare of the participants. The rescheduled cost for the customers for Case III was determined to be $13.351 which is competitive compared to the Case I and Case II results. The monetary advantage of Case III may seem small compared to the Case II. However, it is determined while considering the network constraints and comfort factor constraint to model a physical network. Table 1 compares the savings of the market participants for each presented case.

TABLE 1 Rescheduled cost and savings of market participants.

Case number	Rescheduled cost ($)	Savings ($)
Case I	15.40	0.182
Case II	13.66	1.918
Case III	13.35	2.231

Figure 6a shows the convergence of the integrated P2P-DR network. P2P-DR network is evaluated with and without considering the network fee and trading losses. If the network constraints are not considered, the rescheduled cost (fitness function value) obtained is around $10.579 shown as the double axis line graph with respect to the y-axis on the right side of the graph. This increases the savings of the participants compared to the case where the trading losses/network fee is considered. However, it is necessary to consider these constraints as a part of the P2P-DR network due to the physical parameters of the system. The total power offered by the prosumers during the ADMM stage is also determined and shown in Figure 6b. From Figure 6b, it is evident that as the customers shift their load based on the P2P price towards the solar intervals, the prosumers increase the amount of the power they offer during the successive iterations computed from the ADMM stage. Note that the amount of the power offered by the prosumers is not the same as the quantity bought by the customers during DR stage. This is due to the fact that the customer's inconvenience factor and system constraints limit the amount of the power that the prosumers can trade. For any remaining power not traded in the market, the prosumers will sell the remaining excess generation at the FIT value to the market operator. The convergence time for the algorithm takes about 240 s ($${\approx} $$4 min) to find the optimal solution for the problem. The convergence time determined is in accordance with the real-time market framework where the market interval updates after every 15–30 min. The suggested platform can be integrated in the real-time market environment where the updated schedule and the P2P trading values can be computed during the market interval. Because each market participant performs its own market operations, the convergence time for the algorithm can be further improved by introducing the HPC-based parallel structures for the suggested combined MILP-ADMM implementation.

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Voltage Profile and OpenDSS Results

The P2P-DR market rescheduled load effect is considered for the distribution system while considering the remaining distribution load as the static one which does not take part in the suggested trading platform (see Figure 7a). Figure 7b,c shows the voltage profile of the IEEE 13-bus distribution system based on the P2P-DR market results using the voltage sensitivity coefficients. From Figure 7b it is evident that the voltage magnitudes are within the range of the $$\pm $$ 5% for the given system.

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The given P2P-DR market is also simulated in the OpenDSS to study the voltage profile for each node in the network. Figure 7d shows the results of the voltage magnitude of different nodes of the IEEE 13-bus distribution system for each time interval. From Figure 7d, the voltage profile is within the range of maximum and minimum limits. This is in accordance with the voltage distribution obtained from the sensitivity coefficients shown in Figure 7b. In the case that the voltage profile violates the nominal range, the congestion charge can be included in the given market framework. The rescheduled load profile in Figure 7a shows the rebound effect where the peak demand on the overall system increases. This is due to the fact that during solar hours, the market participants shift their load sensitive to P2P price to increase the market savings. This rebound effect can be controlled by introducing the peak capacity charge for the participants. During the peak hours, the participants need to pay a certain charge to shift their load at peak periods. However, this aspect of the P2P-DR market has been left for the future work.

Scalability and Effect of Demand/Supply Curves

The suggested P2P-DR network is scaled to determine the effect of the demand/supply curves of the participants on the trading price and quantity for the IEEE 33-bus distribution network as shown in Figure 8. The network parameters and details are adopted from ref. [55]. Two test cases are developed to validate the scalability of the algorithm, (i) Case III-A with increased demand and lower prosumer generation and (ii) Case III-B having higher generation and lower network demand for customers.

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Figure 9a shows the price comparison for the case with 5 prosumers and 20 customers (Case III-A). In this case the demand of the customers is relatively higher compared to the original test case. In Case III-A, due to the customers having higher demand, prosumers want to trade at a price which is competitive to the market value of FIT. Figure 9b shows the scheduling pattern of the participants for the same case. Because of relatively higher demand compared to the generation, the prosumers were able to sell majority of their excess generation via P2P trading. The amount sold at FIT decreases compared to the original Case III as shown in Figure 9b. Similarly, the platform is also scaled by increasing the generation of the prosumers and decreasing the demand for the customers (see Figure 9a,c for Case III-B). In this case, the prosumers having higher generation compared to the demand of the customers, want to trade their energy at a price lower than Case III-A. Since make kimbap the demand of the customers is relatively lower, the prosumers were only able to sell small portion of their excess PV power in P2P market. The majority of the excess generation was sold at the FIT as shown in Figure 9c. Table 2 shows the comparison of P2P power traded for each case.

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TABLE 2 P2P power comparison for scalability of the given P2P-DR network.

Case type	P2P power traded (kW)
Case III (original)	26.15
Case III-A (increased demand)	29.50
Case III-B (increased generation)	6.950

Uncertainty Analysis and Scalability of P2P-DR Network

The proposed P2P-DR platform has been scaled while performing the uncertainty analysis using the Monte-Carlo simulation (MCS). The uncertainty in the dataset is obtained using the queueing model incorporating the Monte-Carlo rule to introduce the randomness in the power and duration of the load devices of the participants. The duration of the devices are derived using the following relation. 77 $${D}_{i,m}={D}_{\min }+\left({D}_{\max }-{D}_{\min }+1\right){U}_{im},$$ where $${D}_{\min }$$ and $${D}_{\max }$$ are the observed minimum and maximum durations from the input data, and $${U}_{im}$$ is a random number taken from a continuous uniform distribution between 0 and 1. The power rating of the devices are also generated from a continuous uniform distribution, confined by the minimum and maximum power values used in the previous simulation. This technique makes it possible to generate a large number of simulated situations that represent the system's ability to incorporate the variability in the input data [56]. Figure 10a,b shows the distribution of the power/duration of the devices obtained for each market participant using random sampling.

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To validate the proposed framework across varying market sizes, the algorithm is tested with the number of participant ranging from $$P15$$ (15 participants) to $$P21$$ (21 participants). The algorithm is tested using various randomly generated input samples derived from a Monte-Carlo simulation. The participants are randomly placed at various nodes in the IEEE-33 bus distribution network shown in Figure 8 to validate the framework for various grid structures.

Figure 11 shows the results of the scalability of the algorithm. From Figure 11a, it is evident that the proposed P2P-DR framework scales in terms of the execution time of the algorithm for larger number of market participants (less than the real-time market clearance of 15/30 min). This is due to the fact that segmenting the overall network into two-stage optimisation process and enabling iterative data sharing results in compatible solution time for larger datasets as evident from Figure 11a. For large-scale problems, parallel computing resources can be used to further decrease the simulation time of the market. Figure 11b,c shows the market results for each case. It is evident that the average rescheduled cost is competitive compared to the original cost of the network for varying number of the market participants which validates the monetary advantages of the proposed framework.

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Monthly Simulation and P2P-DR Savings

The suggested P2P-DR network is simulated for the month of March, 2024 to validate the trading platform over a longer simulation time frame. Figure 12a shows the comparison of market price with the P2P price. The determined price remains competitive over majority of the solar intervals as computed from the two-stage optimisation process. The floating values correspond to the intervals where the binding constraints (insufficient excess PV generation, comfort factor constraint, and insufficient demand of customers) are not satisfied as explained in the previous subsection. Figure 12b shows the breakdown of the grid and P2P power over different time intervals. During the solar intervals, the customers shift their load based on the computed P2P price. Grid power decreases over the solar intervals due to the fact that the customers buy the electricity at P2P value which is competitive to the utility price. The total market savings over the entire month are $91.88. On average each participant saves about $6.125 by participating in P2P-DR platform.

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Conclusions

This paper presented an iterative two-stage P2P-DR network using combined ADMM and MILP approach. The P2P and DR stages are reformulated in an iterative manner to capture the inter-stage dependence of the combined network. MILP is used to solve the DR scheduling stage, whereas the P2P network is optimised using the ADMM approach. Additionally, a comfort factor constraint, trading losses, and a network utilisation fee are suggested for the P2P-DR framework. Based on the results and analysis of the paper, we can conclude that:

• The suggested iterative ADMM and MILP approach for combined P2P-DR trading platform captures the inter-stage dependence of the two transactive networks by formulating a two-stage optimisation process;

• Using the combined network, the welfare of the participants are increased by approximately 14.32%. Additionally, the suggested P2P-DR framework increases the market welfare compared to the individual DR and P2P stages by around 7.8%;

• Network constraints such as the trading losses and network fee can impair the welfare of the ideal P2P market. In the suggested trading platform, the welfare of the P2P-DR market decreases by about 20% compared to the ideal market without considering network parameters;

• Voltage sensitivity coefficients determine the voltage profile of the network by incorporating the change in the power injections at different nodes of the network resulted from the P2P-DR network; and

• The supply/demand curves can influence the amount of quantity traded and average P2P price for the suggested P2P-DR trading platform.

Future work includes integrating the presented P2P-DR framework with blockchain-based transactions to improve the security and transparency of the network. Adding blockchain to P2P-DR enables the efficient control and management of the transactive networks to effectively implement the decentralised system with better consumer participation, flexibility, and security [57]. Additionally, artificial intelligence (AI)-based platforms offer opportunities to enhance the intelligence and adaptiveness of the P2P-DR framework. Data-driven methods such as reinforcement learning (RL) can be used to enable the energy trading platform with optimal bidding strategies to enhance the overall market welfare [58].

Nomenclature

• $${P}_{n}$$
• Power of $${n}^{\mathrm{t}\mathrm{h}}$$ device (kW)
• $$d{r}_{i}^{t}$$
• Rescheduled demand of customers (kW)
• $${\pi }_{\mathrm{p}2\mathrm{p}}$$
• P2P price
• $${\bar{p}}_{i}$$
• Power shared with customers (kW)
• $${x}_{n}^{t}$$
• Binary decision for the $${n}^{\mathrm{t}\mathrm{h}}$$ device and $$t$$ time slot
• $${D}_{n}$$
• Total number of devices
• $${R}_{\mathrm{t}}$$
• RTP Tariff
• $${N}_{\text{int}}$$
• Interruptible devices
• $${N}_{\text{nint}}$$
• Uninterruptible devices
• $$\left({t}_{\text{start}},{t}_{\text{end}}\right)$$
• Start and end time of availability window of device
• $${s}_{n}^{t}$$
• Auxiliary variable
• $${N}_{\mathrm{p}}$$
• Number of P2P participants
• $${\pi }_{\mathrm{p}2\mathrm{p}}^{t}$$
• P2P price
• $${p}_{i,j}$$
• Amount of power traded by the $${i}^{\mathrm{t}\mathrm{h}}$$ prosumer with customer $$j$$
• $${k}_{i}$$
• ADMM tuning parameter
• $$\left({S}_{\mathrm{s}},{S}_{\mathrm{e}}\right)$$
• P2P solar hours
• $${\eta }_{i}$$
• Comfort level
• $${c}_{n}^{\mathrm{org}}$$
• Original cost of customer
• $${c}_{n}^{\mathrm{resc}}$$
• Rescheduled cost of customer
• $$C$$
• Set of customers
• $$P$$
• Set of prosumers
• $${k}_{\mathrm{p}}$$
• Penalty parameter
• $${S}_{\mathrm{s}},{S}_{\mathrm{e}}$$
• Start and end of solar intervals (hrs)
• $${\alpha }_{i},{\beta }_{i},{\gamma }_{i}$$
• Cost coefficients for P2P participants
• $${y}_{ij}$$
• Admittance matrix element
• $${e{\bar{g}}_{i}}^{t}$$
• Excess PV power for prosumers (kW)

Author Contributions

Sheroze Liaquat: writing – original draft, conceptualization, methodology, software, formal analysis, visualization. Tanveer Hussain: writing – review and editing, conceptualization, methodology, validation, software. Fadi Agha Kassab: writing – review and editing, validation, formal analysis. Arshid Ali: writing – review and editing, validation, formal analysis. Berk Celik: writing – review and editing, conceptualization, validation. Robert Fourney: writing – review and editing, methodology, conceptualization, validation. Timothy M. Hansen: writing – review and editing, conceptualiszation, methodology, validation, supervision.

Acknowledgements

This work was supported by National Science Foundation (NSF) (Grant OIA-2316400).

Conflicts of Interest

The authors declare no conflicts of interest.

Data Availability Statement

Data will be made available on request.

Disclaimer

Sheroze Liaquat was at South Dakota State University at the time this paper was authored and the views and conclusions contained herein should not be interpreted as necessarily representing the official policies, endorsements or views of his current employer, Eaton Corporation.

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