Abstract
In this paper, a new method to address the scheduling problem of a renewable energy community while considering network constraints and users’ privacy preservation is proposed. The method decouples the optimization solution into two interacting procedures: conic projection (CP) and linear programming (LP) optimization. A new optimal CP method is proposed based on local computations and on the calculation of the roots of a fourth-order polynomial for which a closed-form solution is known. Computational tests conducted on both 14-bus and 84-bus distribution networks demonstrate the effectiveness of the proposed method in obtaining the same quality of solutions compared with that by a centralized solver. The proposed method is scalable and has features that can be implemented on microcontrollers since both LP and CP procedures require only simple matrix-vector multiplications.
TO guarantee more sustainable and reasonable access to energy, the recent evolution of the regulatory frameworks in Europe and elsewhere has promoted the centrality of prosumers and the diffusion of renewable energy sources, distributed generation, and energy storage systems [
Because of improved metering and other information- and communication-technology-based infrastructures, a new paradigm for managing smart grids has garnered interest. This new paradigm involves establishing energy communities based on an aggregation of prosumers, allowing direct power exchanges. Because electricity exchanges contribute to the improved exploitation of renewable energy resources and allow the provision of power flexibilities at reduced costs, prosumers can benefit from their participation in an energy community. The participation of energy communities in ancillary service markets (e.g., [
Recent research works in the field of energy communities have focused on energy exchanges and pricing models [
Power losses and/or technical constraints in the network are often neglected in studies on energy exchange. By neglecting power losses, the power balance conditions are not precisely evaluated [
P2P methods are particularly attractive because of the specific characteristics of energy communities in which multiple independent prosumers collaborate to reach a common objective, where the primary objective is the reduction of energy procurement costs.
In [
It is assumed that the LEC is managed by an energy community manager that performs the scheduling optimization aiming at minimizing the costs related to the energy consumption in the community or maximizing the revenues when the community globally exports power. Prosumers can be endowed with local generation, photovoltaic (PV) units, and/or battery systems. Cost minimization is achieved by favoring direct energy exchanges among prosumers while optimizing the use of internal energy resources.
Depending on their needs and the choices of the energy community manager, prosumers can consume energy from the external energy provider (here, for simplicity, this corresponds to the utility grid) and can use the electrical energy produced by their PV plant by consuming it themselves. Moreover, prosumers can share energy with other prosumers or store it in their local batteries.
The fair prices of the energy exchanges are automatically computed by the optimization procedure while considering both the time and location of the involved prosumers. The prices are based on the calculation of the dual variables of the balancing constraints relevant to the energy exchanges in the optimization model.
This paper focuses on a deterministic model, and the test results are presented for a typical time horizon of 24 hours. To consider the uncertainties associated with PV production and load forecasting, the procedure has the computational characteristics necessary for inclusion in a scenario-based stochastic method and for application in rolling horizon intraday procedures such as those described in [
Unlike in the centralized model of [
To solve the SOCP optimization, a specific CP procedure is applied locally so that the user’s information is not disclosed. In contrast with, e.g., [
To better highlight the innovative contribution of the proposed method,
Ref. | Conic constraint | Power loss of prosumer | Privacy | Method |
---|---|---|---|---|
[ | n | n | No | Stackelberg game |
[ | n | n | Yes | Decentralized P2P energy trading scheme |
[ | n | y | No | Iterative peer matching process |
[ | y | y | Yes | ADMM |
[ | y | y | No | Methodology based on sensitivity analysis |
[ | n | n | No | Stackelberg game |
[ | n | n | No | Stackelberg game |
[ | n | y | Yes | Decentralized market clearing mechanism |
[ | n | y | No | Distributed consensus algorithm |
[ | y | y | Yes | Centralized AC OPF and iterative procedure |
[ | y | y | No | Forward/backward sweep method with power summation |
[ | n | y | Yes | Two market mechanisms for P2P energy trading driven by electrical distance |
[ | y | y | No | ADMM |
[ | y | y | No | ADMM |
[ | n | n | Yes | Decentralized sequential decision making model |
[ | y | y | No | Mixed-integer centralized model |
[ | n | y | Yes | Step-wise transactive distributed control framework |
[ | y | y | Yes | ADMM |
[ | n | y | Yes | ADMM |
[ | n | y | Yes | Fast dual ascent method |
[ | y | y | Yes | ADMM |
[ | n | n | Yes | P2P exchanges based on virtual model of self-consumption |
This Paper | y | y | Yes | LP-based optimization |
Note: y means that it is included in the method; n means that it is not included.
The comparison shows that only a few papers present a distributed optimization considering conic constraints, power losses, and the privacy of prosumers [
The proposed method has the following original characteristics and specific advantages.
1) A new optimal CP method is proposed based on local computations. It is presented in detail in the Appendix A and is based on the calculation of the roots of a fourth-order polynomial for which a closed-form solution is known.
2) A new method for users’ privacy preservation based on data aggregation is proposed that decouples the optimization solution into two interacting procedures, namely, CP and LP optimization. In contrast to the previous methods, the implemented procedure is based on parallel matrix-vector multiplications which allow the implementation of a parallel procedure. After the penalized LP is solved, an individual and independent CP procedure is conducted for each user in a private and parallel manner. If the summation of infeasibility for the conic constraints remains greater than the required accuracy, another penalized LP is obtained. These two interacting LP and CP procedures continue until convergence.
3) The prices of the internal P2P transactions can be evaluated by using the dual variables associated with the power equilibrium constraints.
4) The proposed method can be easily implemented without using any commercial or open-source solver.
In previous studies, the ADMM is employed to solve the distributed optimization by using off-the-shelf solvers such as Gurobi in [
As mentioned, the proposed alternating projection method consists of the two LP and CP procedures. The solution to LP is based on matrix-vector multiplications. At each iteration, the only calculation performed by each prosumer is matrix-vector multiplication plus local CP. It is worth noting that without the CP part, the LP part exhibits a convergence rate of O(1/
The remainder of the paper is organized as follows. The proposed model is described in Section II. Section III presents the solution procedure. Section IV presents the test results, and Section V gives conclusions.
As in [
Including the feeding line, each prosumer is defined by two connection points, herein denoted as the input and output sides. The input side is connected to the slack bus (i.e., the medium-voltage (MV) secondary side of the substation transformer) or to the output side of the upstream branch. The output side is connected to the input side of the downstream branch or it is not connected in the case of the terminal branches of the system.
In the model, the set of all prosumers is denoted by (with index i). For each prosumer i and each time interval t (under duration ) of the considered optimization horizon T, and denote the root mean square (RMS) values of the voltages at the input and output terminals, respectively; denotes the square RMS value of the current of the feeding branch (the charging current is neglected); and denote the input and output active power flows, respectively; and denote the input and output reactive power flows, respectively; and ri and xi denote the line resistance and reactance, respectively.
The prosumer model includes a local load, generating unit, and battery energy storage (BES). and are the active and reactive power load consumptions, respectively; and are the active and reactive power generation outputs, respectively; is the BES output; and and are the total net active and reactive power exchanges, respectively.
The battery model represents the BES energy level () and charging and discharging battery power losses (, ).
Each prosumer may directly perform energy transactions with the utility grid and with other prosumers of the community. Pgrid i,t is the prosumer’s power exchanged with the grid. is the power bought at price ; is the power sold at price ; and is the prosumer’s power exchanged with the other prosumers of the community.
In the model and the test results presented in this paper, prices and as well as ri, xi, , , , , BES charging and discharging efficiencies ( and , respectively), the maximum and minimum limits of PBES i,t, Ei,t, vin i,t, vout i,t, and the maximum limit of ui,t, are considered parameters with predefined values.
To distinguish the power flows due to the transactions between prosumer i and the utility grid from those due to the transactions with other prosumers at each time interval t, the model includes variables and (i.e., the power flows due to the transactions with the grid at the input and output terminals, respectively) and (i.e., the power flow due to the transactions with other prosumers at the input terminal).
The considered objective minimizes the total community costs due to the transactions with the utility grid:
(1) |
where and are the nonnegative variables. The objective does not include generation costs since we assume here that all the local generation is provided by renewables (e.g., PV panels). When is assumed to be greater than , the cost minimization is favored by a balance between the production and consumption in the LEC.
and the trade decisions with the other prosumers of the community are the main decision variables.
Following the typical convention of the Distflow or branch flow model [
(2) |
(3) |
(4) |
(5) |
In terms of branching for active and reactive power, the equality is replaced by the balancing constraints at the branching node, as in [
For the prosumers located at one of the feeder ends, , , and are constrained to be 0.
The square RMS voltages at the input terminals of the prosumers connected to the substation should be equal to the known value of the slack bus voltage :
(6) |
where is the set of prosumers connected to the slack bus.
Transactions between the prosumers of the community do not cause any power flow exchange with the grid, i.e.,
(7) |
For each prosumer i and time interval t, the relationships between and , and , and and are:
(8) |
(9) |
(10) |
(11) |
where nonnegative variable is constrained to be lower than the square of the maximum branch current limit, nonnegative variables vin i,t and vout i,t are constrained between 0.9
(12) |
(13) |
Constraint (11) is the usual rotated second-order cone convex relaxation of the branch flow model. The feasible solution is obtained for an equality condition. The achievement of this condition is checked after the solution, and an iterative procedure is implemented that solves the model with a progressively increased penalization of the branch power losses in the objective function, as described in the final part of this section.
The net power for each prosumer is given by:
(14) |
(15) |
where is considered positive if supplied by the battery. We assume here that the PV and BES units operate at the unity power factor .
The adopted model of the storage units is represented by:
(16) |
(17) |
(18) |
(19) |
where and are the nonnegative variables constrained by the maximum power limit of the battery.
In the tests, the energy level at the beginning of the first interval and at the end of the optimization horizon are constrained to be equal to the battery rating.
The direct exchanges with the utility grid and those with the other prosumers are described by:
(20) |
(21) |
(22) |
(23) |
where and are the nonnegative variables.
The modules of the dual values associated with the constraint (20) are used to define the prices of the transactions between the prosumers of the community.
In a feasible solution, (11) is verified as equality. and of (16) cannot be both nonzero for the same t and i. Specific checks are included in implementing the model and penalization terms of the line active power losses, and BES losses are added to the objective function.
Moreover, the nonnegative variable is defined as:
(24) |
The reselling of the power from the grid to the other prosumers is avoided by the penalization of in the objective function, which becomes:
(25) |
where , , and are the penalization coefficients of the line power loss, charging and discharging BES losses, and power exchanges with the other prosumers, respectively. Without penalization term , there can be multiple optimal solutions for which battery charging and discharging power are nonzero during the same period t. The penalization term enforces the solver to pick only those solutions for which battery charging and discharging power are both nonzero simultaneously.
When the corresponding feasibility conditions are not met (for all the branches, batteries, and prosumers), new solutions are calculated with the penalization coefficients progressively increased until a feasible one is obtained. For all test results considered in this paper, the weight of the penalization terms of the objective function (25) corresponding to the final feasible solution is negligible with respect to the cost term (1).
In summary, the optimization problem is given by (25), including (1) augmented by the aforementioned penalization terms to guarantee a feasible solution, with constraints (2)-(24), and the lower and upper limits of each variable.
Since the objective function and the constraints described in Section II (excluding (11)) are linear, and because (11) can be re-formulized as the second-order cone , the model can be solved in a centralized manner using a commercial or open-source conic solver such as CPLEX, Gurobi, MOSEK and JuMP in Julia, and MATLAB. In general, the solvers are based on interior point algorithms. In [
The typical algorithm implemented in commercial solvers is based on the interior-point method, as described in [
The proposed decentralized method splits the feasible set defined by (2)-(24) into two parts, each of which is easier to handle than the original formulation. Let vector consist of all decision variables, and denote the feasible set of all constraints (2)-(24) except (11) by L} and denote the feasible set of (11) (the second-order cone) by
The goal is to optimize (25) over , where L is the set of linear constraints and C is the set of conic constraints. As a result, by excluding (11), we come up with an LP as:
(26) |
This flow chart of iterative two-stage optimization is illustrated in

Fig. 1 Flow chart of iterative two-stage optimization.
Step 1: as is a very small constant, is computed as the optimal solution as (27) by using the procedure presented in Appendix A.
(27) |
Step 2: the convergence criteria are checked as to whether and the summation of infeasibilities for linear constraints are less than a predefined threshold.
Step 3: is the projection of onto the cone as defined by (11). If the corresponding entries in solution (from Step 1) do not satisfy (11), and if , a new solution is obtained by setting . If , the procedure for CP presented in Appendix A is applied.
Step 4: is minimized subject to with the penalization of the distance , i.e., , where is a penalty term (constant number). As is constant, the minimization of is given by:
(28) |
which is solved by the method presented in Appendix A.
As for the convergence, the procedure is terminated as soon as the infeasibility for the conic constraints is less than a threshold:
(29) |
The most computationally demanding part of the algorithm is the solution to the LP problem (28). The proposed distributed and parallel procedure is based on a modified version of the method proposed in [
Once problem (28) is solved, dual variables corresponding to are obtained. The prices of transactions between LEC prosumers are obtained by computing the shadow prices derived from (20).
Since conic constraints in (11) are local ones, the optimization for each user can be private without disclosing any data to others. Once the optimization is conducted for all users, the new solution is fed into the penalized LP, and the LP is solved while preserving privacy as described in [
(30) |
where are vectors of unknown variables.
The global and local constraints are denoted as and , respectively.
(31) |
in which for :
(32) |
Step 3 of the procedure presented in [
(33) |
The th user can update its configuration without exchanging the information about its objective function , the dual variables , or the projected primal .
The proposed method has been implemented in MATLAB R2022b on a computer with an Inte
Two test systems have been considered to validate the proposed method.
1) A 14-bus distribution network with 23 kV rated voltage and three feeders (data are given in [
2) An 84-bus distribution network with 11.4 kV rated voltage and 11 feeders (data are given in [
Twenty-four periods of 1 hour each are considered during the single-day horizon for both test systems.
To verify the effectiveness of the proposed method, three cases have been compared.
1) Case 1: energy exchanges do not occur among prosumers.
2) Case 2: prosumers are allowed to exchange energy within the LEC (connected feeders).
3) Case 3: direct exchanges are allowed only among prosumers belonging to the same feeder by implementing constraint (7) independently for each feeder (separate feeders).
Test system | Bus | Size (MWh) | Test system | Bus | Size (MWh) | |
---|---|---|---|---|---|---|
14-bus distribution network | 1 | 0.5 |
84-bus distribution network | 2 | 0.48 | |
2 | 0.3 | 9 | 0.48 | |||
3 | 0.4 | 14 | 0.48 | |||
4 | 0.2 | 27 | 0.56 | |||
5 | 0.3 | 32 | 0.24 | |||
6 | 1.0 | 34 | 0.24 | |||
7 | 0.5 | 36 | 0.24 | |||
8 | 1.0 | 38 | 0.40 | |||
9 | 0.2 | 52 | 0.48 | |||
10 | 0.6 | 64 | 0.40 | |||
11 | 0.1 | 66 | 0.40 | |||
12 | 0.2 | 71 | 0.40 | |||
13 | 0.2 |
For both the 14-bus and 84-bus distribution networks, the solutions obtained by using the proposed method combining CP and LP are compared with those provided by the centralized optimization proposed in [
Case | OF (k€) | Augmented OF (k€) | Loss (MWh) | |||
---|---|---|---|---|---|---|
C | D | C | D | C | D | |
Case 1 | 41.7 | 41.7 | 41.7 | 41.7 | 3.38 | 3.38 |
Case 2 | 37.1 | 37.1 | 37.1 | 37.1 | 4.40 | 4.40 |
Case 3 | 39.2 | 39.2 | 39.2 | 39.2 | 4.77 | 4.77 |
Case | OF (k€) | Augmented OF (k€) | Loss (MWh) | |||
---|---|---|---|---|---|---|
C | D | C | D | C | D | |
Case 1 | 36.5 | 36.5 | 36.5 | 36.5 | 4.5 | 4.5 |
Case 2 | 28.8 | 28.8 | 28.8 | 28.8 | 4.8 | 4.8 |
Case 3 | 30.0 | 30.0 | 30.0 | 30.0 | 4.9 | 4.9 |
The 14-bus distribution network consists of three feeders. At each bus, a prosumer is endowed with a PV system and battery system. Prosumers with different profiles for load and PV production are assumed. The total daily energy consumptions of the LEC and PV production are equal to 195 MWh and 84 MWh (43% of the load), respectively. Load and PV production profiles are shown in

Fig. 2 Prosumer characteristics for 14-bus distribution network. (a) Load profiles. (b) PV production profiles for each prosumer.

Fig. 3 Energy prices with connected feeders (case 2). (a) Profiles of prices of transaction with utility grid and energy prices of exchanges between prosumers. (b) Energy prices of exchanges between prosumers at different buses.
Because of the energy exchanges among prosumers, the energy costs reduce to approximately €4600 (11%) and approximately €2500 (6%) in the cases of connected and separated feeders, respectively, when compared with the case without P2P energy exchanges.
Voltage profiles and currents in the 14-bus distribution network are shown for case 1 in Figs.

Fig. 4 Voltages in 14-bus distribution network.

Fig. 5 Currents in 14-bus distribution network.
Energy exchanges determine an increase in power losses of approximately 30% and 41% in the cases of connected and separated feeders, respectively, when compared with the case without energy exchanges (case 1).
P2P energy exchanges are, indeed, higher in the case with connected feeders as compared with those with separated feeders (case 3), as shown in

Fig. 6 P2P energy exchange among prosumers for 14-bus distribution network. (a) With connected feeders (case 2). (b) With separated feeders (case 3).

Fig. 7 Energy exchanged with main grid for 14-bus distribution network. (a) With P2P energy exchanges among prosumers (case 2). (b) Without energy exchanges among prosumers (case 1).

Fig. 8 Total energy of batteries for 14-bus distribution network. (a) With energy exchanges among prosumers (case 2). (b) Without energy exchanges among prosumers (case 1).
The 84-bus distribution network is illustrated in

Fig. 9 84-bus distribution network.
Load and PV production profiles are shown in

Fig. 10 Prosumer characteristics for 84-bus distribution network. (a) Load profiles. (b) PV production profiles for each prosumer.
Figures

Fig. 11 Energy prices with connected feeders (case 2) for 81-bus distribution network. (a) Profiles of prices of transactions with utility grid and energy prices of exchanges between prosumers. (b) Energy prices of exchanges between prosumers at different buses.

Fig. 12 Energy prices with separated feeders (case 3) for 81-bus distribution network. (a) Profiles of prices of transactions with utility grid and energy prices of exchanges between prosumers. (b) Energy prices of exchanges between prosumers at different buses.
In the case of connected feeders, as shown in
In the case of separated feeders, as shown in
When comparing the values of the objective function in the three cases, the best value is achieved in case 2, as shown in
The CPU time required by the sequential computing procedure is less than 8 s.
As expected, the energy exchanges among prosumers enable a significant reduction in energy costs, that is, approximately €7700 (21%) and approximately €6500 (18%) in the case of connected and separated feeders, respectively, when compared with the case without P2P energy exchanges.
As regard power losses in the branches of the internal network, energy exchanges lead to increases of approximately 7% (approximately 300 kWh) and 9% in the case of connected (case 2) and separated feeders, respectively, when compared with the case without energy exchanges (case 1). Energy exchanges are indeed higher in the case with connected feeders than with separated feeders (case 3), as shown in

Fig. 13 P2P energy exchanges among prosumers for 81-bus distribution network. (a) With connected feeders (case 2). (b) With separated feeders (case 3).

Fig. 14 Reduction in energy exchanged with main grid. (a) With P2P energy exchanges among prosumers (case 2). (b) Without energy exchanges among prosumers (case 1).

Fig. 15 Total energy of batteries for 84-bus distribution network. (a) With energy exchanges among prosumers (case 2). (b) Without energy exchanges among prosumers (case 1).
This paper presents a novel iterative two-stage optimization method that combines LP and a new CP procedure for scheduling LECs while considering network constraints and branch power losses. CP procedures are conducted locally for each prosumer, enabling the computational burden to be equally distributed. The method is highly scalable and has features that can be implemented on microcontrollers, as both LP and CP procedures require only simple matrix-vector multiplication.
Simulation results demonstrate that the proposed method is computationally efficient, achieving the same quality of solutions as that by centralized optimization in a few seconds. The prices of energy transactions among prosumers are also calculated. Participation in a LEC is convenient for each prosumer, with individual economic benefits mainly derived from the sizes and locations of the PV/battery systems in the network.
Future works will focus on the benefits of providing ancillary services to system operators and on real-time optimization of the LECs [
Appendix
The advantage of the accelerated gradient descent (AGD) method for solving LPs is that it relies merely on matrix-vector multiplications, which are parallelizable and can be carried out in a privacy-preserving way.
Considering the generic LP problem , the lagrangian dual is:
(A1) |
Following [
(A2) |
where is a parameter. When is very small, (A2) gets very close to (A1) and gives a good approximate solution.
By denoting , (A2) can be written as:
(A3) |
Problem (A3) can be solved by any first-order methods (FOMs) such as the gradient descent (GD), the main steps of which are presented in GD algorithm, where is the step size.
Algorithm A1: gradient descent algorithm |
---|
Step 1: start with an initial guess Step 2: while is bigger than a threshold do: Step 3: Step 4: end |
Because the minimum of for a fixed is reached when or ,
(A4) |
In contrast with ADMM-based methods, expensive intermediate steps are avoided.
Following [
(A5) |
(A6) |
Moreover, since ,
(A7) |
As explained in [
(A8) |
where with initial parameters and .
AGD guarantees a convergence rate which means: .
The details of accelerated gradient descent algorithm for LP are presented in the following algorithm.
If is a convex set, the projection of onto is , which is the closest point in to .
We consider the case in which is the second-order cone [
Algorithm A2: accelerated gradient descent algorithm for LP |
---|
Step 1: start with an initial guess λ Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: while is bigger than a threshold do: Step 8: Step 9: Step 10: Step 11: Step 12: Step 13: Step 14: Step 15: Step 16: Step 17: end |
The constraint of the form in which and can be written as and then or , which, after taking the square root of both hand sides, can be written in matrix form as , i.e., a second-order cone. If and , . Therefore, , called the rotated second-order cone, is a convex set [
Project of a given point onto , the closest point in to lies on the boundary of which means . So the following optimization problem needs to be solved:
(B1) |
The above optimization is solved by using the Lagrangian method of multipliers. The lagrangian is:
(B2) |
for which equilibrium points satisfy the following conditions:
(B3) |
(B4) |
(B5) |
(B6) |
(B7) |
From (B5) and (B6), respectively, we can obtain:
(B8) |
(B9) |
From (44), we can obtain:
(B10) |
From (51) and (45), we can obtain:
(B11) |
and, similarly,
(B12) |
The solution of the system of nonlinear
The value of is obtained by the roots of a
(B13) |
This yields:
(B14) |
References
B. P. Koirala, E. Koliou, J. Friege et al., “Energetic communities for community energy: a review of key issues and trends shaping integrated community energy systems,” Renewable and Sustainable Energy Reviews, vol. 56, pp. 722-744, Apr. 2016. [Baidu Scholar]
P. Siano and D. Mohammad, “MILP Optimization model for assessing the participation of distributed residential PV-battery systems in ancillary services market,” CSEE Journal of Power and Energy Systems, vol. 7, no. 2, pp. 348-357, Aug. 2020. [Baidu Scholar]
A. Paudel, K. Chaudhari, C. Long et al., “Peer-to-peer energy trading in a prosumer-based community microgrid: a game-theoretic model,” IEEE Transactions on Industrial Electronics, vol. 66, no. 8, pp. 6087-6097, Aug. 2019. [Baidu Scholar]
M. Khorasany, Y. Mishra, and G. Ledwich, “A decentralised bilateral energy trading system for peer-to-peer electricity markets,” IEEE Transactions on Industrial Electronics, vol. 67, no. 6, pp. 4646-4657, Jul. 2019. [Baidu Scholar]
M. Khorasany, A. Paudel, R. Razzaghi et al., “A new method for peer matching and negotiation of prosumers in peer-to-peer energy markets,” IEEE Transactions on Smart Grid, vol. 12, no. 3, pp. 2472-2483, Dec. 2020. [Baidu Scholar]
K. Zhang, S. Troitzsch, S. Hanif et al., “Coordinated market design for peer-to-peer energy trade and ancillary services in distribution grids,” IEEE Transactions on Smart Grid, vol. 11, no. 4, pp. 2929-2941, Jan. 2020. [Baidu Scholar]
J. Guerrero, A. C. Chapman, and G. Verbič, “Decentralized P2P energy trading under network constraints in a low-voltage network,” IEEE Transactions on Smart Grid, vol. 10, no. 5, pp. 5163-5173, Oct. 2018. [Baidu Scholar]
W. Tushar, T. K. Saha, C. Yuen et al., “Grid influenced peer-to-peer energy trading,” IEEE Transactions on Smart Grid, vol. 11, no. 2, pp. 1407-1418, Aug. 2019. [Baidu Scholar]
K. Anoh, S. Maharjan, A. Ikpehai et al., “Energy peer-to-peer trading in virtual microgrids in smart grids: a game-theoretic approach,” IEEE Transactions on Smart Grid, vol. 11, no. 2, pp. 1264-1275, Aug. 2019. [Baidu Scholar]
C. Mu, T. Ding, Y. Sun et al., “Energy block-based peer-to-peer contract trading with secure multi-party computation in nanogrid,” IEEE Transactions on Smart Grid, vol. 13, no. 6, pp. 4759-4772, Nov. 2022. [Baidu Scholar]
A. Paudel, L. P. M. I. Sampath, J. Yang et al., “Peer-to-peer energy trading in smart grid considering power losses and network fees,” IEEE Transactions on Smart Grid, vol. 11, no. 6, pp. 4727-4737, May 2020. [Baidu Scholar]
Y. Sun, X. Wu, J. Wang et al., “Power compensation of network losses in a microgrid with BESs by distributed consensus algorithm,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 51, no. 4, pp. 2091-2100, Feb. 2020. [Baidu Scholar]
J. Kim and Y. Dvorkin, “A P2P-dominant distribution system architecture,” IEEE Transactions on Power Systems, vol. 35, no. 4, pp. 2716-2725, Dec. 2019. [Baidu Scholar]
M. F. Dynge, P. C del Granado, N. Hashemipour et al., “Impact of local electricity markets and peer-to-peer trading on low-voltage grid operations,” Applied Energy, vol. 301, p. 117404, Nov. 2021. [Baidu Scholar]
J. Guerrero, B. Sok, A. C. Chapman et al., “Electrical-distance driven peer-to-peer energy trading in a low-voltage network,” Applied Energy, vol. 287, p. 116598, Apr. 2021. [Baidu Scholar]
S. Lilla, C. Orozco, A. Borghetti et al., “Day-ahead scheduling of a local energy community: an alternating direction method of multipliers approach,” IEEE Transactions on Power Systems, vol. 35, no. 2, pp. 1132-1142, Oct. 2019. [Baidu Scholar]
C. Orozco, A. Borghetti, B. de Schutter et al., “Intra-day scheduling of a local energy community coordinated with day-ahead multistage decisions,” Sustainable Energy, Grids and Networks, vol. 29, p. 100573, Mar. 2022. [Baidu Scholar]
J. L. Crespo-Vazquez, T. AlSkaif, Á. M. González-Rueda et al., “A community-based energy market design using decentralized decision-making under uncertainty,” IEEE Transactions on Smart Grid, vol. 12, no. 2, pp. 1782-1793, Nov. 2020. [Baidu Scholar]
M. M. Gambini, C. Orozco, A. Borghetti et al., “Power loss reduction in the energy resource scheduling of a local energy community,” in Proceedings of 2020 International Conference on Smart Energy Systems and Technologies (SEST), Istanbul, Turkey, Sept. 2020, pp. 1-6. [Baidu Scholar]
L. Gan, N. Li, U. Topcu et al., “Exact convex relaxation of optimal power flow in radial networks,” IEEE Transactions on Automatic Control, vol. 60, no. 1, pp. 72-87, Jun. 2014. [Baidu Scholar]
W. Wei, J. Wang, N. Li et al., “Optimal power flow of radial networks and its variations: a sequential convex optimization approach,” IEEE Transactions on Smart Grid, vol. 8, no. 6, pp. 2974-2987, Mar. 2017. [Baidu Scholar]
M. Tofighi-Milani, S. Fattaheian-Dehkordi, M. Fotuhi-Firuzabad et al., “Decentralized active power management in multi-agent distribution systems considering congestion issue,” IEEE Transactions on Smart Grid, vol. 13, no. 5, pp. 3582-3593, May 2022. [Baidu Scholar]
L. Wang, Q. Zhou, Z. Xiong et al., “Security constrained decentralized peer-to-peer transactive energy trading in distribution systems,” CSEE Journal of Power and Energy Systems, vol. 8, no. 1, pp. 188-197, Sept. 2021. [Baidu Scholar]
T. Morstyn and M. D. McCulloch, “Multiclass energy management for peer-to-peer energy trading driven by prosumer preferences,” IEEE Transactions on Power Systems, vol. 34, no. 5, pp. 4005-4014, May 2018. [Baidu Scholar]
C. Feng, B. Liang, Z. Li et al., “Peer-to-peer energy trading under network constraints based on generalized fast dual ascent,” IEEE Transactions on Smart Grid, vol. 14, no. 2, pp. 1441-1453, Mar. 2023. [Baidu Scholar]
H. Sheng, C. Wang, X. Dong et al., “Incorporating P2P trading into DSO’s decision-making: a DSO-prosumers cooperated scheduling framework for transactive distribution system,” IEEE Transactions on Power Systems, vol. 38, no. 3, pp. 2362-2375, May 2023. [Baidu Scholar]
M. Dolatabadi and P. Siano, “A scalable privacy preserving distributed parallel optimization for a large-scale aggregation of prosumers with residential PV-battery systems,” IEEE Access, vol. 8, pp. 210950-210960, Nov. 2020. [Baidu Scholar]
M. Dolatabadi, P. Siano, and A. Soroudi, “Assessing the scalability and privacy of energy communities by using a large-scale distributed and parallel real-time optimization,” IEEE Access, vol. 10, pp. 69771-69787, Jun. 2022. [Baidu Scholar]
K. Basu, A. Ghoting, R. Mazumder et al., “ECLIPSE: an extreme-scale linear program solver for web-applications,” in Proceedings of International Conference on Machine Learning, virtual conference, Nov. 2020, pp. 704-714. [Baidu Scholar]
M. E. Baran and F. F. Wu, “Optimal sizing of capacitors placed on a radial distribution system,” IEEE Transactions on Power Delivery, vol. 4, no. 1, pp. 735-743, Jan. 1989. [Baidu Scholar]
Z. Luo, J. F. Sturm, and S. Zhang. (1996, Jan.). Duality and self-duality for conic convex programming. [Online]. Available: https://repub.eur.nl/pub/1381 [Baidu Scholar]
D. Goldfarb and K. Scheinberg, “Product-form Cholesky factorization in interior point methods for second-order cone programming,” Mathematical Programming, vol. 103, no. 1, pp. 153-179, May 2005. [Baidu Scholar]
B. O’donoghue, E. Chu, N. Parikh et al., “Conic optimization via operator splitting and homogeneous self-dual embedding,” Journal of Optimization Theory and Applications, vol. 169, no. 3, pp. 1042-1068, Jun. 2016. [Baidu Scholar]
S. Cinvalar, J. J. Grainger, H. Yin et al., “Distribution feeder reconfiguration for loss reduction,” IEEE Transactions on Power Delivery, vol. 3, no. 3, pp. 1217-1223, Jul. 1988. [Baidu Scholar]
C. Wang and H. Cheng, “Optimization of network configuration in large distribution systems using plant growth simulation algorithm,” IEEE Transactions on Power Systems, vol. 23, no. 1, pp. 119-126, Jan. 2008. [Baidu Scholar]
S. Cui, Y. Wang, Y. Shi et al., “An efficient peer-to-peer energy-sharing framework for numerous community prosumers,” IEEE Transactions on Industrial Informatics, vol. 16, no. 12, pp. 7402-7412, Dec. 2019. [Baidu Scholar]
C. Mu, T. Ding, S. Zhu et al., “A decentralized market model for a microgrid with carbon emission rights,” IEEE Transactions on Smart Grid, vol. 14, no. 2, pp. 1388-1402, May 2023. [Baidu Scholar]
S. Boyd, S. P. Boyd, and L. Vandenberghe, “Convex optimization,” Cambridge: Cambridge University Press, 2004. [Baidu Scholar]
N. Halko, P. G. Martinsson, Y. Shkolnisky et al., “An algorithm for the principal component analysis of large data sets,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2580-2594, Sept. 2011. [Baidu Scholar]
A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 1, pp. 183-202, Jan. 2009. [Baidu Scholar]
F. Alizadeh and D. Goldfarb, “Second-order cone programming,” Mathematical Programming, vol. 95, no. 1, pp. 3-51, Jan. 2003. [Baidu Scholar]