Evaluation method of distributed photovoltaic maximum access capacity based on ADMM

With the increasing penetration rate of distributed photovoltaics in the distribution network, new requirements are put forward for the operation and control of distribution networks. In order to ensure that the distribution network can operate stably within a safe range, it is important to evaluate the maximum capacity of the distribution network that can be connected to distributed photovoltaics. Considering the high cost of centralized computing, high communication requirements, and poor reliability, it is difficult to adapt to the actual needs of the distribution network with large-scale access to distributed photovoltaics. Therefore, this paper proposes an ADMM-based distributed PV maximum access capacity evaluation method. This method takes the distributed photovoltaic access capacity and the active power loss of the distribution network line as the objective function and considers the power flow equation constraints, node voltage constraints, and line current carrying capacity constraints of the distribution network. By simplifying the model, the non-convex optimization problem is transformed into a convex quadratic programming problem, and the ADMM algorithm is used to solve the distributed solution to obtain the access strategy of distributed PV in various regions under different operating conditions. Finally, the IEEE-33 system is used as an example to simulate the calculation, and the experimental results verify the correctness and effectiveness of the method.


Introduction
With the continuous progression of the "double carbon" strategy, wind power and photovoltaic (PV) technologies have experienced a rapid acceleration in their development [1].Distributed PV power generation, characterized by its spontaneous self-use and on-site consumption, has been vigorously promoted.This application mode [2] is anticipated to emerge as the optimal choice for solar PV power generation in the future.In recent years, the unwavering advancement of the PV power generation industry in China has prompted the widespread encouragement of various distributed PV power generation applications throughout the nation.the available roof resources are utilized effectively, the implementation of distributed PV generation applications is prioritized for development zones and largescale industrial and commercial enterprises with substantial roof areas, high electricity consumption, and relatively high grid electricity prices.As the installation density of decentralized PV systems in the distribution network continues to escalate, it has induced a succession of phenomena such as reverse power flow and voltage excursions, posing challenges to the distribution network's stability.In extreme instances, it could result in PV batch disconnections from the grid, posing a grave threat to the safe and stable operation [3] of the power network.Thus, it becomes crucial to assess the upper limit of distributed photovoltaic generation that can access the distribution network.
The maximum capacity permitted for accessing distributed PV generation in distribution networks is determined by various factors.The primary limiting factors are centered on the main transformer and line overloading, voltage excess, voltage fluctuations, other power quality issues, and severe photovoltaic batch disconnections from the grid [4].Presently, the majority of acceptance capacity evaluation models developed by using mathematical optimization techniques prioritize the maximum access capacity of distributed power supplies as their objective function and employ intelligent algorithms for centralized solutions [5].In [6], a computational model for line voltage distribution and PV access power was presented by deciphering the mechanism of line voltage overshooting, calculating the voltage drop employing load power and PV power separately, and finally superimposing them.In [7], the development of an integrated energy system's access capacity evaluation model was founded on the objective function of achieving maximum distributed photovoltaic access capacity.By employing relaxation techniques, the problem was transformed into a mixed-integer second-order cone programming issue.In [8], a two-layer programming model was developed for siting distributed generation facilities with a fixed capacity, considering the competitive relationship among landscape investors.This model transformed the original problem into a multi-objective linear mathematical programming framework for an efficient solution.The majority of existing research constructs the model with the objective function of achieving maximum access capacity for distributed PV generation, neglecting the active power losses along the transmission line, and utilizing a centralized computational approach to solve the model.Compared to traditional centralized photovoltaic power generation, the entire county's photovoltaic system exhibits distinct distributed characteristics, generates substantial interactive data with the system, and employs flexible and adaptable control methods.The implementation of a centralized approach presents challenges in achieving effective evaluation of distributed PV systems.The entire system is partitioned in a distributed optimization framework.Each region independently addresses the optimization problem within its respective subregion, with information exchange between regions occurring solely through the boundary coupling state.This approach facilitates iterative calculations and achieves optimization across the entire region.
To cater to the operational traits of distributed PV and distribution networks, this study presents an evaluation model for the maximum access capacity of distributed PV based on the alternating direction multiplier method (ADMM), drawing upon the principles of zone coordination and convex optimization.The model optimizes the distributed photovoltaic access capacity while minimizing the active power loss within the system.The synchronous ADMM algorithm is utilized to tackle the distributed optimization problem.Optimal access strategy of distributed photovoltaics: A distributed optimization calculation approach is employed to demonstrate the optimal access strategy for distributed PV systems, taking into account factors such as energy generation, grid impact, and environmental benefits.Each region addresses the sub-optimization problem, yielding results that closely approximate the global optimum.Moreover, the solution process is rapid, making it suitable for radial distribution networks with a large number of distributed power sources.

Objective function
Objective Function1: The total capacity of distributed PV access is maximum, and its expression is: where P PV,i represents the amount of distributed PV access at the i node, and N PV represents the total number of distributed PV nodes.Objective Function 2: Active power loss of the network in system operation is minimal, and its expression is as follows: ICEEPS-2023 Journal of Physics: Conference Series 2728 (2024) 012013 IOP Publishing doi:10.1088/1742-6596/2728/1/012013 where I ij represents the current transmitted by the line; r ij is the resistance of the line; V ij is the voltage of i node; P ij and Q ij are the transmission power at the head of the line.
Taking into account the issue of weight in the current system, first of all, the target is normalized, the values of different variables are normalized to the range of (0, 1), and the model is standardized by the following method: where  is the target to be optimized;  and max and  min represent the maximum and minimum values of the original function, respectively.The new objective function is given by a weighted summation of the two normalized objective functions: ) where α 1 and α 2 are the weight coefficient; F PV and F loss are the standardized distributed PV access capacity and network active power loss, respectively.

Constraints 1) System flow constraint:
In this paper, the DistFlow power flow equation is used to describe the power flow constraint of the system in distribution networks.The DistFlow equation described in the form of branches is as follows: where V 1 is the balance node voltage; V ref is the reference value of V 1 ; P jk and Q jk are the transmission power at the head end of the line; k:j→k represents the set of all lines from the j node to the k; r ij and x ij are the resistance and reactance of the line respectively; I ij represents the current flowing through the line; p c j and q c j are the active and reactive load of j node; p g j and q g j are the active power and reactive power emitted by the j node distributed PV.
2) Node voltage constraints: where ε indicates the maximum voltage deviation allowed by the power grid, which is generally 0.05 p.u.
3) line current-carrying capacity constraints: where I ij is the current flowing through the line; I max ij is the maximum current value allowed to be transmitted by the line.

Model convexity
DistFlow power flow (5) in the above PV maximum access capacity evaluation model involves complex quadratic terms, which makes the optimization problem a non-convex model.This paper has carried out a series of reasonable simplifications of the DistFlow power flow equation to achieve the purpose of the convex of the original optimization problem.First, it is assumed that the active power transmitted on the line is greater than the loss.Second, it is assumed that the voltage of the node is much greater than the voltage deviation between the individual nodes.The main purpose of power flow calculation is to study the steady-state operation of the power system, and thus the deviation of the power flow equation caused by the above simplification can be ignored.At this time, the voltage amplitude V i in the objective function can be expressed by the voltage amplitude V i of the balance node, and the quadratic term in the DistFlow power flow equation can also be ignored.Therefore, the evaluation model of the maximum PV access capacity of the system can be simplified as follows: : .
where U=V.After the above simplification and variable substitution, the quadratic constraint in the original problem is transferred into a linear constraint.At this point, the convexity of the model is completed, and the optimal solution to the problem can be obtained by the distributed optimization algorithm.

Synchronous ADMM distributed solution
ADMM provides a framework for solving optimization problems with linear equality constraints, which is suitable for solving distributed convex optimization problems.The core of the algorithm is the augmented Lagrange method of the original duality algorithm.The Lagrange function is an optimization problem that solves multiple constraints, and this method can solve an optimization problem with n variables and k constraints.The augmented Lagrange method in the original duality method is the Lagrange method with the addition of a penalty term to make the algorithm converge faster.
where n is the number of iterations; λ a and λ b are the dual variables of Region a and Region b, containing 4 elements respectively, and each element corresponds to the state of the coupling branch; For example, the state of the coupling branch of Region a can be expressed as X a,ij ={P a, ij, Qa, ij, Ua, i, U a,j }; ρ is the penalty parameter of the ADMM algorithm.X n Ka and X n Kb select the average value of the branch coupling state obtained by taking the nth iteration of Region a and Region b, which is as follows: , , After identifying sub-optimization problems in each region, a distributed solution of the model is carried out in each region according to the flow chart shown in Figure 2. The maximum number of iterations n max is shown in Figure 1, which is set to 300 in this paper.Specific steps are as follows: Reference value of region a   2) The fixed reference value of the next iteration is calculated.It is the average value of the coupled branch state, as shown in Equation (20).
3) Each region updates the dual variable separately, and the equation is as follows:   The criterion of algorithm convergence is that the coupled branch state variables in adjacent regions are equal, that is, the boundary residuals tend to be zero.Boundary residuals are defined as follows: where δ is convergence precision.When the above conditions are met, the algorithm converges and the iteration ends.

Access testing for distributed photovoltaics
In this section, all nodes are tested as access points of distributed PV, and the access capacity of each node is calculated as shown in Figure 3 when the power factor is 1.The total capacity of distributed PV accessed by the system is 4.88 MW, and the total active power loss of the system is 76.87 kW.Table 1 shows the access capacity and active power loss of distributed PV under different power factors, and Figure 4 shows the node voltage when the access capacity of distributed PV of each node is optimal under different power factors.As can be seen from the chart, in the case of a system without voltage overtripping, the smaller the power factor is, the larger the distributed PV capacity is acceptable to the system, the lower the active power loss on the line is, and the voltage level of each node has also been significantly improved.From Table 2, it can be seen that compared with the centralized calculation, the ADMM distributed optimization algorithm improves the access capacity of distributed PV to a certain extent.Meanwhile, it can reduce the active power loss on the line.

Conclusion
Based on the ADMM algorithm, this paper studies the access level of distributed PV in the distribution network system and analyzes its constraints.This study minimizes the volume of interactive data between systems, mitigates the demands on data transmission hardware, comprehensively enhances the photovoltaic access capacity, and reduces active power network losses.Optimizing the access point and access capacity of distributed PV systems under varying operating conditions, we take the IEEE-33 node system as a case study.The ADMM algorithm has been demonstrated to enhance the access capacity of distributed photovoltaic systems and mitigate active power losses along transmission lines.
,i indicates the access capacity of distributed PV of the i node; S max DPV indicates the maximum capacity of distributed PV allowed to be accessed by the node.5) Power flow reverse constraint: where P re indicates the power transmitted by the low-voltage side and high-voltage side of the main transformer of the distribution network.
In this section, two partitions (a and b) are taken as examples to introduce the solving process of ADMM in detail.The sum of the augmented Lagrange function L a (x a , X n Ka ,λ n a ) and L b (x b , X n Kb ,λ n b ) corresponding to the objective function of the region a and the region b sub-optimization problem is established, and the appropriate transformation is performed.The transformed augmented Lagrangian function is as follows:

Figure 3 .
Figure 3.The access capacity of each node.Figure 4. The voltage curve during decentralized access.

Figure 4 .
Figure 3.The access capacity of each node.Figure 4. The voltage curve during decentralized access.

Table 1 .
Disposable Access to Optimal Capacity and Contribution Loss.

Table 2
shows distributed PV access capacity and active power loss obtained by centralized solution and ADMM distributed solution under different power factors λ.

Table 2 .
Distributed Access Capacity and Active Power Loss Under Different Calculation Methods.