Table of contents
1.2 Innovations and related work
4 Case and Matlab code implementation
1 Overview
Power flow is a nonlinear problem that needs to be solved by Newton's method for a DC microgrid with constant power terminals. In this paper, sufficient conditions for the quadratic convergence of Newton's method in such meshes are proposed. The classical Newton method as well as the approximate Newton method are analyzed in both master-slave and island operation with droop control. The requirements for convergence starting from voltages close to 1 pu and for the existence and uniqueness of the solution are presented. Computational results complement this theoretical analysis.
Microgrids are expected to play a fundamental role in future smart grid concepts. In particular, DC microgrids are gaining increasing attention due to their advantages in efficiency, reliability, and controllability. DC microgrids can achieve high efficiency and simplified control due to the absence of reactive power or frequency control; high reliability due to the ability to operate in islands; and since many generation and storage technologies are already DC (such as solar photovoltaic , battery), so simple integration is possible. Also, most household appliances can be adapted for DC operation. In a typical DC microgrid, power electronic converters can operate as constant current or constant power. In the latter case, the model of the grid becomes non-linear, requiring a static state analysis with power flow algorithms. This problem is non-linear/non-convex and requires numerical algorithms to solve it. Of course, due to the nonlinear nature of the problem, the convergence of such algorithms is not always guaranteed. An algorithm may even diverge or converge toward an unrealistic solution.
Therefore, it is necessary to establish precise conditions under which the power flow algorithm converges to a unique and realistic solution. On the other hand, there are two main control methods for DC microgrids, namely master-slave control and vertical flow control.
In master-slave control , the converter fixes the voltage across the grid; this is the most common operation for grid-connected microgrids.
For island mode operation, the power in the grid is modified by droop control to achieve a stable equilibrium point.
Both modes of operation require power flow algorithms. It is important to note that there is linearization in AC grids, also known as DC power flow. The name comes from the analogy between the angle in an AC grid and the voltage in a linear DC grid. This article is not about these types of analogies or linearizations. Due to the presence of power converters, we are interested in grids that are practically DC and non-linear.
1.1 Power flow in DC grid
Power flow in DC microgrids has some similarities to the AC situation, therefore, we refer to related work in AC power flow in the literature review below. However, there are some important differences worth highlighting : first , the problem still exists in the set of real numbers, allowing analysis with real numbers rather than complex numbers (note, for example, that the dc equations are analytic, while the ac equations are not). Second , grids can operate in islands (i.e., without slack nodes), so it is necessary to include droop control in power flow problems. Third , DC grids have properties that do not exist with their AC counterparts, e.g., are always diagonally dominant,
but monotonic (note that AC grids with high capacitive effects may have non-existent
pairs corner dominant). Therefore, power flows in a DC microgrid require analyzes that cannot be extrapolated from the AC situation.
1.2 Innovations and related work
In this paper, we analyze the convergence of Newton's and approximate Newton's methods for power flow in DC microgrids. This analysis is important for two main reasons: First , the power flow algorithm needs to be executed many times during the operation and planning of the microgrid. In operation, guaranteed convergence is a desirable feature in a smart grid environment with less human oversight. In planning, power flows may be part of other algorithms, especially in heuristic optimization problems. Therefore, the quadratic convergence and uniqueness of the solution are key conditions. Second , the power flow gives the equilibrium point of the grid dynamic model. Finding the balance is the first step in most studies related to the dynamics and stability of microgrids.
The convergence of the Gauss-Seidel method in master-slave operations was recently analyzed. Here we demonstrate this result by defining the exact conditions for the convergence of Newton's method in master-slave and island operations. In addition, we analyze the convergence of an approximate Newton's method. This approach is similar to the fast decoupling load flow of AC grids. We use Kantorovitchs theory for the former and contraction mapping theory for the latter. Although these theories are classic results in practical analysis, no one has used them to analyze these problems before. As expected, Newton's method has quadratic convergence, albeit in a small basin of attraction, while the approximate Newton's method has a guaranteed linear convergence.
1.3 Structure of this paper
The rest of the paper is organized as follows: Section II describes the grid model in master-slave and island operation. The third section analyzes Newton's method, and the fourth section analyzes approximate Newton's method. Afterwards, numerical simulations are performed, and then conclusions and references are drawn.
2 Math/Network Models
2.1 Master-slave operation
2.2 Island operation
3 Newton's method
Numerical simulations are carried out in the DC microgrid shown in Fig. 3, and its parameters are given in Table I. Node 1 is voltage controlled, while nodes 3, 7, 10 and 14 are step nodes (thus eliminated by Kron reduction). Experiments show that assumption A2 is easily satisfied in our test system (ie |Pm ax | < 1/ρ = 40.9 pu).
Figure 3. Diagram of DC microgrid with 21 nodes
Table 1 Parameters of the proposed test system
This is because a well-designed DC microgrid has a small resistance value for high efficiency. Power flows are calculated in four cases: using Newton's method and approximate Newton's method, master-slave operation and island operation (meaning the switch in 3 is open).
The values of ‖F (vk )‖ are given in Fig. 4 for each iteration. We can see that the first iterations of Newton's and Approximate Newton's methods are the same. However, with the implementation of the algorithm, the error of Newton's method is shown in Figure 1. Table 2. Summary of Convergence Measures for Master-Slave Operations. Table 3. Summary of Convergence Measures for Island Operations. In both island and master-slave operations, the error decreases quadratically, while the error of the approximate Newton method decreases linearly. This behavior is consistent with Propositions 1 to 4; in fact, these conclusions, drawn from numerical simulations, can be drawn directly using these prepositions. Tables II and III summarize these results. It should be noted that the conditions proposed in this paper are sufficient but necessary. Therefore, our results are conservative but realistic convergence bounds. This means that we can guarantee the convergence of the method if the conditions are met. However, if certain conditions are not met, it does not mean that the algorithm diverges. This is the case for the approximate Newton method for islanding operations, which in our example does not satisfy the conditions of Proposition 4 (see Table III), but, as we can see in Figure 4, the algorithm achieves convergence .
Figure 4. Error as a function of iteration
Table 2 Summary of Master-Slave Operation Convergence Measures
Table 3 Summary of Convergence Measures for Intra-Island Operations
