Abstract
For enhancing the stability of the microgrid operation, this paper proposes an optimization model considering the small-signal stability constraint. Due to the nonsmooth property of the spectral abscissa function, the droop controller parameters’ optimization is a nonsmooth optimization problem. The Sequential Quadratic Programming with Gradient Sampling (SQP-GS) is implemented to optimize the droop controller parameters for solving the nonsmooth problem. The SQP-GS method can guarantee the solution of the optimization problem globally and efficiently converges to stationary points with probability of one. In the current iteration, the gradient of the nonsmooth function can be evaluated on a set of randomly generated nearby points by computing closed-form sensitivities. A test on the microgrid system shows that the optimality and the efficiency of the SQP-GS are better than those of the heuristic algorithms.
1. Introduction
Microgrid is an alternative solution to integrate distributed energy resources (DERs). It concludes a cluster of loads, distributed generation (DG), and energy storage systems, just like a small-scale power supply network [1, 2]. Due to the negligible physical inertia of power-electronic converters, the microgrid operation is susceptible to random disturbances, such as variability of DG and random load fluctuations [3], like charging for electric vehicles [4], especially in an islanded mode. The small-signal stability of an islanded microgrid is critical for its reliable operation. In an islanded microgrid, droop control, following a predefined droop coefficient, is widely used to maintain the proportional load sharing between DERs [5]. The small-signal stability of the microgrid in islanded mode is strongly affected by the droop controller parameters. Therefore, it is essential to optimize the controller parameters to enhance the small-signal stability of an islanded microgrid.
Particle swarm optimization (PSO) method is often used to optimize the controller parameters of the microgrid [6–9]. In [6], the maximum of the real part of eigenvalues, also called the spectral abscissa, was minimized to enhance the stability in grid-connected mode. On the other hand, a nonlinear time-domain simulation-based objective function was employed to minimize the error in measure power in islanded mode. The study in [7] proposed a method for optimizing the controller parameters of dynamic droop control for a microgrid in islanded mode and used a spectral abscissa objective function to consider small-signal stability. The study in [8] tried to optimize proportional-integral (PI) current controller parameters considering three design criteria and multiple operating conditions which include grid-connected mode and different islanded modes by a weighted sum of multiple objective functions. The work in [9] proposed an optimization model with a nonlinear time-domain simulation-based objective function for a microgrid in islanded mode. In addition to PSO methods, the study in [10] used the Genetic Algorithms (GA) method to optimize the controller parameters by minimizing the power difference-integrating during the changing process between grid-connected mode and islanded mode.
All the studies above adopt the heuristic algorithms to optimize the control parameters of the microgrid due to the nonsmooth property of the optimization model. Although those algorithms can obtain fast and feasible solutions, they are hard to deliver an optimal solution to the vast majority of cases [11, 12]. The solutions derived by the heuristic algorithms are not deterministic and reproducible. These disadvantages will make the debugging and correctness checking difficult, impeding the industrial application. Concerning the optimization model, the study in [6] did not consider the small-signal stability in islanded mode. As for considering stability, the small-signal stability analysis was performed in [8–10] to obtain the optimum ranges of the controller parameters, while the study in [7] employed spectral abscissa as an objective function to enhance small-signal stability better. The study in [8] considered multiple operating conditions in the optimization model, which may cause the certain index of the design criteria to be poor due to the considering way.
To tackle the weakness mentioned above, this paper proposes an optimization model considering the spectral abscissa in multiple islanded mode operating conditions. Besides, an SQP method combined with GS (SQP-GS) [13] is presented to solve the proposed optimization model. Therefore, droop controllers with optimized parameters can enhance the small-signal stability of the microgrid in islanded mode. As for the SQP-GS method, it had been adopted to optimize the parameters of PSSs and PODs [14].
The major contributions can be summarized as follows:(1)Based on SQP-GS, a mathematical programming method is presented to solve the proposed nonsmooth optimization model and obtain an optimal and deterministic solution without any convergence problem.(2)The small-signal stability in different operating conditions is enhanced using a nonsmooth objective function in the optimization model. It can ensure that optimized parameters are robust under multiple operating conditions.
The remainder of this paper is organized as follows: Section 2 presents the small-signal stability model of the microgrid. Section 3 proposes the optimization model for optimizing the droop controller parameters of the microgrid. Section 4 introduces the SQP-GS method to solve the optimization model. In Section 5, the test on the microgrid system under multiple operating conditions is presented. Conclusions are drawn in Section 6.
2. The Small-Signal Stability Model of the Microgrid
The small-signal stability model of the microgrid can be divided into three parts: inverter, network, and load. These small-signal stability models are deduced in reference frame [10, 15].
2.1. The Model of the Inverter
The small-signal stability model of the inverter contains two parts: the droop controller and LC filter and coupling inductor. Moreover, the dynamic model of the droop controller consists of the power controller, voltage controller, and current controller.
2.1.1. The Model of the Power Controller
The real and reactive power can be obtained when the instantaneous active and reactive power, calculated from the measured output voltage and output current in frame, passes through the low-pass filters. The small-signal stability model can be given bywhere is the state variable with respect to the time, is the state variable or the algebraic variable after linearization, are the real and reactive power, respectively, are the output voltage and current in frame, respectively, and is the cut-off frequency of the low-pass filters.
The frequency and voltage can be set by the droop gain; and the small-signal stability model can be given bywhere is the frequency, are the droop gains related to frequency and voltage amplitude, respectively, and are the output voltage reference in frame, respectively.
The first frame of the inverter is set as the common frame; and all the variables from other inverter frames can be converted to the common frame. The small-signal stability model is given bywhere is the frequency of the common frame taken by the first inverter and is the angle between an inverter frame and the common frame.
2.1.2. The Model of the Voltage Controller
The voltage controller contains the PI regulator and the feed-forward terms; and the small-signal stability model of the voltage controller can be given bywhere are and , respectively, are the output current reference in frame, respectively, are the integral and proportional gains of the PI regulator in voltage controller, respectively, is the nominal frequency, is the feed-forward gain, and is the per-phase capacitance.
2.1.3. The Model of the Current Controller
The current controller includes the PI regulator; and the small-signal stability model of the current controller can be given bywhere are and , respectively, are the output inverter voltage reference in frame, respectively, are the sampled filter current in frame, respectively, are the integral and proportional gains in PI regulator of the current controller, respectively, and is the per-phase inductance.
2.1.4. The Model of the LC Filter and Coupling Inductance
The LC filter and coupling inductance can remove the harmonic wave near the switch frequency; and the small-signal stability model of the LC filter and coupling inductance are given bywhere are the bus voltage in frame, respectively, are the inverter voltage in frame, respectively, are the steady state value at the initial operating point, and are the resistance, inductance, and capacitance in the microgrid, respectively.
2.1.5. The Model of the Individual Inverter
Through the transformation matrix, other inverter variables can transfer to the common frame. The linearization models of the transformation matrix are obtained:where are the output current and bus voltage of the th inverter converted to the common frame, respectively, and is the angle between the inverter frame and the common frame.
Combined by the linearization models (1)-(22), the linearization state equation and the linearization output equation of the th inverter in the common frame are given bywhere .
2.2. The Model of the Network and Load
The linearization models of the network and load are obtained:where , , and are the numbers of the lines and loads, respectively.
2.3. The Model of the Microgrid
To define the node voltage clearly, the virtual resistor is assumed between each node and ground. The small-signal stability model is given bywhere , is the number of nodes in the microgrid, maps the inverter connection points onto network nodes, maps the load connection point onto the network nodes, and maps the connection lines onto nodes.
Combined with (23)-(27), the small-signal stability model of the microgrid is given bywhere , is the number of inverters, and is the state matrix of the microgrid model.
3. The Proposed Optimization Model
The dynamic system is stable in small-signal stability sense when all the real parts of the eigenvalues of are negative. The largest real part of all eigenvalues is called the spectral abscissa , which stands for the power system’s stability margin. The value of the spectral abscissa is smaller, and the rate of decay is faster when the system suffers a small disturbance. Therefore, the dynamic system is more stable in the small-signal stability sense. Based on the above small-signal stability model of microgrid, the eigenvalues of can be obtained. Therefore, the spectral abscissa can be given bywhere represents all eigenvalues of . is the real part of all eigenvalues, and is the eigenvalue with largest real part.
Since the load model is contained in the small-signal stability model of the microgrid, the variation of the load can affect the spectral abscissa of the microgrid exceedingly. Also, load condition variation, such as random load fluctuation, frequently happens in the microgrid operation. Although the microgrid can work well at one load condition, also called operating conditions, it cannot guarantee the small-signal stability at other operating conditions. Therefore, it is crucial to consider multiple operating conditions in the proposed optimization model to achieve better robustness.
The spectral abscissas of different operating condition can be represented as , which stands for the spectral abscissa of the th operating conditions, , and is the number of the operating conditions.
In [7, 9, 15], through the eigenvalue analysis, the dominant eigenvalues near the imaginary axis are largely sensitive to and the eigenvalues of medium frequency are largely sensitive to , , , and . The value of , even the system stability, will vary with droop controller parameters.
Therefore, the critical droop controller parameters of each inverter are considered as the variables to be optimized:where is the total number of inverters.
According to (29) and (30), the spectral abscissa is actually an implicit function of the controller parameters:
The spectral abscissas of the worst operating condition, also called the largest spectral abscissa, can be represented as :
According to (32), the objective function can be given to achieve maximum stability of the microgrid in the worst operating condition. This objective function also ensures the maximum stability of microgrid in other operating conditions:
The constraints of controller parameters of the th inverter can be listed as the constraint of optimization model:
However, is not locally Lipschitz. Fortunately, has been proved to be locally Lipschitz and continuously differentiable on open dense subsets of [16], which means that it is continually differentiable almost everywhere. Also, its gradient can be obtained by calculating spectral abscissa sensitivities. Besides, the small-signal stability constraint with the spectral abscissa of multiple operating conditions can be added to the optimization model to ensure that the microgrid works well in the multiple operating conditions; and, due to the small-signal stability constraint in the optimization model, minimizing is equivalent to minimize .
Through the above discussion, the complete optimization model can be given bywhere is the upper limit of spectral abscissa; and the constraints of the optimization model can be listed as follows:where and are the upper and lower limits of controller parameters and .
The spectral abscissa of multiple operating conditions moves toward the left as far as possible to ensure maximum stability through the optimization model. Thus, the microgrid with optimized controller parameters can operate stably in multiple operating conditions.
According to (31), the spectral abscissa cannot be deduced as an analytical and nonlinear or linear expression of controller parameters; and the spectral abscissa with respect to the parameters has been proved to be nonsmooth [17]. The optimization model in (35)-(42) is a nonsmooth programming problem, which cannot be tackled by the methods for linear or nonlinear programming. It is hard to solve the nonsmooth programming in mathematical programming until the breakthrough in [13].
4. The SQP-GS Method for Solving the Optimization Problem
4.1. The SQP-GS Method
According to the Clarke theorem, if any subgradient is zero, which belongs to the subdifferential of , there is a stationary point in this nonsmooth function [18].
The nonsmooth problem is hard to be solved until a GS method was proposed in 2005 [18], which can approximate the subgradients for minimizing a nonsmooth objective function. In [18], if the nonsmooth function or constraints are locally Lipschitz continuous, a series of the independent and random gradients of sample point near the nonsmooth point can be evaluated as convex hull to express the subgradients of .
The traditional SQP method can solve a QP subproblem to obtain a descending direction. It only solves the smooth problem but fails for the nonsmooth problem. In 2012, the Sequential Quadratic Programming with Gradient Sampling (SQP-GS) method was proposed in [13]. It can converge globally to stationary points with probability of one when the objective function or constraints are locally Lipschitz and continuously differentiable on open dense subsets of .
The SQP-GS method is developed to solve the optimization model in the following form:where the objective function and the inequality constraint functions .
The SQP-GS method can compute a search direction in the th iteration by solving a QP dual subproblem:where is the vector of variables in the th iteration, is the approximate Hessian of the Lagrangian optimization model of (43), is a penalty parameter, and and are the approximated minimum-norm elements of the subdifferential of and , respectively, and are the vectors of the sample points of and , respectively, andare sets of independent and identically distributed random sample points sampled from the closed ball centered at , and
For reflecting the iteration process, infeasibility value can be defined as
The following model reduction is also defined in terms of primal and dual infeasibility:
When the SQP-GS method converges to the optimal solution, these optimality certificates, such as , , can be 0. The sample size can be set to 0 when the SQP-GS method is solving a smooth function, in which case the SQP-GS method reduces to the general SQP method [14].
4.2. Gradient Evaluation of Spectral Abscissa Function
The gradient of the spectral abscissa is crucial for implementing the SQP-GS method; and the spectral abscissa with respect to the controller parameters, also called the spectral abscissa sensitivity, reflects the variation of the variable on the impact of system stability. Therefore, the gradient of spectral abscissa can be represented as spectral abscissa sensitivity. The spectral abscissa sensitivity can be calculated by the numerical differentiation method, which is numerical spectral abscissa sensitivity.
The numerical spectral abscissa sensitivity can be obtained bywhere is the spectral abscissa of the state matrix at the equilibrium point, is the small quantity of the variable variation, is the variable, and is the spectral abscissa of perturbed state matrix.
As for different variables, the small quantity range is not defined clearly. It causes the accuracy of numerical spectral abscissa sensitivity to be uncertain. When calculating the spectral abscissa sensitivity for each variable, the eigenvalue of needs to be recalculated more than once in each iteration. It decreases the efficiency of the optimization.
Hence, the closed-form eigenvalue sensitivity can be an alternative option [19]. It is a mathematical eigenvalue derived at an equilibrium point; and the eigenvalue sensitivity with respect to variable can be expressed aswhere is the state matrix of microgrid at the equilibrium point, and are the left and right eigenvectors of at an equilibrium point, respectively, and can be obtained on matrix calculus.
The closed-form spectral abscissa sensitivity can be expressed by
In (56) and (57), , , and are required to calculate only once in each iteration. Besides, the closed-form spectral abscissa sensitivity is more accurate than numerical spectral abscissa sensitivity in each iteration. Therefore, the closed-form spectral abscissa sensitivity is used to compute the gradient of the spectral abscissa.
4.3. Solving the Droop Controller Parameters Optimization Problem by SQP-GS
Before employing the SQP-GS method to solve a QP subproblem to get a descending direction, the gradients of objective function and constraints in the optimization model with respect to variable should be obtained. The constraint functions in (37)-(42) are smooth; and the gradient of these constraints can be obtained easily. Therefore, the sample size reduces to zero when the SQP-GS method solves the smooth function. Objective function (35) and the small-signal stability constraint (36) are nonsmooth. The state matrix in (28) for each point in is established. The most critical eigenvalue and the corresponding left and right eigenvectors and for each can be obtained in each iteration. According to (56), the most critical eigenvalue sensitivity with respect to the th variable for each point is calculated. Therefore, the gradient with respect to the th variable in (57) for each point can be obtained. The initial values of droop controller parameters need to be set in the range of controller parameters limit. Otherwise, they may not converge. The Algorithm 1 is given as follows [14].
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5. Test Variation
The following test system in Figure 1 is used to demonstrate the effectiveness of the proposed SQP-GS method. The simulations are carried out on a computer with a four-core 3.70 GHz processor and 16 GB RAM. Also, the SQP-GS method is implemented in MATLAB R2012a. As for the SQP-GS method, those parameters are selected as follows: , , , , , and .
5.1. Microgrid System Parameters and Four Operating Conditions
We design four operating conditions of the microgrid to verify the effectiveness of the SQP-GS method. The test system in Figure 1 contains three inverters, two connection lines, and two loads. Load 1 is in bus 1, and the other one is in bus 3; and the microgrid system parameters are given in Table 1. Also, the initial condition of the microgrid system is given in Table 2.
To ensure that the droop controller parameters are robust after optimization, similar to multiple operating conditions in [6], we design different operating conditions. They are shown in Table 3. of 25 and 20 per phase is equal to the resistive load of and [13]. The objection of this paper is to investigate the optimized parameters in droop controllers that can enhance the small-signal stability of all four operating conditions.
The limit of controller parameters can be shown in Table 4. It is selected based on the trace of the eigenvalues varying with the controller parameters variation [9, 10]. When the droop controller parameters vary in the limit of the variable, the critical eigenvalue will move within the stable region. Meanwhile, the stability of the microgrid operation can be guaranteed. According to the above rule, we can obtain the value domain of the variables from the root locus of the critical eigenvalues with the variable variation.
5.2. Optimization Method Verification
In the last subsection, the four operating conditions have been designed; and we implement the SQP-GS method in MATLAB to optimize the droop controller parameters in microgrid.
From Table 5, we can see the comparison of spectral abscissas between the unoptimized and the optimized values in the four operating conditions. In Case 1 and Case 3, the unoptimized spectral abscissa is 72.3025 and 72.3112, respectively. It stands that the microgrid system is susceptible to encounter oscillation when random load fluctuation occurs. After the optimization, the optimized spectral abscissa is −16.1651 and −16.1212, respectively. It proves that the small-signal stability of the microgrid in four operating conditions is enhanced through SQP-GS method optimization. Therefore, it demonstrated that the SQP-GS method is effective for optimizing the droop controller parameters.
The original parameters and the optimized parameters of the controller are given in Tables 6 and 7. As for the original parameters, the droop controller parameter is designed by the traditional method [15]. Also, the voltage and current controller parameters are selected based on the small-signal stability analysis [9, 10]. For the closed-form spectral abscissa sensitivities of , , , and are fairly small, these controller parameters stay the same nearly in the iteration process.
From Figure 2, there are two eigenvalues with the positive real part in the original system of Case 1. One positive eigenvalue is 72.3025, and the other one is 15.9420. It means that the original system is unstable. After optimizing the controller parameters by the SQP-GS method, all the optimized eigenvalues lie on the left plane. The spectral abscissa of Case 1 is −16.1651. It proves that the stability of the system is improved after the optimization.
In Figure 3, the spectral abscissa of Case 1 decreases in the iteration process with the SQP-GS method. After 83 iterations, the spectral abscissa of Case 1 varies from 72.3025 to −16.1651. It means that the microgrid system with optimized droop controller parameters is stable in the small-signal stability sense.
In Figure 4, the upper limit of the spectral abscissa is varied from −15 to −16.061 in the iteration process. It means that the spectral abscissas of all the cases are less than −16.061. The stability margin of the system is improved.
Figure 5 depicts the sampling radius variation, which is one of the convergence indicators in SQP-GS method. After 83 iterations, the sampling radius decreases from to ; and it proves that the SQP-GS method arrives at convergence.
5.3. Comparison with GA Method
The SQP-GS and GA methods are called nine times, respectively, for optimizing the droop controller parameters; and the mean value of , spectral abscissa in Case 4, and the computing time are listed in Table 8. As the table shows, the mean value of optimized by the GA method is −15.31. Besides, optimized by the SQP-GS method is −16.06. In the term of the spectral abscissa, the best value optimized by the GA method is −16.05, while the value optimized by the SQP-GS method is −16.12. As for the computing time, the longest time for the GA method to converge is 0.709 hours, while the shortest time is 0.116 hours. However, the max computing time of the SQP-GS method is 0.102 hours. These results demonstrate that the optimality and efficiency of the SQP-GS method are better than those of the GA method. Also, Table 8 shows that the SQP-GS can be globally convergent to stationary points with probability of one.
6. Conclusion
This paper proposes the SQP-GS method, which is a mathematical programming method to optimize the spectral abscissa of the microgrid. Combined with the proposed optimization model and SQP-GS method, the microgrid system is better robust in different operating conditions. Moreover, the SQP-GS method’s effectiveness and efficiency are better than those of the GA method.
Data Availability
The microgrid system parameters and the initial condition of microgrid system which support the findings of this study are available in [15].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant no. 51967001, in part by the Natural Science Foundation of Guangxi under Grant no. 2018JJA160164, in part by Guangxi Special Fund for Innovation-Driven Development under Grant no. AA19254034, and in part by Guangxi Key Laboratory of Power System Optimization and Energy Technology Research Grant.