Robust Stability of DC Microgrid Under Distributed Control

Compared with AC microgrid, DC microgrid has attracted more and more attention due to their high reliability and simple control. In this paper, we analyze the existence and stability of equilibrium of DC microgrid under distributed control. Firstly, the power-flow equation of the DC microgrid with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> DGs and <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> CPLs is built. On this basis, under the limitation of currents of DGs, the sufficient solvability conditions are obtained based on Brouwer’s fixed-point theorem. Secondly, we build the small-signal model to forecast the system qualitative behavior around equilibrium. The eigenvalue analysis based on inertial theorem provides the analysis basis for system stability, then we obtain the robust stability condition based on the properties of a special matrix. The obtained stability conditions provide a reference for establishing a stable DC microgrid. Lastly, simulation results verify the correctness of the proposed theorems.

instability and the loss of equilibrium (voltage collapse) [5]. 23 Therefore, the control objectives of DC microgrid can be 24 briefly summarized as the following three objectives: current 25 sharing, voltage recovery and maintaining system stability. 26 For these three objectives, scholars at home and abroad have 27 done a lot of research and made great progress. In order to 28 realize current sharing and voltage recovery of DC micro-29 grid, decentralized control and distributed control are usually 30 adopted. The traditional droop control is a typical decen-31 tralized control, which can realize current sharing roughly. 32 The associate editor coordinating the review of this manuscript and approving it for publication was Yilun Shang.
However, this control method has two drawbacks: voltage 33 drop and biased current sharing [6], [7]. 34 To solve this problem, scholars have proposed a distributed 35 control strategy based on consensus algorithm. The core idea 36 of this method is to achieve global current sharing and voltage 37 recovery by keeping consistent with adjacent DGs [8], [9]. 38 However, in the above study, only the stability under con-39 ventional resistive load is considered. DC microgrid usually 40 contains a large number of CPLs which easily lead to the 41 instability of the system. Therefore, the design of DC micro-42 grid should overcome the instability of CPLs. 43 Aiming at the stability problem of DC microgrid under 44 current sharing control, the research contents can be divided 45 into four categories according to the differences of topol-46 ogy and control method: Star DC microgrid (Multiple DGs 47 with single CPL) under droop control; Star DC microgrid 48 under distributed control; Meshed DC microgrid (Multiple 49 DGs with multiple CPLs) under droop control; Meshed DC 50 microgrid under distributed control. At present, the stability 51 analysis of the first three types of systems has been reported, 52 the fourth type is more complex and rarely reported. 53 Small-signal analysis is a typical method to analyze sys-54 tem stability when subject to small disturbances. Existing 55 approaches for the small-signal stability of power elec-56 tronic systems are mainly based on impedance-based and 57 to establish a reliable microgrid. CPLs will lead to system 114 unstable due to its negative impedance characteristic, and 115 the access of distributed control will bring difficulties to the 116 analysis of system stability. In this study, we supplement the 117 existing research about the solvability conditions of power 118 flow equation and introduce a distributed control aiming 119 at current sharing and voltage regulation. Compared with 120 the stability analysis through impedance-based method in 121 [10], [12], and [11], the stability analysis in this paper can 122 determine the stability of the entire system. What's more, 123 high precision current sharing and voltage regulation are 124 considered. Compared with decentralized control in [5], the 125 topology of DC microgrid in this paper is more general, and 126 the distributed control method in this paper can achieve more 127 accurate current sharing.

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The main contributions of this paper can be summarized as 129 follows:

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• A distributed control method aims at current sharing 131 and voltage regulation is proposed, which can overcome the 132 instability of CPLs. On this basis, the sufficient conditions of 133 the existence of equilibrium considering current constraint is 134 obtained based on Brouwer's fixed-point theorem.

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• The small signal model of the system with n DGs and m 136 CPLs under distributed control is obtained. Through the anal-137 ysis of system Jacobian matrix based on inertia theorem, the 138 robust stability condition independent of the loads real-time 139 information is obtained.

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• We give a stability analysis method for a class of matrix 141 AB + F (A is semi-positive definite, B has negative eigenval-142 ues, and F is a rank-one matrix), which usually appears in the 143 stability analysis of DC microgrid under distributed control. 144 The rest of this paper is organized as follows: Section II 145 introduces the distributed control framework. The sufficient 146 condition for the existence of equilibrium of the system is 147 obtained in section III. The stability analysis and the robust 148 stability condition are introduced in section IV. In section V, 149 simulation results verify the correctness of the proposed 150 theorems. Conclusions are made in section VI. Section VI 151 introduces the preliminaries and notations used in this paper. 152

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The graph theorem has been introduced in [15], and this 155 omitted here. It should be noted that for a connected graph, 156 its Laplacian matrix £ has at least one spanning tree, and 157 ker(£) = span(1 n ), i.e., £1 n = 0 n .  According to Kirchhoff's and Ohm's laws, we have where u S i , δi i and δu i represent the output voltage, current-186 correction term and the voltage-correction term, respectively, 187 for the ith DG.

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The current-correction term is designed as follows: where q i is the current sharing proportionality coefficient, To realize voltage regulation, the voltage-correction term 196 is expressed as follows where v ref andũ S = 1 n 1 T n u S represent the reference voltage 199 and average voltage of DGs, respectively. Current sharing and voltage regulation are two objectives of 209 the DC microgrid, which are considered in this paper.

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Objective 1: In the DC microgrid, current sharing means 211 the output current of DGs is divided equally in proportion in 212 the steady state, that is, 213 i 1 : i 2 : · · · : i n = q 1 : q 2 : · · · : q n (7) 214 Objective 2: In the DC microgrid, average voltage regula-215 tion means in the steady state, following equation holds Next, we will prove that in the steady state, two control 218 objectives are realized.

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Proof: Above all, following equation is satisfied when 220 system is stable Clearly, Q is invertible, then there is a matrix Q −1 such that 223 graph, then £ has a zero eigenvalue and its corresponding 226 eigenvector is 1 n , that is, 1 T n L = 0 T n .

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Left multiplied by 1 T n , (10) becomes By substituting (8) into (9), we obtain Q£Qi S = 0 n , 232 i.e. Qi S ∈ span(1 n ), then (7) is obtained. In this section, we analyze the properties of the sub-network, 241 which is necessary for the analysis of power-flow equation.

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The main results are as follows.

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Theorem 1: For a microgird with strongly physical network 244 and sub-network, the following statements hold: Proof: For the statement (1) is also a symmetric matrix, i.e., E T B LL E can be written as Thus, the sub-network induced by load nodes is divided 255 into two independent networks which is contrary to the 256 assumption that the sub-network is strongly connected. Then 257 B LL is a irreducible matrix.

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According to Lemma 2, since B LL is an irreducible The statement (1) is proved.
In steady state, u S is a known positive constant vec-279 tor. Let u * S denote the steady-state value of u S . Define 280 (12) is equivalent to 282 the following form To obtain the solvability condition of the power flow equa-285 tion, two mainly methods are proposed, namely ''weighted 286 sum of sub-equations'' [11], and ''fixed point theorem'' [12]. 287 The main ideas of these methods are as follows. For the first 288 method, if there exists a diagonal matrix H such that (15) is 289 not solvable, then (14) has no solution.
In fact, (15) is the sum of sub-equations of (14), conse-292 quently, (15) must be solvable if (14)  For the second method, if there is a compact set D such 299 that f (u) is a continuous self-mapping, contraction mapping 300 and concave increasing self-mapping on D, according to 301 Brouwer's, Banach's and Taski's fixed-point theorem, there 302 is a u * ∈ D such that f (u * ) = u * .

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In [10], the compact set D is constructed as By invoking Banach's fixed-point theorem, there exists a 307 unique vector u * ∈ D that is a solution to (5). Considering unique vector that is a solution of (14). Moreover, a stronger 318 explicit solvability condition than (17) is obtained as follows where η max and η min are the maximum and minimum Perron 321 eigenvalue of A, respectively.

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The results in (19) are also obtained in [21] based on nested Proof: Clearly, S is a compact convex set. If G(u L ) is a 368 self-mapping in S, according to Brouwer's fixed-point theo-369 rem, (13) admits a unique solution in S. 370 Firstly, we prove G(u L ) ∈ S for any u L ∈ S. 371 Since M > 0, then for any u L ∈ S, G(u L ) < v ref , and when 372 condition (26) holds we obtain Then we obtain Then we have for any u L ∈ S, G(u L ) ∈ S, i.e., G(u L ) is 380 self-mapping. where R L = u 2 L p is the equivalent impedance matrix of loads. 391 Linearizing (1) and (6), the following is obtained Then, we obtain i S = B eq u S , where B eq = B SS − 394 Combine (27) and (28), the dynamic of the system is 396 described as follows

B. ROBUST STABILITY CONDITIONS
Jacobian matrix of the system is given as follows B eq is N 0 − matrix. Moreover, we obtain B −1 eq < O according we obtain η T B −1 eq η = c 2 1 T n Q −1 B eq Q −1 1 n < 0. Then, the 437 focus of the problem is proving that when (29) holds, B eq is  444 then we obtain t = ρ(A) < ρ(A + R) according to Lemma 5.

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The proof is finished.

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If (29) holds, the eigenvalues of J 1 are all nonnegative, 458 however, how is it going when a rank-one matrix J 2 is added? 459 In the following, we answer this question through two Theo-460 rems proposed in the following.

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For convenience, note P 5 = P 1 P 3 P 4 and 3 = 2 ⊕ 0. 481 Define J 5 = P −1 5 JP 5 , then J 5 is cospectral with J . Note η = 482 P −1 5 1 n and ξ = P T 5 1 n , then J 5 = 3 + ηξ T . If J 5 is positive 483 definite, we can make the empirical conclusion that J 5 will 484 be positive stable if the absolute value of b 2 is small enough. 485 Unluckily, 3 has a zero root. Consequently, it will need more 486 math tricks to obtain the stability conditions.

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Theorem 5: If (29) is satisfied, system is robust stable when 488 loads change in ψ.

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The proof is accomplished.

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Remark 3: We obtain the robust stability condition through 529 three steps:

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Thus J is expressed as the sum of a semi-positive stable 532 matrix and a rank-one matrix, which shows that J may be 533 positive stable under some conditions.

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Step 2: Prove that J 1 is diagonalizable if (29) holds, which 535 plays a crucial role in the next step.

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Step 3: Prove that when (29) holds, all eigenvalues of J 5 537 (the sum of a semi-positive definite diagonal matrix and a 538 rank-one matrix) have positive real parts. Thus, if (29) holds 539 the system is robust stable when loads change in ψ.

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Through the above three steps, the robust stability con- inequality (29), the system is robust stable when loads change 544 in ψ. Thus, Q2 is answered. To verify the existence condition for the power-flow equa-564 tion, we design two cases.  According to Theorem 1 and 3 in [18], if (13) is solv-570 able, Newton-Raphson Method in Feasible Power-Flow is 571 convergent as long as the power-flow equation is solvable. 572 Thus, we will verify whether (13) has a solution according to 573 Newton-Raphson Method.

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The simulation results are in FIGURE 4 (a) and (b).    The definitions and lemmas used in this paper are shown as 616 follows.

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Definition 1: For a positive vector x=[x 1 x 2 · · · x n ] T , 618 we define Let O be a matrix 619 whose elements are all 0. Define 1 n (0 n ) as the n-dimensional 620 vector which all entries are 1(0) and I is the unit matrix with 621 appropriate dimension. Lemma 1: Brouwer's fixed-point theorem [19]. Let D ∈ R n 637 be a nonempty compact set and G(x) : R n →R n satisfies the 638 condition : G(x) is a self-mapping, i.e., ∀x ∈ D, G(x) ∈ D. 639 Then, there is a unique vector x * ∈D such that x * =G(x * ).

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(2) If A is a nonsingular principal submatrix of M , then 657 (M /A) is also N 0 − matrix.