Jump-Diffusion Modeling of a Markov Jump Linear System with Applications in Microgrids Gilles Mpembele, Member, IEEE, and Jonathan W

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Jump-Diffusion Modeling of a Markov Jump Linear System with Applications in Microgrids Gilles Mpembele, Member, IEEE, and Jonathan W. Kimball, Senior Member, IEEE

Abstract—This paper proposes a jump-diffusion model for the the derivation of the ordinary differential equations that govern analysis of a microgrid operating in grid-tied and standalone the evolution of the system statistics. modes. The model framework combines a linearized power The application of stochastic models to the analysis of the system dynamic model with a continuous-time Markov chain (CTMC). The power system has a stochastic input modeled with dynamics of power systems is relatively new [4], [8]. This a one-dimensional Wiener process and multiplicative diffusion representation is particularly relevant for a class of stochastic coefficient. The CTMC gives rise to a compound Poisson pro- processes called Stochastic Hybrid Systems (SHSs). In this cess that represents jumps between different operating modes. framework, the linearized power system dynamic model is Starting from the resulting stochastic differential equation (SDE), combined with discrete transitions of the system variables the proposed approach uses Itoˆ calculus and Dynkin’s formula to derive a system of ordinary differential equations (ODEs) triggered by stochastic events [9], [10], [11], [4], [5], [7]. The that describe the evolution of the moments of the system. The linearized model is represented by a system of differential approach is validated with the IEEE 37-bus system modified to algebraic equations (DAEs) that includes the evolution of form a microgrid. The results match a Monte Carlo simulation the dynamic states, and of the output variables expressed as but with far lower computational complexity. In addition, the a linear combination of the dynamic states and the control ODEs are amenable to further analysis, such as stability analysis and determination of operational bounds. variables [4], [7], [5]. In a particular class of SHSs called Markov Jump Linear Systems (MJLSs), the stochastic discrete Index Terms—Jump-Diffusion, Markov Jump Linear Systems transitions of the system are explicitly modeled as continuous (MJLSs), Monte Carlo simulations, Power System statistics, Stochastic Differential Equation (SDE), Stochastic Hybrid Sys- time Markov chains. The jump-diffusion model developed tems (SHSs). in this study is based on the Markovian property of the jumps, appropriately supplied by a Poisson process, while the diffusion component is supplied by a Wiener process. I.INTRODUCTION For a microgrid, a jump in a dynamic state is defined as a HE stochastic jump-diffusion model is well suited to variation of significant magnitude that cannot be represented T dynamic systems subject to random disturbances of a lim- by a linear drift or a noise perturbation. For instance, when ited amplitude, as well as to random and abrupt perturbations an abrupt change occurs in the loading condition or in the of a relatively higher magnitude [1]. Disturbances of limited characteristics of distributed power sources, and because of the (small) amplitude correspond to the diffusion component of finite and limited inertia of the system, the bus voltages and the model and are described using the Wiener process. Higher the system frequency can experience drastic variations that, in (large) magnitude disturbances apply to the jump component the absence of adequate control, could result in instabilities of the model and are represented by the Poisson process and even in the collapse of the entire system [12], [7]. In the [1], [2], [3]. The objective of this study is to develop such context of this paper, discrete jumps result in the switching of jump-diffusion model for a microgrid that oscillates between the microgrid between one grid-tied operating mode and two standalone (islanded) and grid-tied operating modes. Preced- distinct standalone (islanded) operating modes. The difference ing works focused on deterministic systems controlled by between the two standalone modes results from a difference stochastic inputs of the jump type [4], [5] and, in other in load conditions and corresponds to two distinct sets of cases, on stochastic systems subject to switching behaviors dynamic states and output variables [7], [13]. of a pure-jump type [6], [7]. This work extends previous Figure 1 shows bus voltages and load currents at an arbitrary models: 1) the entire system is considered switching between bus (labeled 26) of the IEEE 37-bus microgrid. This plot distinct stochastic modes, 2) this switching behavior involves was obtained from a Simulink R /PLECS R simulation of the all dynamic and algebraic states, and not just the system IEEE 37-bus microgrid and from results from an experimental inputs as in [4], [5], 3) the modeling of the stochastic process testbed [14]. includes a Wiener process as well Poisson jump processes in The derivation of a small-signal model of an inverter-based microgrid system was carried out in [14]. For the IEEE 37- G. Mpembele is with the Department of Electrical and Computer Engi- bus microgrid, the resulting state space model is of the 225th neering, Missouri University of Science and Technology, Rolla, MO, 65409, USA. He is also with the Boeing Company, Weapons and Missile Systems, order and represent a two-time scale system where states with St. Charles, MO 63301 USA, email: [email protected]. slow dynamics are mixed with those with fast dynamics [15], J. W. Kimball is with the Department of Electrical and Computer Engi- [7]. A reduction method based on the singular perturbation neering, Missouri University of Science and Technology, Rolla, MO, 65409, USA, e-mail: [email protected]. technique was presented in [16], [15], [17]. In the case of Manuscript received XXXX. the 225th order system (IEEE 37-bus system), the technique 2

could be deterministic, time dependent and non linear, in general. In accordance with power system models, this study considers deterministic, state dependent and linear coefficient functions. Furthermore, the existence and uniqueness of the solution to the SDE (1) are guaranteed only if the drift, diffusion and jump coefficient functions satisfy the Lipschitz conditions, as well as the linear growth conditions, the definition of which can be found in [1], [19], [20]. When given in the differential form, the SDE (1) is considered an informal or short hand notation for its integral form. The integral form yields the Fig. 1. Voltage and Load current at bus 26 solution to (1) and is called an Itoˆ process with jumps or a jump-diffusion process Z t Z t allowed to reduce the system to the 56th order by eliminating X(t) =X0 + f(s, X(s))ds + g(s, X(s))dW (s) the fast states. Accuracy of this method was evaluated in [15], 0 0 where the system’s dynamic response was evaluated in grid- N(t) X tied and islanded modes, as well as during transitions between + h τk,X(τk−) (2) the two modes. Results obtained using the reduced model were k=1 found to match experimental results from a hardware testbed The first integral is an ordinary Riemann integral. The [15]. second integral is an Itoˆ integral with respect to the Wiener A SHS model for a power system was developed in [4] to process W (t). The third term represents a compound Poisson represent the dynamic behavior of a system subject to stochas- process where, as noted above, {N(t), t > 0} is a Poisson tic inputs. The resulting ODEs representing the evolution of process with intensity λ. the conditional moments were derived using stochastic inputs In (2), {h(τk,Xτk−), k > 0} is a family of independent and and did not include random variations of the Wiener type. This identically distributed (i.i.d) random variables (independent of model was used in [7] for a system switching between different N) called jump sizes. equilibrium points. It was shown in [5], [18], [7] that a matrix The notation Xτk − corresponds to the almost sure left-hand representation of the system can be used along with analytical limit of X(t) at time τk, and if a jump occurred at time τk, tools to compute the system statistics. In this study, the model the jump size is defined as is adapted for a slow-switching system, and includes a Wiener ∆X(τ ) = X(τ ) − X(τ −) process component. The derivation of the model is carried k k k (3) out directly from the jump-diffusion stochastic differential The generalization of (1) and (2) to a multidimensional equation (SDE), rather than from the SHS framework as in system is straightforward. The evolution of the stochastic [4], [6], [7]. vector X = {X(t) ∈ Rn, t ≥ 0}, is described by an n- This paper is organized as follows. In section II, the dimensional SDE with jumps [1] development of a jump-diffusion model of a power system is dX(t) = f(t, X(t))dt + g(t, X(t))dW (t) presented. In section III, the Dynkin’s formula is applied to the d Itoˆ formula of the Stochastic Differential Equation to derive X ` ` the ODEs that represent the evolution of the system statistics. + h (t, X(t−))dN (t) (4) In section IV, a numerical application of the jump-diffusion `=1 n model to the IEEE-37 bus microgrid system is discussed. for t ∈ [0,T ], with initial value X0 ∈ R , an q-dimensional A conclusion and avenues for future work are presented in Wiener process W = {W (t) = W 1(t), ..., W q(t)}, t ∈ [0,T ] Section V. and d independent compound Poisson processes, N `, with ` parameters λ`, ` = 1, ..., d, and jump sizes h (t, X(t−)). The integral form is derived from (4) as II.JUMP-DIFFUSION MODEL OF A POWER SYSTEM Z t Z t Let X = {X(t), t ≥ 0} be a one-dimensional continuous X(t) = X0 + f(s, X(s))ds + g(s, X(s))dW (s) stochastic process. In the most general sense, a jump-diffusion 0 0 process can be represented by a stochastic differential equation d N `(t) X X ` with jumps + h τk, X(τk−) (5) `=1 k=1 dX(t) = ft, X(t)dt + gt, X(t)dW (t) To develop a jump-diffusion model for a microgrid system, + h t, X(t−) dN(t) (1) the conventional dynamic model needs to be cast into the stochastic differential equation with jumps defined by (4). The for t ≥ 0 with initial value X(0) = X . Here, W = 0 deterministic dynamic model is represented by the differential {W (t), t ≥ 0} represents a standard Wiener process and algebraic equation (DAE) [5], [4] N = {N(t), t ≥ 0} a Poisson process with intensity λ. The processes f, g, and h are called respectively drift, diffusion, x˙ = f(x, y, u) (6) and jump coefficient functions. These coefficient functions 0 = g(x, y, u) 3 where x(t) ∈ Rn is referred to as the dynamic states; y(t) ∈ III.DERIVATION OF THE CONDITIONAL MOMENTS v R denotes the algebraic states or outputs of the system; and The procedure to derive the conditional moments include w u(t) ∈ R represents the inputs of the system. the following steps: The linearization of (6) around a given stable equilibrium • Derivation of the Itoˆ formula for the SDE with jumps point results in a linear affine model [5], [4] • Application of the Dynkin’s formula to the Itoˆ formula x˙ = Ax + Bu + C • Derivation of the conditional moments dynamics (7) y = Dx + Eu + F • Derivation of a matrix representation Following [7], [14], [15], the equilibrium points correspond to the steady-state operating points of a microgrid operating A. Itoˆ formula for the SDE with jumps in grid-tied mode (X1) or in two distinct standalone modes The aim of this sub-section is to derive the expression of the (X2, X3). The difference between the two standalone modes stochastic differential equation for a process that is function is based on different load conditions on the system. The affine of the solution of equation (4). In a more formal way, the term, C, is defined in such a way that small variations are problem can be stated as follows: modelled around the equilibrium points X1,X2,X3 Given a stochastic differential equation with jumps of the type (4), and a process ψ(t) which is a function of X(t) C = −AiXi, i = 1, 2, 3 (8) In the output equation, F is set to zero: the outputs depend ψ = ψ(t, X(t)) (14) on the dynamic states and input vector. where the function ψ(t, X(t)) is continuously differentiable The first step in developing a stochastic model for a power in t and twice continuously differentiable in X, determine the system is to convert the deterministic variables in equation (7) stochastic differential equation for the process ψ(t) into stochastic variables dX(t) = (A(t)X(t) + B(t)U(t) + C(t)) dt dψ(t, X(t)) = f˜(t, X(t))dt +g ˜(t, X(t))dW (t) (9) Y (t) = D(t)X(t) + E(t)U(t) + h˜(t, X(t−))dN(t) (15) To establish a correspondence between (4) and (9), the state Equation (15) is called the Itoˆ formula (or Itoˆ rule) and equation of the dynamic model is cast as the drift term of the the definitions of the coefficient functions f˜, g˜, h˜ are based SDE with jumps, i.e., on the rules of the stochastic calculus. The Itoˆ formula is the f(X(t), t) = A(t)X(t) + B(t)U(t) + C(t) (10) equivalent to the chain rule in classical calculus. It is used to quantify the changes in ψ(t, X(t)) that are caused by changes The diffusion term is modeled as a multiplicative diffusion in X(t). component, with a noise generated ripple proportional to The Itoˆ formula for a multidimensional process X(t) is the dynamic state. The diffusion coefficient is a constant n adequately chosen to limit the ripple to about 10% of the ∂ψ t, X(t) X ∂ψ t, X(t) dψ(t, X(t)) = + f it, X(t) steady-state value of the dynamic state, ∂t ∂xi i=1 n 2 g(X(t), t) = βX(t) (11) 1 X ∂ ψ t, X(t) i,j + g(t, X(t))gT (t, X(t)) dt 2 ∂xixj where β = 0.1 . i,j=1 The jump size is defined as n X ∂ψ t, X(t) + git, X(t)dW (t) h t, X(t−) = X(t) − X(t−) (12) ∂xi i=1 where X(t) is set to the steady-state operating point corre- d X sponding to the destination mode (i.e. mode the system has + ψt, X(t) − ψt, X(t−) dN `(t) (16) jumped to), and X(t−) is the value of the dynamic state just `=1 prior to the jump event where X(t) is a n-dimensional continuous stochastic vector With all the terms of the SDE (4) defined, the jump- process X(t) = {X1(t),X2(t), ..., Xn(t), t ≥ 0}, and f and diffusion model of a power system based on the deterministic g are n-dimensional drift and diffusion coefficient functions, model (7) can now be represented by the SDE with jumps respectively, f(t, X(t)) = {f 1(t, X(t)), ..., f n(t, X(t)), t ≥ 1 n dX(t) = Ai(t) X(t) − Xi(t) + BiU(t) dt + βX(t)dW (t) 0} and g(t, X(t)) = {g (t, X(t)), ..., g (t, X(t)), t ≥ 0}. As d mentioned above, W (t) is a one-dimensional Wiener process, X ` th + X(t) − X(t−)dN `(t) (13) and N (t) is the ` component of a d-dimensional Poisson `=1 process, with ` = 1, ..., d. where W (t) is a one-dimensional Wiener process1 and N `, ` = 1, ..., d, represents the `th Poisson process with B. Application of Dynkin’s formula intensity λ`, and i ∈ {1, 2, 3} represents the system operating In stochastic analysis, Dynkin’s formula is a theorem that mode. relates the expectation of a function of a jump-diffusion pro- 1A one-dimensional Wiener process is used here to simplify the equations, cess and a functional of the backward jump-diffusion operator without loss of generality. [3]. It can be interpreted as a stochastic generalization of the 4 fundamental theorem of calculus. For the jump-diffusion SDE For a power system jump-diffusion model (13), jump sizes of the type (4), the Dynkin’s formula consists in taking the correspond to jumps in dynamic and algebraic states when the expectation of (16) system switches between one equilibrium point to another one. They can subsequently be represented as follows dE[ψt, X(t)] = n ψ(t, X(t)) − ψ(t, X(t−)) = ψ (t, X(t)) − ψ (t, X(t)) (23) ∂ψ t, X(t) X ∂ψ t, X(t) j i E + f it, X(t) ∂t ∂xi where ”i” represents the origin mode and ”j” the destination i=1 n 2 mode. Therefore, these poissonian jumps between different 1 X ∂ ψ t, X(t) i,j + g(t, X(t))gT (t, X(t)) dt modes of the stochastic system can be characterised by a set 2 ∂xixj i,j=1 of jump parameters (or transition rates) λij, i, j = 1, ..., N. d The full transition rate matrix for a stochastic system with N X + E ψ t, X(t) − ψ t, X(t−) λ`dt (17) stochastic modes is defined as `=1 Λ = {λij, i, j = 1, ..., N} (24) Notice the term with dW (t) has disappeared from (17). In fact, the expected value of the Wiener process E[dW (t)] = 0. where, in accordance with the Markov chain theory, the self- 2 In addition, the differential term dN `(t) has been replaced by transitions N its expected value λ`dt where λ` represents the parameter of X a Poisson process N `(t). λii = − λij, ∀i = 1, ..., N (25) j=1 j6=i C. Derivation of the conditional moments dynamics The jump term in (22) can now be written as The Dynkin’s formula, combined with the law of total N expectation allows for a derivation of the ODEs that describe X E ψ t, X(t) − ψ t, X(t) λ = the evolution of the conditional moments. j i ij The law of total expectation can simply be defined as i,j=1 N follows: given a set of stochastic events Ai, i = 1, ...N, the X E ψ t, X(t) λ − E ψ t, X(t) λ (26) expectation of a random variable X equals the sum of the j ij i ii j=1 expectations of X given Ai j6=i N X The function process ψi(t, X(t)) of a stochastic variable E(X) = E(X|Ai)P r(Ai) (18) X(t) has not been defined yet. It can be expressed in many i=1 0 different ways. For instance, for ψi(t, X(t)) = X(t) i , where P r(Ai) denotes the probability of event Ai. equation (22) corresponds to the 0th order moment dynamics The conditional moment of a process function ψ(t) given a (0) 1 and will be denoted by µ˙ i (t). For ψi(t, X(t)) = X(t) i , stochastic event Ai is expressed as equation (22) represents the evolution of the conditional mo- (1) µi(t) = E[ψ(t, X(t))|Ai]P r(Ai) (19) ment of the first order (mean) and will be denoted by µ˙ i (t), 2 and for ψ (t, X(t)) = X(t) , equation (22) represents the It follows from (18) that the total expectation of ψ(t, X(t)) i i evolution of the conditional moment of the second order (also equals the sum of all conditional moments given the events called uncentered second moment) and will be denoted by Ai (2) N µ˙ (t). X i µ(t) = E[ψ(t, X(t))] = µi(t) (20) In general, conditional moments of order m will be de- (m) i=1 noted by µi and for an n-dimensional vector X(t), m = and the evolution of a conditional moment: (m1, m2, ..., mn), and

d (m) (m1,m2,...,mn) µ˙ (t) = E[ψ(t, X(t))|A ]P r(A ) (21) µi (t) = µi (t) (27) i dt i i th The right hand side of equation (21) represents the Dynkin’s It results, for the 0 order moment formula (17), expressed here with respect to a stochastic mode (0) (0,...,0) µi (t) = µi (t) i, where i = 1, ..., N 0 0 0 = E X1 X2 ...Xn (28) µ˙ i(t) = n th ∂ψ t, X(t) X ∂ψ t, X(t) For the k element of X(t), Xk, the ﬁrst moment is deﬁned E + f it, X(t) ∂t ∂xi as i=1 (1) (0,..,1,...0) n 2 µi (t) = µi (t) 1 X ∂ ψ t, X(t) T k,j + g(t, X(t))g (t, X(t)) 0 1 0 2 ∂xkxj = E X1 ...Xk ...Xn (29) k,j=1 d 2To not confuse with diagonal elements of a transition probability matrix X N + E ψ t, X(t) − ψ t, X(t−) λ (22) P ` of a Markov chain, λii = λij `=1 j=1 j6=i 5

The uncentered second moment can be deﬁned with respect ODEs (33) is described in equation (34). This corresponds to to one element (kth) of X(t) or two elements, (kth) and (`th) the expression that was obtained in [5], [7], [18]

(2) (0,..,2,...,0) (0) T (0) µi (t) = µi (t) µ˙ = Λ µ (34) 0 2 0 = E X1 ...Xk ...Xn (30) where ΛT is the transpose of the full transition rate matrix (24). Equation (34) corresponds to the well-known Chapman- (2) (0,..,1,...,1,...,0) Kolmogorov equation. µi (t) = µi (t) 2) The 1st conditional moments are the statistical means of = EX0...X1...X1...X0 (31) 1 k l n the stochastic distribution. The general form of the 1st moment It follows from (28), (29), (30), (31) and from the jump- ODEs derived in [5], [7] was as follows diffusion model of a power system in (13) µ˙ (1) = G(1)µ(1)(t) + H(1)µ(0)(t) (35) µ˙ (m)(t) = i where the evolution of the first moments is function of the n (m) X ∂ψ (t, X(t)) moments of the 1st and 0th orders. Using Matlab notation, E i · A(t)X(t) + B(t)U(t) + C(t) ∂xr the coefficient functions G(1) and H(1) in [5], [7] were then r=1 defined as n 2 (m) 1 2 X ∂ ψi (t, X(t)) 2 + β E X (t) 1 T 2 ∂xrxs G = blkdiag(A1, ..., AN ) + (Λ ⊗ I(N)) (36) r,s=1 H1 = blkdiag(−A X , ..., −A X ) (37) N 1 1 N N X + E ψj t, X(t) λij − E ψi t, X(t) λii (32) Because of the way the jump term is defined in the jump- j=1 (1) (1) j6=i diffusion model developed here, the matrices G and H are modified as follows A more detailed derivation of the conditional moments of 1 T a stochastic hybrid system is described in [4]. However, the G = blkdiag(A1, ..., AN ) + diag(Λ ⊗ I(N)) (38) 1 resulting model in [4] is different from the one developed H = blkdiag(−A1X1, ..., −AN XN ) in this paper, for the reason that the power system in [4] is + [X1; X2; X3] ◦ ΛT ⊗ 1 (39) subjected to stochastic inputs whereas in this study, the entire trans n×1 system is switching stochastically between distinct equilibrium where ◦ represents the Hadamard product (element-wise mul- points. tiplication, Matlab ‘.*’), Λtrans = Λ − diag(Λ), and 1n×1 is In (32), ψj(t, X(t) is set to the equilibrium operating point an n × 1 matrix of all ones (Matlab ones(n, 1)). of the destination mode (slow switching system) after a jump 3) The 2nd conditional moments represent the uncentered event has occurred. In addition, the second term of the right second moments. From [5], [7] m < 2 hand side of (32) is equal to zero for . (2) (2) (2) (2) (1) The resulting system of ODEs that describes the evolution µ˙ = G µ (t) + H µ (t) (40) of the conditional moment of the jump-diffusion model of a where the evolution of the second moment is function of power system is the moments of the 2nd and 1st orders. The structure of the n n coefficient functions G(2) and H(2) is pretty complex and was (m) X X m−ep+er m−ep µ˙ i (t) = mp aprµi (t) + µi (t)vp described in length in [5]. However, for the jump-diffusion p=1 r=1 model, because of the way the jump term is defined and of n n 1 X X (m) the inclusion of the Wiener term, the equation of the ODEs + β2 m (m − 1) + m m µ (t) 2 p p p r i representing the evolution of the second moment is as follows p=1 r=1 r6=p N (2) (2) (2) (2) (1) (2) (0) X (m) (0) (m) µ˙ = G µ (t) + H µ (t) + J µ (t) (41) + λijXi µj (t) − λiiµi (t) (33) j=1 Matrix G(2) contains the contributions from the state matri- j6=i ces Ai, i = 1, 2, 3, from the diagonal elements of the transition th ∀ i = 1, ..., N, where vp represents the p element of the rate matrix (λii, self-transitions), and from the square of the 1 2 (2) vector Bu + C and ep, er are two unit vectors with a 1 at the Wiener coefficient 2 β . Each block entry of G is defined pth, rth entry, respectively. as

D. Matrix representation (2) 0 Gi = (Tx + Ty) ∗ (eye(n) ⊗ Ai) ∗ (Tx) 0 The derivation of the matrix representation of equation (33) + diag(Λ ⊗ eye(n + nz)) follows the procedure described in [5], [7], [18]. + β2 ∗ eye(n ∗ (n + 1)/2) (42) 1) The 0th moments represent the occupational probabilities, which are the probabilities for the system to be in a where nz is the binomial coefﬁcient of n and 2, Tx,Ty (and Tz particular operating mode. The matrix representation of the below) are reordering matrices. These matrices are not unique, 6 they ensure the elements of G(2) are ordered in a desired fashion. Matrix H(2) is constructed with the afﬁne term in the drift component of the SDE (13), −AiXi + Biu, i = 1, 2, 3. Each bloc entry is (2) Hi = (Tx + Ty) ∗ (eye(n) ⊗ (−AiXi + Biu)) (43) Matrix J (2) is a Kronecker product of the transition rates (non-diagonal elements of Λ) with the steady-state vectors. The block elements of J (2) are of the form λ J (2) = ij ∗ T ∗ (Xj ⊗ Xj) (44) ij 2 z where i, j = 1, 2, 3 and i 6= j. Fig. 2. Zeroth Moments of the stochastic system (occupational probabilities)

IV. NUMERICAL APPLICATION 37-bus power system [21], [18], [14], [15]. The plots in red In this section, a numerical application of the jump-diffusion represent the averaged Monte Carlo simulations. They closely model to the IEEE 37-bus microgrid is presented and dis- match the jump-diffusion plots in dashed-black. The error cussed. The ODEs representing the conditional moments are observed in Fig. 3 (b) and (h) can be considerably reduced by solved using the matrix representation of the jump-diffusion increasing the number of Monte Carlo runs, but at the expense model. Computation of the conditional moments is limited of the execution time. to the lower order moments (zeroth, first and second orders). 3) Second moment results These lower order moments correspond respectively to the oc- To obtain the solutions for the uncentered second moments, cupational probabilities, the statistic means and the uncentered the moment order is set as m = 2 in matrix ODEs (41). It second moments, and are in general sufficient to characterise results from (41), G(2) is 4788×4788, H(2) is 4788×168, and a probability density function of a distribution. Conceivably, J (2) is a 4788 × 3. Following the argument developed for the moments of higher order can be calculated, but the analytical first order moments, the uncentered second moments for the expression of the system of ODEs becomes too complex dynamic states is a 1596 × 1, and the second moments for the and the solutions too computational expensive, they are not algebraic states is a 1540 × 1 vector. The results are shown in addressed in this paper. Fig. (4) for the dynamic states (inverter 2). An examination of The results presented in this study were limited to 20,000 Fig. (4) indicates the averaged Monte Carlo simulation results runs and the computer execution time was 55 hr 20 min 23 s, converge to the jump-diffusion results. The comments related on a PC with a 3.2 GHz Intel R CoreTM i7-800 CPU processor to the comparison between the first moments obtained from with 32GB memory in the MATLAB’s environment. The the Jump-Diffusion model and form the MJLS model from [7] computer execution time increases significantly as the number are also valid for the second moments and the variances. of runs is increased. In this regard, the jump-diffusion method developed here is vastly superior to the averaged Monte Carlo V. CONCLUSIONAND FUTURE WORK method. The execution time to compute the zeroth, first and second order conditional moments was 2 hr 18 min 54 s, a This study presented a jump-diffusion model to analyze the reduction of 95.8% relative to the Monte Carlo approach. dynamics of a microgrid operating in grid-tied and standalone 1) Zeroth moment results modes. For a switching system, a statistical approach based For the zeroth moments, the jump-diffusion model corre- on the computation of the conditional moments is suitable for sponds to the Chapman-Kolmogorov equations, the differential the analysis of the system dynamics. The SDE representing expression of which is provided in (34). The solutions for the the system included the Wiener process as well as Poisson zeroth order moments are identical to those obtained in [7], processes to represent the stochastic jumps between different they are represented here again, in Fig. 2, for completeness. operating modes of the power system. The solution of the 2) First moment results jump-diffusion SDE led to a system of ODEs that represent the The solutions for the first moments are obtained by solving evolution of the conditional moments of the system. A matrix the system of ODEs represented by the matrix equation (35), representation of the system of ODEs, and the numerical where the moment order m = 1. In (35), matrix G(1) is 168× solutions were shown to converge to the results of the averaged 168 and matrix H(1) is 168 × 3. To obtain the means for the Monte Carlo simulation. Future work will include using the 56 dynamic states, the solutions corresponding to the three Jump-Diffusion model developed in this study to analyse the stochastic modes and embedded in the column vector µ(1) are stability of a microgrid system subjected to random and abrupt summed up, in accordance with the law of total expectation transitions between different operating modes. (1) µ[56×1] = [eye(56) eye(56) eye(56)]µ (45) The first moments represented in Fig. (3) correspond to the dynamic states associated with Inverter 2 of the modified IEEE 7

Fig. 3. First Moments of the dynamic states

Fig. 4. Variances of the dynamic states 8

REFERENCES Gilles Mpembele (S’98–M’07) received the B.S. degree in electrical engineering from the Univer- [1] E. Platen and N. Bruti-Liberati, Numerical solution of stochastic dif- sity of Kinshasa, in the Democratic Republic of ferential equations with jumps in ﬁnance, ser. Stochastic modelling and Congo, and the M.S. degree in electrical engineering applied probability. Springer-Verlag, 2010, vol. 64. from the University of Bridgeport, CT, USA. He [2] A. Seierstad, Stochastic control in discrete and continuous time. is currently pursuing the Ph.D. degree in electrical Springer, 2009. engineering with the Missouri University of Science [3] F. B. Hanson, Applied Stochastic Processes and Control for Jump- and Technology, Rolla, MO, USA. He was with the Diffusions: Modeling, Analysis and Computation, ser. Advances in Hemlock Semiconductor Corporation from 2010 to Design and Control. Society for Industrial and Applied Mathematics, 2013, as a Senior Process Electrical Engineer. He is 2007. currently with the Boeing Company, as an Electrical [4] S. V. Dhople, Y. C. Chen, L. DeVille, and A. D. Dominguez-Garcia, and Electronics System Design and Analysis Engineer. His current research “Analysis of power system dynamics subject to stochastic power injec- interests include distributed power generation and microgrid stability analysis. tions,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 12, pp. 3341–3353, 2013. [5] J. A. Mueller and J. W. Kimball, “Modeling and analysis of dc microgrids as stochastic hybrid systems,” IEEE Transactions on Power Electronics, 2021. Jonathan W. Kimball (M’96–SM’05) received the [6] S. Dhople, L. DeVille, and A. Dominguez-Garcia, “A stochastic hybrid B.S. degree in electrical and computer engineering systems framework for analysis of markov reward models,” Reliability from Carnegie Mellon University, Pittsburgh, PA, Engineering & System Safety, vol. 123, pp. 158–170, 2014. USA, in 1994, and the M.S. degree in electrical [7] G. Mpembele and J. Kimball, “Markov Jump Linear System Analysis engineering and the Ph.D. degree in electrical and of a Microgrid Operating in Islanded and Grid Tied Modes,” TechRxiv, computer engineering from the University of Illi- 2020. nois at Urbana-Champaign, Champaign, IL, USA, in [8] Y. Xu, F. Wen, H. Zhao, M. Chen, Z. Yang, and H. Shang, “Stochastic 1996 and 2007, respectively. He was with Motorola, small signal stability of a power system with uncertainties,” Energies, Phoenix, AZ, USA, from 1996 to 1998, where he vol. 11, no. 11, p. 2980, 2018. was involved in designing insulated gate bipolar [9] J. Hespanha, “Modelling and analysis of stochastic hybrid systems,” transistor modules for industrial applications. He IEE Proceedings - Control Theory and Applications, vol. 153, no. 5, then joined Baldor Electric, Fort Smith, AR, USA, where he designed pp. 520–535, 2006. industrial adjustable-speed drives ranging 1–150 hp. In 2003, he joined the [10] J. a. P. Hespanha, “A model for stochastic hybrid systems with applica- University of Illinois at Urbana-Champaign as a Research Engineer, where he tion to communication networks,” Nonlinear Analysis: Theory, Methods then became a Senior Research Engineer. In 2003, he co-founded SmartSpark & Applications, vol. 62, no. 8, pp. 1353–1383, 2005. Energy Systems, Inc., Champaign, IL, USA, where he was the Vice President [11] ——, “Modeling and analysis of networked control systems using of Engineering. In 2008, he joined the Missouri University of Science and stochastic hybrid systems,” Annual Reviews in Control, vol. 38, no. 2, Technology, Rolla, MO, USA, where he is currently an Associate Professor. pp. 155–170, 2014. Dr. Kimball is a member of Eta Kappa Nu, Tau Beta Pi, and Phi Kappa Phi. [12] J. A. Mueller, M. Rasheduzzaman, and J. W. Kimball, “A model modiﬁ- He is a licensed Professional Engineer in the state of Illinois. cation process for grid-connected inverters used in islanded microgrids,” IEEE Transactions on Energy Conversion, vol. 31, no. 1, pp. 240–250, 2016. [13] T. Green and M. Prodanovic, “Control of inverter-based micro-grids,” Electric Power Systems Research, vol. 77, pp. 1204–1213, 2007. [14] M. Rasheduzzaman, J. A. Mueller, and J. W. Kimball, “An accurate small-signal model of inverter- dominated islanded microgrids using dq reference frame,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 2, no. 4, pp. 1070–1080, 2014. [15] ——, “Reduced-order small-signal model of microgrid systems,” IEEE Transactions on Sustainable Energy, vol. 6, no. 4, pp. 1292–1305, 2015. [16] L. Luo and S. V. Dhople, “Spatiotemporal model reduction of inverter- based islanded microgrids,” IEEE Transactions on Energy Conversion, vol. 29, no. 4, pp. 823–832, 2014. [17] V. Mariani, F. Vasca, J. C. Vasquez, and J. M. Guerrero, “Model order reductions for stability analysis of islanded microgrids with droop control,” IEEE Transactions on Industrial Electronics, vol. 62, no. 7, pp. 4344–4354, 2015. [18] G. Mpembele and J. Kimball, “A matrix representation of a markov jump linear system applied to a standalone microgrid,” in 2018 9th IEEE International Symposium on Power Electronics for Distributed Generation Systems (PEDG), June 2018, pp. 1–8. [19] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes. North-Holland, 2011. [20] P. E. Protter, Stochastic Integration and Differential Equations. Springer-Verlag, 2013. [21] G. Mpembele and J. Kimball, “Analysis of a standalone microgrid stability using generic markov jump linear systems,” in 2017 IEEE Power and Energy Conference at Illinois (PECI), Feb 2017, pp. 1–8.