Doctoral Dissertations

Keywords and Phrases

Jump-Diffusion; Markov Jump Linear Systems; Microgrid; Microgrid Stability; Power Systems; Stochastic Hybrid Systems


"This research discusses stochastic models for a microgrid operating between standalone and grid-tied modes. The transitions between different modes are modeled as a continuous-time Markov chain (CTMC). In each operating mode, the system is modeled using conventional differential algebraic equations (DAEs), linearized around some equilibrium point.

In Topic-I, a model is developed using the Stochastic Hybrid Systems (SHSs) formulation. The microgrid is modeled as a Markov jump linear system (MJLS), which is a type of SHS in which the discrete events evolve according to a Continuous Time Markov Chain (CTMC). The model allows for the derivation of Ordinary Differential Equations that represent the evolution of the conditional moments of the stochastic system, and subsequently the derivation of a matrix representation of these ODEs. The validation of the model relies on comparing numerical results obtained from the simulation of the IEEE 37-bus microgrid system to the conventional averaged Monte Carlo simulation.

The jumps in Topic-I are impulsive and large overshoots can occur. In Topic- II, a jump-diffusion model is developed based on a stochastic differential equation with jumps. The Jump component is modeled as a compound Poisson process, and the resulting conditional moments converge with greater accuracy to the Monte Carlo simulation results. A key advantage of this method is that it is far less computationally expensive than the conventional averaged Monte Carlo simulation.

To analyze the stability of the jump-diffusion model, methods based on the mean square stability are used in Topic-III. The jump-diffusion model is converted into a martingale to allow for the use of the Burkholder-Davis-Gundy (BDG) inequality. The method consists in computing the quadratic variation process and using the BDG inequality to derive bounds on the conditional moments of the system"--Abstract, page iv.


Kimball, Jonathan W.

Committee Member(s)

Crow, Mariesa
Ferdowsi, Mehdi
McMillin, Bruce M.
Shamsi, Pourya


Electrical and Computer Engineering

Degree Name

Ph. D. in Electrical Engineering


The author is thankful to the Boeing Company for funding his PhD program.


Missouri University of Science and Technology

Publication Date

Spring 2021

Journal article titles appearing in thesis/dissertation

  • Markov jump linear system analysis of a microgrid operating in islanded and grid-tied modes
  • Jump-diffusion modeling of a Markov jump linear system with applications in microgrids
  • Mean square stability of a microgrid’s jump diffusion model


xvi, 114 pages

Note about bibliography

Includes bibliographic references.


© 2021 Gilles Mpembele, All rights reserved.

Document Type

Dissertation - Open Access

File Type




Thesis Number

T 11843

Electronic OCLC #