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An energy management system for off-grid power systems

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Abstract

Next generation power management at all scales will rely on the efficient scheduling and operation of both generating units and loads to maximize efficiency and utility. The ability to schedule and modulate the demand levels of a subset of loads within a power system can lead to more efficient use of the generating units. These methods become increasingly important for systems that operate independently of the main utility, such as microgrid and off-grid systems. This work extends the principles of unit commitment and economic dispatch problems to off-grid power systems where the loads are also schedulable. We propose a general optimization framework for solving the energy management problem in these systems. An important contribution is the description of how a wide range of sources and loads, including those with discrete states, non-convex, and nonlinear cost or utility functions, can be reformulated as a convex optimization problem using, for example, a shortest path description. Once cast in this way, solution are obtainable using a sub-gradient algorithm that also lends itself to a distributed implementation. The methods are demonstrated by a simulation of an off-grid solar powered community.

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Algorithm 1
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Notes

  1. At times we will also refer to the “disutility” of loads. In this way, we aim to minimize the disutility.

  2. They can be included for completeness by defining, for example, \(x_{i}^{g}(t+1) = f_{i}^{g}(g_{i}(t),x_{i}^{g}(t), u_{i}^{g}(t))=0\).

  3. A similar formulation can be used for the loads.

  4. In this way, the transition cost reflects a change in state at time t to time (t+1); the transition cost could similarly be defined to reflect the change from time (t−1) to t, representing a slightly different interpretation for the total cost at time t.

  5. For an equality constraint, as in (2.7), the multiplier is unconstrained. If, however, the power balance constraint was written as an inequality constraint, the multiplier update (3.8) would have to be projected onto the positive orthant.

  6. Note that for the remainder of this example we refer to ‘disutility’ as opposed to ‘utility.’ This is without loss of generality and translates only to a sign change for the load segment of (2.1).

Abbreviations

G,L :

number of generating units, number of loads

\(\mathcal{G},\mathcal{L}\) :

set of generating units, set of loads

T :

time horizon

\(\mathcal{T}\) :

set of time indices

i,j,t :

index for generating units, loads, and time

g i (t),y j (t):

power level of generating unit \(i \in \mathcal{G}\) and demand of load \(j \in\mathcal{L}\) at time \(t \in\mathcal{T}\)

\(x_{i}^{g}(t), u_{i}^{g}(t)\) :

state and control variables for generating unit \(i \in\mathcal{G}\) at time \(t \in\mathcal{T}\)

\(x_{j}^{l}(t), u_{j}^{l}(t)\) :

state and control variables for load \(j\in\mathcal{L}\) at time \(t \in\mathcal{T}\)

\(f_{i}^{g}(g_{i}(t),x_{i}^{g}(t),u_{i}^{g}(t))\) :

dynamic evolution of generating unit variables for unit \(i \in\mathcal{G}\)

\(f_{j}^{l}(y_{j}(t),x_{j}^{l}(t),u_{j}^{l}(t))\) :

dynamic evolution of load variables for load \(j \in\mathcal{L}\)

\(C_{i}(g_{i}(t),x_{i}^{g}(t),u_{i}^{g}(t))\) :

operating cost of generator unit \(i \in\mathcal{G}\) at time \(t \in\mathcal{T}\)

\(U_{j}(y_{j}(t),x_{j}^{l}(t),u_{j}^{l}(t))\) :

utility of load \(j \in\mathcal {L}\) at time \(t \in\mathcal{T}\)

\(\mathcal{S}_{i}^{g}(t),\mathcal{S}_{j}^{l}(t)\) :

abstract constraint set for generator unit \(i \in\mathcal{G}\) and load \(j \in\mathcal{L}\) at time \(t \in \mathcal{T}\)

\(\mathcal{X}_{i}^{g}(t),\mathcal{U}_{i}^{g}(t)\) :

abstract constraint set for generator load state and control variables at \(t \in\mathcal{T}\)

\(\mathcal{X}_{j}^{l}(t), \,\mathcal{U}_{j}^{l}(t)\) :

abstract constraint set for load state and control variables at \(t \in\mathcal{T}\)

L(⋅),q(λ):

Lagrangian and dual functions

λ t ,v(λ t ),α :

Lagrange multiplier and sub-gradient at time \(t \in\mathcal{T}\), step size

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Acknowledgements

The authors would like to thank the associate editor and reviewers for the in-depth comments and discussions regarding this work.

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Authors and Affiliations

  1. Institute for Systems Theory and Automatic Control, Universität Stuttgart, 70550, Stuttgart, Germany

    Daniel Zelazo

  2. Department of Aeronautics and Astronautics, University of Washington, Seattle, WA, 98195-2400, USA

    Ran Dai & Mehran Mesbahi

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  1. Daniel Zelazo
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Correspondence to Daniel Zelazo.

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Zelazo, D., Dai, R. & Mesbahi, M. An energy management system for off-grid power systems. Energy Syst 3, 153–179 (2012). https://doi.org/10.1007/s12667-012-0050-4

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