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A System of Systems Engineering Framework for Modern Power System Operation

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Sustainable Interdependent Networks II

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 186))


In the literature, numerous definitions have been proposed for system of systems (SoS). The concept of system of systems has been widely used in defense applications. In addition, it has been applied to other domains, e.g., healthcare, robotics, global communication systems, transportation, space exploration, and power system operation.

In this chapter, we discuss about the possibility of modeling active distribution grids (ADGs), which are composed of several microgrids, based on the concept of system of systems. Although various energy management/control functions, such as long-term planning and day-ahead scheduling, can be investigated based on the concept of SoS, the main focus of this chapter is on the short-term ADG operation and the optimal power flow (OPF) problem. The whole ADG is considered to be an SoS in which distribution system operators (DSOs) and microgrids (MGs), which are autonomous entities, are modeled as self-governing systems. The information privacy of DSOs and MGs is explained, and a local OPF is formulated for each system taking into account the mutual interactions between DSOs and MGs. A distributed optimization algorithm, which is based on the augmented Lagrangian relaxation, is presented to find the optimal operating point of ADG while respecting information privacy of the subsystems. A test system is designed to provide a tutorial for readers on how to formulate local OPF problems of ADGs and MGs and how to implement the distributed optimization algorithm.

This work was supported in part by the Louisiana Board of Regents under Grant LEQSF(2016-19)-RD-A-10 and the National Science Foundation under Grant ECCS-1711850.

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a :

Index for autonomous systems

b, b :

Border buses between two neighboring systems

i, j :

Index for buses

k :

Index for outer loop iterations

l :

Index for levels in the hierarchical structure

m, n :

Index for subproblems

q :

Index for inner loop iterations

u :

Index for generating units

Ω i :

Set of all buses connected to bus i

\( {\varOmega}_L^n \) :

Set of all branches in system n

\( {\varOmega}_G^n \) :

Set of all generating units located in system n

\( {N}_B^n \) :

Number of buses in system n

d i :

Power demand at bus i

p u :

Power generated by unit u

PL ij :

Power flow in line ij

Ɵ :

Bus voltage angles

f n(·):

Local objective function of system n

f(p u):

Generation cost function of unit u

f la(·):

Local objective function of system a at level l

g n(·):

Compact form of inequality constraints of system n

h n(·):

Compact form of equality constraints of system n

g la(·):

Set of local inequality constraints of system a at level l

h la(·):

Set of local equality constraints of system a at level l

θ b :

Voltage angle of bus b

θ b, m :

Voltage angle of bus b determined by system m

θ b, n :

Voltage angle of bus b determined by system n

\( {\theta}_{2a}^{\ast } \) :

Target values that are constants in microgrids’ optimization

\( {\theta}_{2a}^{\prime \ast } \) :

Response values that are constants in DSO’s optimization

x n :

Local variables of system n

\( {\overset{\sim }{x}}_{la} \) :

Set of local variables of system a at level l

t 2 a :

Target variables of system a at level 2

r 2 a :

Response variables of system a at level 2

\( \overline{P_u} \) :

Upper bound for generating unit u

\( \underset{\_}{P_u} \) :

Lower bound for generating unit u

\( \overline{PL_{ij}} \) :

Upper bound for power flow in line ij

\( \underset{\_}{PL_{ij}} \) :

Lower bound for power flow in line ij


Penalty function

λ T :

Lagrange multiplier

ω :

Penalty multiplier

β :

Step size


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Mohammadi, A., Safdarian, F., Mehrtash, M., Kargarian, A. (2019). A System of Systems Engineering Framework for Modern Power System Operation. In: Amini, M., Boroojeni, K., Iyengar, S., Pardalos, P., Blaabjerg, F., Madni, A. (eds) Sustainable Interdependent Networks II. Studies in Systems, Decision and Control, vol 186. Springer, Cham.

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