Towards Optimal Primary- and Secondary-control Design for Networks with Generators and Inverters Manish K. Singh D. Venkatramanan and Sairaj Dhople

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Towards Optimal Primary- and Secondary-control

Design for Networks with Generators and Inverters

Manish K. Singh, D. Venkatramanan, and Sairaj Dhople

Department of Electrical & Computer Engineering

University of Minnesota

Minneapolis, MN USA

{msingh, dvenkat, sdhople}@umn.edu

Abstract—For power grids predominantly featuring large syn-

chronous generators (SGs), there exists a signiﬁcant body of work

bridging optimization and control tasks. A generic workﬂow in

such efforts entails: characterizing the steady state of control

algorithms and SG dynamics; assessing the optimality of the

resulting operating point with respect to an optimal dispatch

task; and prescribing control parameters to ensure that (under

reasonable ambient perturbations) the considered control nudges

the system steady state to optimality. Well studied instances of

the aforementioned approach include designing: i) automatic gen-

eration control (AGC) participation factors to ensure economic

optimality, and ii) governor frequency-droop slopes to ensure

power sharing. Recognizing that future power grids will feature

a diverse mix of SGs and inverter-based resources (IBRs) with

varying control structures, this work examines the different steps

of the optimization-control workﬂow for this context. Considering

a representative model of active power-frequency dynamics of

IBRs and SGs, a characterization of steady state is put forth

(with and without secondary frequency control). Conditions on

active-power droop slopes and AGC participation factors are

then derived to ascertain desired power sharing and ensure

economically optimal operation under varying power demands.

I. INTRODUCTION

A large and complex interconnected power system can be

viewed as one massive machine with sophisticated controls

enabling its secure operation [1]. In most power systems,

various parts are owned, operated, and frequently replaced, by

loosely coordinating stakeholders. Such ﬂexibility is enabled

by an elegant and robust hierarchical control scheme [2]. These

schemes allow integration of heterogeneous synchronous gen-

erators, as long as they comply with certain local dynamic

requirements referred to as primary control; and respond to

control signals from a supervisory scheme known as secondary

control. Being central to the operation of large power grids,

these schemes have been actively researched since incep-

tion [3]–[8]. However, there has been a renewed interest in

their analysis and design in response to the ongoing transfor-

mations in power systems [9]–[12]. These transformations are

mainly due to (and to accommodate) the increasing share of

inverter-based renewable generation.

Several indicators suggest that inverter-based resources will

be commonplace across transmission and distribution networks

in future grids [13], [14]. Vendors of inverter technology are

This work was supported by the U.S. Department of Energy (DOE) Ofﬁce

of Energy Efﬁciency and Renewable Energy under Solar Energy Technologies

Ofﬁce (SETO) through the award numbers EE0009025 and 38637 (UNIFI

consortium), respectively.

Fig. 1. Overview of the considered hierarchical scheme for system operation

and control.

quick to innovate, but incorporating uncountable such devices

into the operation and control architecture of power grids is not

(and cannot be expected to be) seamless. Utilities and system-

operators require easily veriﬁable benchmarks of anticipated

operation and performance. Establishing universal benchmarks

is challenged by the different operation timescales, the com-

plex set of nonlinear resource dynamics, and the lack of

available models for inverters [15]–[17].

We are interested in capturing all salient features of power-

system operation without delving deep into the minutiae

pertaining to modeling. To that end, we develop a general

multi-agent state-space model for a collection of resources

(generators and inverters) with outputs (powers) that are

intended to supply an external demand. The supply-demand

difference is set up to excite the dynamics of a system-

wide performance metric (frequency). A perfect match in

total supply and demand maps to desired system performance

(frequency equals nominal). The resources can respond lo-

cally to changes in the performance metric. For higher-level

coordination, we have a trip planner that is engaged in optimal

resource allocation, as well as a supervisory secondary-control

layer that systematically allocates the optimal trajectories to

the resources while executing a closed-loop control maneuver

to compensate the error in the performance metric. The above

architecture mimics all three operational layers of the power

system. It is set up with a level of generality to capture a

arXiv:2210.05841v1 [math.OC] 12 Oct 2022

wide range of reported secondary-control schemes as well as

macro-level salient features of resource dynamics. See Fig. 1

for an illustration of the overall system architecture. With this

generalized state-space model, we obtain the following:

•Steady-state characterization for power system dynamics

with and without secondary-control.

•Conspicuous dependence of system steady-state on spe-

ciﬁc design parameters.

•Rationale behind conventional design choices for

primary- and secondary-control handles.

Our development of a generalized system state-space model

to mirror fundamentals of power-system architecture is driven

by the limited attention bestowed by classical texts on in-

tegrated system modeling (and therefore, system-theoretic

analysis and design of desirable equilibria and control han-

dles) [18]–[20]. In general, there is an astonishingly vast col-

lection of work on redesigning secondary control, and this has

largely mirrored fashionable control synthesis tools du jour [7],

[9], [10], [21], [22]. Curiously, there is limited literature on

bench-marking the performance of the classical system—as it

is understood to be implemented in practice—from the point

of view of dynamic and steady-state performance. Notable

exceptions to this include [23] where a connection between

dispatch and secondary-control participation factors is estab-

lished to justify a common design practice for the latter, [24]

where the equilibria of the system dynamics are teased out and

precisely mapped to power-ﬂow equations, and [12] where a

theoretical stability analysis of automatic generation control

is presented with a formal treatment of time-scale separation

between primary and secondary control.

Of relevance is also a wide body of literature on microgrid

control and optimization [25]–[29]. In many such efforts,

there is a tendency to mirror the operational layers of the

bulk power system. However, given the increased ﬂexibility

on offer in synthesizing control and optimization schemes

(untethered by adherence to utility and system-operator speci-

ﬁcations), several advanced methods involving distributed and

decentralized schemes have appeared in the literature. While

not directly capturing all features of bulk-system operation

(and constraints), these effort, nonetheless, throw up several

interesting insights that can be translated.

The remainder of the paper is organized as follows: Sec-

tion II presents the considered abstract dynamic models and

overviews their relevant power-system instantiations. Sec-

tion III provides an analysis of the modeled systems, with

and without secondary control while particularly emphasizing

steady-state characterization. Having elucidated the impact of

prominent parameter choices on the system steady state, Sec-

tion IV outlines prudent design choices to harness engineering-

centric performance objectives. Summarizing remarks and

forward-looking discussions conclude the paper.

Notation. Throughout the paper, Rdenotes the set of real

numbers, while R>0and R<0are the subsets of positive and

negative real numbers, respectively. Upper- and lower-case

boldface letters denote matrices and column vectors. Given

a vector a, its n-th entry is denoted by an;diag(a)represents

a diagonal matrix with abeing the principal diagonal. The

symbol (·)>stands for transposition and ˙

ximplies the time-

derivative of x. The symbol 1denotes a vector of all ones, 0

denotes a vector or a matrix of all zeros, and Iis the identity

matrix; all with appropriate dimensions.

II. MODELING

For majority of the technical exposition of this paper, we

employ a generic notation that considers a resource supply-

demand setup with explicit dynamical structure assumed for

individual agents, and for a shared supervisory control. The

assumed schemes are inspired by prevalent primary- and

secondary-control architectures in power systems. The generic

structure will allow us to obtain representative results while

avoiding cumbersome notational overhangs. Nevertheless, per-

tinent remarks delineating the mapping from the considered

generic model to speciﬁc power system instantiations will be

suitably provided.

A. System Dynamical Model

Consider a supply-demand setup of a certain commodity

where a demand dis to served collectively by Nagents.

Denote the supply by vector x∈RNsuch that xnis the output

of supplier n. We will be treating vector xas a dynamic state,

the governing model of which will shortly follow. Consider a

system-wide mismatch indicator ythat is a proxy for supply-

demand mismatch under steady-state, necessitating speciﬁcally

1>xss =d⇐⇒ yss = 0, where xss and yss denote the

steady-state values of xand y, respectively.

Figure 1 illustrates the overall system we study. It is

composed of four subsystems. The ﬁrst captures the dynam-

ics of the system-wide performance metric, y; the second

encapsulates the dynamics of the agents (executing control

at a primary-control level); the third is a supervisory system

effecting control at a secondary level; and the fourth is a long-

horizon trip planner implementing a tertiary-level optimiza-

tion. We discuss the salient dynamics of each subsystem next.

1) System-wide Performance Metric: The dynamics of state

yis assumed to be governed by:

M˙y+Dy =1>x−d, (1)

where, M > 0and D > 0represent cumulative inertia and

damping, respectively.

2) Agents & Primary Control: The agents are assumed to

be ﬁrst-order integrators with the capacity to respond locally to

the mismatch. In particular, they are characterized by dynamics

diag(τ)˙

x=xref −x+ry, (2)

where, τ∈RN

>0captures the time-constants of the agents’

response, xref ∈RNdenotes the reference command being

provided from the supervisory control layer, and r∈RN

denotes the vector of local regulation gains. Since a positive

yis indicative of an oversupply (cf. the steady-state (1)),

local corrective action is engineered for negative feedback by

picking entries of rto be negative.

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TowardsOptimalPrimary-andSecondary-controlDesignforNetworkswithGeneratorsandInvertersManishK.Singh,D.Venkatramanan,andSairajDhopleDepartmentofElectrical&ComputerEngineeringUniversityofMinnesotaMinneapolis,MNUSAfmsingh,dvenkat,sdhopleg@umn.eduAbstractForpowergridspredominantlyfeaturinglargesyn-chron...

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Towards Optimal Primary- and Secondary-control Design for Networks with Generators and Inverters Manish K. Singh D. Venkatramanan and Sairaj Dhople

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