1 Optimal Design of V oltV AR Control Rules for Inverter-Interfaced Distributed Energy Resources

2025-04-30 0 0 2.48MB 12 页 10玖币

侵权投诉

Optimal Design of Volt/VAR Control Rules for

Inverter-Interfaced Distributed Energy Resources

Ilgiz Murzakhanov, Member, IEEE, Sarthak Gupta, Graduate Student Member, IEEE,

Spyros Chatzivasileiadis, Senior Member, IEEE, and Vassilis Kekatos, Senior Member, IEEE

Abstract—The IEEE 1547 Standard for the interconnection

of distributed energy resources (DERs) to distribution grids

provisions that smart inverters could be implementing Volt/VAR

control rules among other options. Such rules enable DERs to

respond autonomously in response to time-varying grid loading

conditions. The rules comprise afﬁne droop control augmented

with a deadband and saturation regions. Nonetheless, selecting

the shape of these rules is not an obvious task, and the default

options may not be optimal or dynamically stable. To this end,

this work develops a novel methodology for customizing Volt/VAR

rules on a per-bus basis for a single-phase feeder. The rules are

adjusted by the utility every few hours depending on anticipated

demand and solar scenarios. Using a projected gradient descent-

based algorithm, rules are designed to improve the feeder’s

voltage proﬁle, comply with IEEE 1547 constraints, and guar-

antee stability of the underlying nonlinear grid dynamics. The

stability region is inner approximated by a polytope and the

rules are judiciously parameterized so their feasible set is convex.

Numerical tests using real-world data on the IEEE 141-bus feeder

corroborate the scalability of the methodology and explore the

trade-offs of Volt/VAR control with alternatives.

Index Terms—Dynamic stability; second-order cone; nonlinear

dynamics; project gradient descent; voltage proﬁle.

I. INTRODUCTION

Motivated by climate change concerns and rising fossil

fuel prices, countries around the globe are integrating large

amounts of solar photovoltaics and other distributed energy

resources (DERs) into the grid. Unfortunately, the uncertain

nature of photovoltaics and DERs can result in undesirable

voltage ﬂuctuations in distribution feeders. Inverters equipped

with advanced power electronics can provide effective voltage

regulation through reactive power compensation if properly

orchestrated. This work aims at designing the Volt/VAR con-

trol rules for inverters, as recommended by the IEEE 1547.8

Standard [1], on a quasi-static basis to ensure their dynamic

stability and real-time voltage regulation performance.

Inverter-based voltage regulation has been extensively stud-

ied and adopted approaches can be classiﬁed as central-

ized,distributed, and localized.Centralized approaches entail

Manuscript received October 22, 2022; and revised February 10, and

April 23, 2023; accepted May 20, 2023. This work was supported in

part by the ID-EDGe project, funded by Innovation Fund Denmark, Grant

Agreement No. 8127-00017B, and the US National Science Foundation

grant 2034137. I. Murzakhanov and S. Chatzivasileiadis are with the De-

partment of Wind and Energy Systems, Technical University of Denmark.

E-mails: {ilgmu, spchatz}@dtu.dk. S. Gupta and V. Kekatos are with the

Bradley Dept. of ECE, Virginia Tech, Blacksburg, VA 24061, USA. E-mails:

{gsarthak,kekatos}@vt.edu

Color versions of one or more of the ﬁgures is this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer XXXXXX

communicating instantaneous load/solar data to the utility,

solving an optimal power ﬂow (OPF) problem to obtain

optimal setpoints [2], and communicating back to inverters.

Although centralized approaches are able to compute optimal

setpoints, they may incur high computation and communica-

tion overhead in real time. Distributed approaches partially

address these concerns by sharing the computational burden

across inverters [3], [4]. However, they may need a large

number of iterations to converge, which leads to delays in

obtaining setpoints. Real-time OPF schemes where setpoints

are updated dynamically have been shown to be effective [5],

[6] by computing fast an approximate solution, yet two-way

communication is still necessary.

Controlling inverters using control rules has been advocated

as an effective means to reduce the computational overhead.

In such a scheme, inverter setpoints are decided as a (non)-

linear function of solar, load, and/or voltage data; see e.g., [7],

[8] and references therein. Although such approaches reduce

the computational burden, they still have high communication

needs if driven by non-local data. To this end, there has been

increased interest in local rules, i.e., policies driven by purely

local data [9]. Perhaps not surprisingly, local control rules lack

global optimality guarantees as established in [10], [11], yet

they offer autonomous inverter operation.

As a predominant example of local control rules, the IEEE

1547.8 standard provisions that inverter setpoints can be

selected upon Volt/VAR, Watt/VAR, or Volt/Watt rules [1].

The recommended rules take a parametric, non-increasing,

piecewise afﬁne shape, equipped with saturation regions and

a deadband. Albeit easy to implement, designing the exact

shape of control curves is not an obvious task. Among the

different control options, Volt/VAR rules could be considered

most effective as voltage is the quantity to be controlled

and also carries non-local information. Watt/VAR curves have

been optimally designed before in [12], [13]. The resulting

optimization models involve products between continuous

and binary variables, which can be handled exactly using

McCormick relaxation (big-M trick) as in [13]. On the other

hand, designing Volt/VAR curves is more challenging as they

incur a closed-loop dynamical system, whose stability needs to

be enforced. Moreover, designing Volt/VAR curves gives rise

to optimization models involving products between continuous

variables, which are harder to deal with.

Although Volt/VAR rules have been shown to be stable

under appropriate conditions, their equilibria may not be

optimal in terms of voltage regulation performance [14], [15],

[16]. This brings about the need for systematically designing

arXiv:2210.12805v2 [eess.SY] 23 May 2023

Volt/VAR curves and customizing their shapes based on grid

loading conditions on a per-bus basis. Volt/VAR dynamics

exhibit an inherent trade-off between stability and voltage

regulation. In view of this, several works have suggested

augmenting Volt/VAR rules with a delay component so that

the reactive power setpoint q(t)at time tdepends on voltage

v(t)as well as the previous setpoint q(t−1). These so-termed

incremental rules have been studied and designed in [17],

[18], [19], [20], [21]. Here we focus on non-incremental

Volt/VAR curves to be compliant with the IEEE 1547.8

Standard. Reference [22] designs stable Volt/VAR curves to

minimize the worst-case voltage excursions when loads and

solar generation lie within a polyhedral uncertainty set. This

design task can be approximated by a quadratic program;

however, Volt/VAR rules are oversimpliﬁed as afﬁne, ignor-

ing their deadband and/or saturation regions. For example,

reference [23] simultaneously optimizes afﬁne Volt/VAR and

nonlinear (polynomial) Volt/Watt rules. The proposed opti-

mization program incorporates stability and adopts a robust

uncertainty set design. Reference [24] considers the detailed

model of Volt/VAR curves and integrates them into a higher-

level OPF formulation to properly capture the behavior of

Volt/VAR-driven DERs; nevertheless, here curve parameters

are assumed ﬁxed and are not designed.

This work considers the optimal design of Volt/VAR control

rules in single-phase distribution grids. Using a dataset of

grid loading scenarios anticipated for the next 2-hr period,

the goal is to centrally and optimally design Volt/VAR con-

trol curves to attain a desirable voltage proﬁle across a

feeder. The contributions are on three fronts: i) Develop a

scalable projected gradient descent algorithm to ﬁnd near-

optimal Volt/VAR control curves. The computed curves are

customized per inverter location, comply with the detailed

form and constraints provisioned by the IEEE 1547 Standard,

and ensure stable Volt/VAR dynamics (Section IV); ii) Provide

a polytopic representation for the dynamic stability region

of Volt/VAR control rules (Section II-C); and iii) Select a

proper representation of the Volt/VAR rule parameters so that

stability and the IEEE 1547-related constraints are expressed

as a convex feasible set (Section III). The proposed design

scheme is evaluated using numerical tests using real-world

load and solar generation data on the IEEE 141-bus feeder.

The paper is organized as follows. Section II reviews an

approximate feeder model, the IEEE 1547 Volt/VAR rules,

their steady-state properties, and expands upon their stability.

Section III states the task of optimal rule design and selects

a proper parameterization of the rules. Section IV presents an

iterative algorithm based on projected gradient descent to cope

with the optimal rule design task. Numerical tests are reported

in Section V and conclusions are drawn in Section VI.

Notation: Column vectors (matrices) are denoted by lower-

(upper-) case letters. Operator dg(x)returns a diagonal matrix

with xon its diagonal. Symbol (·)⊤stands for transposition;

and INis the N×Nidentity matrix.

II. FEEDER AND CONTROL RULE MODELING

A. Feeder Modeling

Consider a single-phase radial distribution feeder with N+1

buses hosting a combination of inelastic loads and DERs.

Buses are indexed by set N:= {1, . . . , N}. Let vndenote

the voltage magnitude at bus n, and pn+jqnthe complex

power injected at bus n∈ N . Let vectors (v,p,q)collect the

aforesaid quantities across all buses. To express the depen-

dence of voltages on power injections, we adopt the widely

used linearized grid model [25]

v≃Rp +Xq +v01(1)

where v0is the substation voltage. Matrices (R,X)are sym-

metric positive deﬁnite with positive entries. They depend on

the feeder topology and line impedances, which are assumed

ﬁxed and known for the control period of interest. Symbol 1

denotes a vector of all ones and of appropriate length.

The vectors of power injections can be decomposed as

p=pg−pℓand q=qg−qℓ

where pg+jqgis the complex power injected by inverter-

interfaced DERs, and pℓ+jqℓis the complex power consumed

by uncontrollable loads.

Volt/VAR control amounts to adjusting qgwith the goal

of maintaining voltages around one per unit (pu) despite

ﬂuctuations in (pg,pℓ,qℓ). With this control objective in

mind, let us rewrite (1) as

v=Xqg+˜

v=Xq +˜

v(2)

where with a slight abuse in notation, we will henceforth

denote qgsimply by q. Moreover, vector ˜

v:= R(pg−

pℓ)−Xqℓ+v01captures the effect of current grid loading

conditions on voltages, and will be referred to simply as the

vector of grid conditions.

B. Volt/VAR Rules

The IEEE 1547 standard provisions DERs to provide re-

active power support according to four possible modes: i)

constant reactive power; ii) constant power factor; iii) active

power-dependent reactive power (watt-var); and iv) voltage-

dependent reactive power (volt-var) mode. Mode i) is invariant

to grid conditions. Modes ii) and iii) do adjust reactive injec-

tions, yet adjustments depend solely on the active injection

of the individual DER. On the contrary, mode iv) adjusts the

reactive power injected by each DER based on its voltage

magnitude. Albeit measured locally, voltage carries non-local

grid information and constitutes the quantity of control interest

anyway. Hence, our focus is on Volt/VAR control rules.

Per the IEEE 1547 standard [1], a Volt/VAR control rule is

described by the piecewise afﬁne function shown on Fig. 1(a).

This plot shows the dependence of reactive injection on local

voltages over a given range of operating voltages [vℓ, vh]. The

curve is described by voltage points (v1, v2, vr, v3, v4)and

reactive power points (q1, q4). To simplify the presentation,

let us suppose the Volt/VAR curve is odd symmetric around

the axis v= ¯v=vr. The simpliﬁed rule shown in Fig. 1(b)

Fig. 1. (a) IEEE 1547 Volt/VAR rule [1]; and (b) its symmetric version.

can be described by four parameters: the reference voltage ¯v;

the deadband voltage ¯v+δ; the saturation voltage ¯v+σ; and

the saturation reactive power injection ¯q. Per the standard, the

tuple (¯v, δ, σ, ¯q)expressed in pu is constrained as

0.95 ≤¯v≤1.05 (3a)

0≤δ≤0.03 (3b)

δ+ 0.02 ≤σ≤0.18 (3c)

0≤¯q≤ˆq. (3d)

Constraint (3d) ensures the extreme reactive setpoints are

within the reactive power capability ˆqof the DER. It is worth

emphasizing here that the Volt/VAR curve can be designed

to saturate at ¯qthat can be lower than ˆq. The standard also

speciﬁes default settings as δ= 0.02,σ= 0.08,¯v= 1, and

¯q= ˆq= 0.44¯p, where ¯pis the per-unit kW rating of the DER.

The rule segment over [¯v+δ, ¯v+σ]can also be written as

q=−α(v−¯v−δ)where α=¯q

σ−δ>0.(4)

The rule segment over [¯v−σ, ¯v−δ]is q=−α(v−¯v+δ).

We will see later that the slope αis crucial for stability.

Although the standard offers ﬂexibility in the design of

Volt/VAR curves, it is not clear to utilities and software

vendors how to optimally tune such control settings. The

tuple (¯v, δ, σ, ¯q)can be customized on a per bus basis. Let

qn=fn(vn)denote the Volt/VAR rule for bus n. The rule

for DER nis parameterized by (¯vn, δn, σn,¯qn). Let vectors

(¯

v,δ,σ,¯

q)collect the rule parameters across all buses.

When Volt/VAR-controlled DERs interact with the electric

grid, they give rise to the nonlinear dynamical system [14]

vt=Xqt+˜

v(5a)

qt+1 =f(vt).(5b)

where the n-th entry of (5b) denotes the Volt/VAR rule qn=

fn(vn)for DER n. Given the nonlinear dynamics of (5), three

questions arise: q1) Is the system stable? In other words, if

initiated at some v0for t= 0, DERs run Volt/VAR rules qn=

fn(vn)for all n, and grid experiences loading conditions ˜

do the nonlinear dynamics of (5) reach an equilibrium where

vt=veq and qt=qeq for all t>T for some T? During the

interval t∈[0, T ], the grid loading conditions ˜

vare assumed

time-invariant assuming Volt/VAR dynamics are faster than

load/solar variations. q2) If stable, what is its equilibrium? and

q3) Is that equilibrium useful for voltage regulation? These

questions have been addressed in [14], [15]. We review and

build upon those answers.

C. Stability of Volt/VAR Rules

Regarding q1), let vector αcollect all slope parameters αn

and deﬁne the diagonal matrix A:= dg(α). The nonlinear

dynamics in (5) are stable if ∥AX∥2<1, where ∥AX∥2is

the maximum singular value of AX; see [14], [15], [16] for

proofs of local and global exponential stability. If DERs are

installed only on a subset G ⊆ N of buses, the condition

becomes ∥AXGG ∥2<1, where XGG is obtained from Xby

keeping only its rows/columns associated with the buses in

G. To simplify the presentation, we will henceforth assume

G=N, and elaborate when needed otherwise.

Because it is hard to satisfy ∥AX∥2<1as a strict

inequality, one may want to tighten the constraint as [22]

∥AX∥2≤1−ϵ(6)

for some positive ϵ. Constraint (6) can be expressed as a linear

matrix inequality (LMI) on αas

(1 −ϵ)I AX

XA (1 −ϵ)I≻0.

This is because by Schur’s complement, the LMI is equiv-

alent to (1 −ϵ)2I⪰XA2X, or equivalently, ∥AX∥2=

pλmax(XA2X)≤1−ϵ. To avoid the computational complex-

ity of an LMI, reference [14] surrogated the LMI for ϵ= 0

by the linear inequality ∥α∥∞· ∥X1∥∞<1. This inequality

may be conservative as it upper bounds all slopes αnby the

same constant ∥X1∥−1

∞. We next propose a tighter restriction

of the LMI involving only linear inequality constraints.

Lemma 1. The dynamical system in (5) is stable if

Xα≤(1 −ϵ)·1(7a)

α≤(1 −ϵ)·[dg(X1)]−11(7b)

with the inequalities understood entrywise.

Proof: Matrices Aand Xhave non-negative entries. By

a rendition of H¨

older’s inequality, it holds that

∥AX∥2

2≤ ∥AX∥1· ∥AX∥∞

where ∥AX∥1is the maximum column-sum and ∥AX∥∞

is the maximum row-sum of matrix AX. It is not hard to

verify that (7a) implies ∥AX∥1≤1−ϵ, while (7b) implies

∥AX∥∞≤1−ϵ, so (6) follows.

Note that previous works have suggested that (7b) alone is

sufﬁcient to ensure stability [15], [21]. The next counterexam-

ple shows that (7a) is needed as well. Suppose a toy feeder

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

1OptimalDesignofVolt/VARControlRulesforInverter-InterfacedDistributedEnergyResourcesIlgizMurzakhanov,Member,IEEE,SarthakGupta,GraduateStudentMember,IEEE,SpyrosChatzivasileiadis,SeniorMember,IEEE,andVassilisKekatos,SeniorMember,IEEEAbstract—TheIEEE1547Standardfortheinterconnectionofdistributedenergyr...

展开>> 收起<<

1 Optimal Design of V oltV AR Control Rules for Inverter-Interfaced Distributed Energy Resources.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

1 Optimal Design of V oltV AR Control Rules for Inverter-Interfaced Distributed Energy Resources

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: