#  Exploring reactive power limits on wind
farm collector networks with
convex inner approximations

Nawaf Nazir  Energy and Environment Directorate
Pacific Northwest National Laboratory
Richland, USA
<EMAIL_ADDRESS>Ian A. Hiskens  Department of EECS
University of Michigan
Ann Arbor, USA
<EMAIL_ADDRESS>Mads R. Almassalkhi N. Nazir and M. Almassalkhi were
supported by the U.S. Department of Energy’s Advanced Research Projects
Agency-Energy Award DE-AR0000694 and the National Science Foundation (NSF)
Award ECCS-2047306. Department of EBE
University of Vermont
Burlington, USA
<EMAIL_ADDRESS>
###### Abstract

A wind farm can provide reactive power at sub-transmission and transmission
buses in order to support and improve voltage profiles. It is common for the
reactive power capability of a wind farm to be evaluated as the sum of the
individual turbine ratings. However, such an assessment does not take into
account losses over the collector network, nor the voltage constraints imposed
by the turbines and network. In contrast, the paper presents a method for
determining the range of reactive power support that each turbine can provide
whilst guaranteeing satisfaction of voltage constraints. This is achieved by
constructing convex inner approximations of the non-convex set of admissible
reactive power injections. We present theoretical analysis that supports the
constraint satisfaction guarantees. An example illustrates the effectiveness
of the algorithm and provides a comparison with a fully decentralized approach
to controlling wind farm reactive power. Such approaches have the potential to
improve the design and operation of wind farm collector networks, reducing the
need for additional costly reactive power resources.

##  I INTRODUCTION

Reactive power support from wind farms can play on important role in
maintaining power system reliability. Specifically, the reactive power
capability of type 3 and type 4 wind turbines can be used to regulate the grid
voltage at the point of common coupling (PCC) [1]. Hence, it is important for
wind farm operators to characterize and control their reactive power
capability so that this resource is available to the transmission system
operator (TSO). Early work characterizing the reactive power capability did
not account for the collector network that interconnects the wind turbines
[2]. Consequently, the impact of voltage limits could not be assessed [3].
This issue was partially addressed by the decentralized control scheme
proposed in [4, 5], which controls wind turbine reactive power to regulate the
PCC voltage but does not offer a priori assessment of the available reactive
power capability. More recent work provides voltage support from wind farms by
using a sensitivity-based approach to rank reactive power loading for wind
farms and their turbines [6].

A alternative approach to account for the wind farm network while dispatching
turbine reactive power, is to explicitly consider the wind farm’s radial
(balanced) network within an optimal power flow (OPF) setting. However, that
requires solving a non-convex OPF problem, which is NP-hard [7]. The technical
challenges associated with the non-convex formulation could be overcome by
considering either linear approximations or convex relaxations [8]. For
example, traditional methods for solving the OPF problem in (radial) networks
include the LinDist model, which neglects the losses in the network to arrive
at a simplified linear model. Much of the previous work on wind farm
optimization utilizes the LinDist model as it offers computational benefits.
However, ignoring line losses (both reactive and active) can lead to unmodeled
voltage violations under certain operating conditions [9]. Since convex
relaxations of the OPF problem can provide solutions with zero duality [10],
they have become popular proxies for the underlying network physics. However,
in the case of a wind farm providing a desired value of reactive power at the
PCC, convex relaxations can engender optimal solutions with so-called
fictitious losses whose realized dispatch can cause voltages to exceed their
limits [11].

This paper overcomes previous shortcomings by employing convex inner
approximations (or convex restrictions) to determine a wind farm’s realizable
reactive power capacity, and devise a feedback control scheme for regulating
the PCC voltage. The control strategy disaggregates the time-varying reactive
power reference among the wind turbines in a manner that guarantees network
conditions always remain within limits. Unlike convex relaxations (i.e., outer
approximations) and linearized approximations, convex inner approximations
(CIAs) ensure that feasible solutions are also physically realizable. Of
course, inner approximation may beget conservative solutions that can reduce
performance.

Previously, CIAs have been employed in the optimization of dispatching
(discrete) mechanical grid assets [12] and (continuous) distributed energy
resources (DERs) [13]. In this work, we adapt CIAs to determine practical
reactive power bounds for each turbine (i.e., at each node) for a given wind
power scenario. Within these nodal reactive power bounds, we can guarantee
that any combination of turbine reactive power dispatch will ensure that
network voltages are within their limits. Based on these nodal bounds, a real-
time disaggregation control loop is formulated that can dispatch turbines and
deliver desired reactive power to support grid operations.

The paper is organised as follows: Section II develops the mathematical model
of a wind farm network and illustrates the concept of nodal reactive
capacities on a simple 3-node wind farm. Section III develops the convex inner
approximation for the non-convex optimization problem that defines the wind
farm’s reactive power capability. Section IV develops a real-time control
algorithm that provides grid voltage support while ensuring satisfaction of
wind farm network voltage constraints. Section V concludes the paper and
highlights future research directions.

##  II Mathematical modeling and nodal reactive capacity

###  II-A Wind farm model

In this section, we present the model of a wind farm, where a balanced, radial
network often couples the turbines to the PCC, as shown in Fig. 1. Thus, we
can use the nonlinear DistFlow formulation to model the wind farm network.
Consider an undirected graph
$\mathcal{G}=\\{\mathcal{N}\cup\\{0\\},\mathcal{L}\\}$ consisting of a set of
$N+1$ nodes with $\mathcal{N}:=\\{1,\ldots,N\\}$ and a set of $N$ branches
$\mathcal{L}:=\\{1,\ldots,N\\}\subseteq\mathcal{N}\times\mathcal{N}$, such
that $(i,j)\in\mathcal{L}$, if nodes $i,j$ are connected. Node $0$ is assumed
to be the head node (i.e., PCC) with a fixed voltage $V_{0}$. Let
$B\in\mathbb{R}^{(N+1)\times N}$ be the incidence matrix of $\mathcal{G}$
relating the branches in $\mathcal{L}$ to the nodes in
$\mathcal{N}\cup\\{0\\}$, such that the $(i,k)$-th entry of $B$ is $1$ if the
$i$-th node is connected to the $k$-th branch and, otherwise, $0$. Without
loss of generality, $B$ can be organized to form an upper-triangular matrix.
If $V_{i}$ and $V_{j}$ are the voltage phasors at nodes $i$ and $j$ and
$I_{ij}$ is the current phasor in branch $(i,j)\in\mathcal{L}$, then define
$v_{i}:=|V_{i}|^{2}$, $v_{j}:=|V_{j}|^{2}$ and $l_{ij}:=|I_{ij}|^{2}$. Let
$P_{ij}$ ($Q_{ij}$) be the active (reactive) power flow from node $j$ to $i$,
let $p_{j}$ ($q_{j}$) be the active (reactive) power generations into node
$j$, and let $r_{ij}$ ($x_{ij}$) be the resistance (reactance) of branch
$(i,j)\in\mathcal{L}$, which means that the branch impedance is given by
$z_{ij}:=r_{ij}+jx_{ij}$. Then, for a radial wind farm, the relation between
node voltages and power flows is given by the DistFlow equations
$\forall(i,j)\in\mathcal{L}$:

$\displaystyle v_{j}=$ $\displaystyle
v_{i}+2r_{ij}P_{ij}+2x_{ij}Q_{ij}-|z_{ij}|^{2}l_{ij}$ (1a) $\displaystyle
P_{ij}=$ $\displaystyle p_{j}+\sum_{h:h\rightarrow j}(P_{jh}-r_{jh}l_{jh})$
(1b) $\displaystyle Q_{ij}=$ $\displaystyle q_{j}+\sum_{h:h\rightarrow
j}(Q_{jh}-x_{jh}l_{jh})$ (1c) $\displaystyle l_{ij}(P_{ij},Q_{ij},v_{j})=$
$\displaystyle\frac{P_{ij}^{2}+Q_{ij}^{2}}{v_{j}},$ (1d)

Figure 1:  Nomenclature for a radial wind farm network [14].

The goal of this work is to maximize the range of reactive power output from
the wind farm, i.e., $Q_{10}$, such that all voltages $v_{j}$ and currents
$l_{ij}$ are within their respective limits (i.e.,
$v_{j}\in[\underline{v}_{j},\overline{v}_{j}]\,\,\forall j\in\mathcal{N}$ and
$l_{ij}\in[\underline{l}_{ij},\overline{l}_{ij}]\,\,\forall(i,j)\in\mathcal{L}$).
However, finding such a range is challenging due to the non-linear nature of
(1d). For clarity, we provide definitions of the following key terms used in
the manuscript.

######  Definition II.1 (AC Admissibility).

A solution of a convex OPF problem is AC admissible, if the solution applied
to the original, non-convex AC OPF, which uses (1), is feasible.

######  Definition II.2 (Nodal reactive capacity).

Nodal reactive capacity is the range of AC admissible reactive power dispatch
$\Delta q_{j}:=[q_{j}^{-},q_{j}^{+}]\,\forall j\in\mathcal{N}$ with lower and
upper bounds $q_{j}^{-}\leq 0$ and $q_{j}^{+}\geq 0$, respectively. That is,
for all nodes $j$, any dispatch $q_{j}\in\Delta q_{j}$ is AC admissible.

Next, we consider the nodal reactive capacity of a simple 3-node wind farm
network in Fig. 2 to motivate the approach.

###  II-B Motivating example on nodal reactive capacity

Fig. 2 represents a simple, balanced wind farm network with two turbines at
nodes $2$ and $3$, and $V_{0}=1$ pu. Each (positive sequence) branch of the
network has impedance $z=0.228+0.092j$ pu. Nodes $2$ and $3$ have generation
$s_{\text{g},2}=0.005-0.02j$ pu and $s_{\text{g},3}=0.01-0.015j$ pu,
respectively. Only the reactive power injections at nodes $2$ and $3$
(labelled $q_{\text{g},2}$ and $q_{\text{g},3}$) are assumed to be
controllable. Based on the AC power flow solutions obtained with Matpower
[15], by varying $q_{\text{g},2}$ and $q_{\text{g},3}$, Fig. 3 shows the
feasible set of the AC OPF for the 3-node system. The figure shows that the
admissible set is non-convex and contains a “hole” due to a voltage
constraint. Hence, it is important when dispatching $q_{\text{g},2}$ and
$q_{\text{g},3}$ to choose a trajectory that ensures AC admissibility.
Specifically, Fig. 3 shows that trajectory A is contained in the admissible
set and, hence, the resulting network voltages are within their limits as this
dispatch trajectory is traversed. However, dispatch trajectory B passes
through the “hole” and results in voltage violations. Even though trajectory A
is AC admissible it requires $q_{\text{g},2}$ and $q_{\text{g},3}$ to be
coordinated (i.e., stay on the trajectory) to ensure admissibility. This means
that any change to one requires a change in the other and, thus, they are not
considered nodal reactive capacities. This simple example shows the need to
develop methods that compute nodal reactive capacities for wind farms. This
can avoid communication requirements between turbines in a wind-farm, paving
the way for fast real-time control. Towards that objective, the next section
develops a convex inner approximation of the non-convex DistFlow formulation
in (1).

Figure 2:  A simple 3-node wind farm network. Figure 3:  Analysis of 3-node
example. (Top) The set of admissible injections is non-convex. (Bottom)
Voltage profiles along admissible (A) and inadmissible (B) trajectories

##  III Convex inner approximation and optimization formulation

###  III-A Convex inner approximation

In this section, we first present a compact matrix representation of the
linear components (1a)-(1c). Then, we bound the nonlinear branch current terms
in (1d), $l_{ij}(P_{ij},Q_{ij},v_{j})$, by a convex envelope, which leads to a
convex inner approximation of (1).

First, define vectors $P:=[P_{ij}]_{(i,j)\in\mathcal{L}}\in\mathbb{R}^{N}$,
$Q:=[Q_{ij}]_{(i,j)\in\mathcal{L}}\in\mathbb{R}^{N}$,
$V:=[v_{i}]_{i\in\mathcal{N}}\in\mathbb{R}^{N}$,
$p:=[p_{i}]_{i\in\mathcal{N}}\in\mathbb{R}^{N}$,
$q:=[q_{i}]_{i\in\mathcal{N}}\in\mathbb{R}^{N}$, and
$l:=[l_{ij}]_{(i,j)\in\mathcal{L}}\in\mathbb{R}^{N}$ and matrices
$R:=\text{diag}\\{r_{ij}\\}_{(i,j)\in\mathcal{L}}\in\mathbb{R}^{N\times N}$,
$X:=\text{diag}\\{x_{ij}\\}_{(i,j)\in\mathcal{L}}\in\mathbb{R}^{N\times N}$,
$Z^{2}:=\text{diag}\\{z_{ij}^{2}\\}_{(i,j)\in\mathcal{L}}\in\mathbb{R}^{N\times
N}$, and $A:=[0_{N}\quad I_{N}]B-I_{N}$, where $I_{N}$ is the $N\times N$
identity matrix and $0_{N}$ is a column vector of $N$ rows. Then, directly
applying [14], we get expressions for $P$, $Q$ and $V$:

$\displaystyle V=v_{\text{0}}\mathbf{1}_{N}+M_{\text{p}}p+M_{\text{q}}q-Hl,$
(2) $\displaystyle P=Cp-D_{\text{R}}l,\qquad Q=Cq-D_{\text{X}}l,$ (3)

where matrices $M_{\text{p}}:=2C^{T}RC$, $M_{\text{q}}:=2C^{T}XC$,
$H:=C^{T}(2(RD_{\text{R}}+XD_{\text{X}})+Z^{2})$ and $C:=(I_{N}-A)^{-1}$,
$D_{\text{R}}:=(I_{N}-A)^{-1}AR$, and $D_{\text{X}}:=(I_{N}-A)^{-1}AX$
describe the network topology and impedance parameters. Note that in [9], it
is proven that the matrix $(I_{N}-A)$ is non-singular for radial, balanced
distribution networks.

Clearly, (2) and (3) represent linear relationships between the nodal power
injections, $(p,q)$, the branch power flows, $(P,Q)$, and node voltages $V$.
However, setting $l=0$ and neglecting (1d), as done with the commonly used
LinDist approximation, can result in overestimating the nodal reactive
capacities [9]. Next, we present methods for bounding the nonlinearity
$l_{ij}(P_{ij},Q_{ij},v_{j})$ from above and below.

Based on the description of voltages in (2) and branch flows in (3), denote
$l_{\text{lb}}$ and $l_{\text{ub}}$ as lower and upper bounds on $l$. Then, we
can define the corresponding upper $(.)^{+}$ and lower $(.)^{-}$ bounds of
$P$, $Q$ and $V$ as follows:

$\displaystyle P^{+}(p):=$ $\displaystyle Cp-D_{\text{R}}l_{\text{lb}}$ (4a)
$\displaystyle P^{-}(p):=$ $\displaystyle Cp-D_{\text{R}}l_{\text{ub}}$ (4b)
$\displaystyle Q^{+}(q):=$ $\displaystyle Cq-
D_{\text{X}_{+}}l_{\text{lb}}-D_{\text{X}_{-}}l_{\text{ub}}$ (4c)
$\displaystyle Q^{-}(q):=$ $\displaystyle Cq-
D_{\text{X}_{+}}l_{\text{ub}}-D_{\text{X}_{-}}l_{\text{lb}}$ (4d)
$\displaystyle V^{+}(p,q):=$ $\displaystyle
v_{\text{0}}\mathbf{1}_{n}+M_{\text{p}}p+M_{\text{q}}q-H_{+}l_{\text{lb}}-H_{-}l_{\text{ub}}$
(4e) $\displaystyle V^{-}(p,q):=$ $\displaystyle
v_{\text{0}}\mathbf{1}_{n}+M_{\text{p}}p+M_{\text{q}}q-H_{+}l_{\text{ub}}-H_{-}l_{\text{lb}},$
(4f)

where $D_{\text{X}_{+}}$ and $H_{+}$ include the non-negative elements of
$D_{\text{X}}$ and $H$, respectively, and $D_{\text{X}_{-}}$ and $H_{-}$ are
the corresponding negative elements. For example, if the network is purely
inductive, then $D_{\text{X}_{-}}=H_{-}=0$ and the formulation reduces to the
one presented in [12]. These upper and lower bounds in (4) satisfy $P^{-}\leq
P\leq P^{+}$, $Q^{-}\leq Q\leq Q^{+}$ and $V^{-}\leq V\leq V^{+}$. Note that
the bounds $l_{\text{lb}},l_{\text{ub}}$ in (4) effectively allow us to
neglect the nonlinear (1d). Thus, if we can find convex representations of
these bounds, the corresponding OPF formulation will be a convex inner
approximation. This is described next.

######  Remark.

While this work uses a network where the wind turbines’ inductive transformers
overcome the lines’ capacitances, the approach presented herein readily
extends to networks with arbitrary inductance/capacitance impedances as
presented in [13].

Equation (4) provides a linear formulation for bounding the AC power flow
equations in terms of bounds $l_{\text{lb}},l_{\text{ub}}$ and controllable
generations. Next, we summarize the derivation of these bounds and leverage
them to formulate a novel convex inner approximation of the AC OPF to
determine the nodal reactive capacities for the wind farm network.

Based on any nominal or predicted operating point
$x^{0}_{ij}:=\text{col}\\{P_{ij}^{0},Q_{ij}^{0},v_{j}^{0}\\}\in\mathbb{R}^{3}$,
the second-order Taylor series approximation for (1d) can be expressed as:

$\displaystyle l_{ij}$ $\displaystyle\approx
l_{ij}^{0}+\mathbf{J_{ij}^{\top}}\mathbf{\delta_{ij}}+\frac{1}{2}\mathbf{\delta_{ij}^{\top}}\mathbf{H_{\text{e},ij}}\mathbf{\delta_{ij}}$
(5)

where $l_{ij}^{0}:=l_{ij}(P_{ij}^{0},Q_{ij}^{0},v_{j}^{0})$ are branch current
flows at the operating point and
$\mathbf{\delta_{ij}}(P_{ij},Q_{ij},v_{j},x_{ij}^{0})$, the Jacobian
$\mathbf{J_{ij}}$ and the Hessian $\mathbf{H_{\text{e},ij}}$ are defined
below:

$\displaystyle\mathbf{\delta_{ij}}:=\begin{bmatrix}P_{ij}-P_{ij}^{0}\\\
Q_{ij}-Q_{ij}^{0}\\\
v_{j}-v_{j}^{0}\end{bmatrix}\qquad\mathbf{J_{ij}}:=\begin{bmatrix}\frac{2P^{0}_{ij}}{v^{0}_{j}}\\\
\frac{2Q^{0}_{ij}}{v^{0}_{j}}\\\
-\frac{(P^{0}_{ij})^{2}+(Q^{0}_{ij})^{2}}{(v^{0}_{j})^{2}}\end{bmatrix}$ (6)
$\displaystyle\mathbf{H_{\text{e},ij}}:=\begin{bmatrix}\frac{2}{v^{0}_{j}}&&0&&\frac{-2P^{0}_{ij}}{(v^{0}_{j})^{2}}\\\
0&&\frac{2}{v^{0}_{j}}&&\frac{-2Q^{0}_{ij}}{(v^{0}_{j})^{2}}\\\
\frac{-2P^{0}_{ij}}{(v^{0}_{j})^{2}}&&\frac{-2Q^{0}_{ij}}{(v^{0}_{j})^{2}}&&2\frac{(P^{0}_{ij})^{2}+(Q^{0}_{ij})^{2}}{(v^{0}_{j})^{3}}\end{bmatrix}$
(7)

The expression in (5) holds if we can neglect the third order term, i.e., the
expression is cubic order accurate or the order of accuracy is
$\mathcal{O}(||\delta||_{\infty}^{3})$.

Furthermore, [12] shows that $\mathbf{H_{\text{e},ij}}$ is positive semi-
definite, which, together with (5), means that the lower and upper bounds of
$l_{ij}$ for all $(i,j)\in\mathcal{L}$ are given by:

$\displaystyle l_{ij}=|l_{ij}|$
$\displaystyle\approx|l_{ij}^{0}+\mathbf{J_{ij}^{\top}}\mathbf{\delta_{ij}}+\frac{1}{2}\mathbf{\delta_{ij}^{\top}}\mathbf{H_{\text{e},ij}}\mathbf{\delta_{ij}}|$
(8)
$\displaystyle\leq|l_{ij}^{0}|+|\mathbf{J_{ij}^{\top}}\mathbf{\delta_{ij}}|+|\frac{1}{2}\mathbf{\delta_{ij}^{\top}}\mathbf{H_{\text{e},ij}}\mathbf{\delta_{ij}}|$
(9) $\displaystyle\leq
l_{ij}^{0}+\max\\{2|\mathbf{J_{ij}^{\top}}\mathbf{\delta_{ij}}|,|\mathbf{\delta_{ij}^{\top}}\mathbf{H_{\text{e},ij}}\mathbf{\delta_{ij}}|\\}$
(10) $\displaystyle\implies l_{ij}$ $\displaystyle\leq
l_{ij}^{0}+\max\\{2|\mathbf{J_{ij+}}^{\top}\mathbf{\delta_{ij}^{+}}+\mathbf{J_{ij-}}^{\top}\mathbf{\delta_{ij}^{-}}|,\mathbf{\psi_{ij}}\\}=:l_{\text{ub},ij}$
(11) $\displaystyle l_{ij}$ $\displaystyle\geq
l_{ij}^{0}+\mathbf{J_{ij+}}^{\top}\mathbf{\delta_{ij}^{-}}+\mathbf{J_{ij-}}^{\top}\mathbf{\delta_{ij}^{+}}=:l_{\text{lb},ij}$
(12)

where $\mathbf{J_{ij+}}$ and $\mathbf{J_{ij-}}$ includes the positive and
negative elements of $\mathbf{J_{ij}}$,
$\mathbf{\delta_{ij}^{+}}:=\mathbf{\delta_{ij}}(P_{ij}^{+},Q_{ij}^{+},v_{j}^{+},x_{ij}^{0})$
and
$\mathbf{\delta_{ij}^{-}}:=\mathbf{\delta_{ij}}(P_{ij}^{-},Q_{ij}^{-},v_{j}^{-},x_{ij}^{0})$,
and
$\mathbf{\psi_{ij}}:=\max\\{\mathbf{(\delta_{ij}}^{+,-})^{\top}\mathbf{H_{\text{e},ij}}(\mathbf{\delta_{ij}}^{+,-})\\}$,
which represents the largest of eight possible combinations of $P/Q/v$ terms
in $\mathbf{\delta_{ij}}$ with mixed $+,-$ superscripts. Thus, from (4), (11)
and (12) we have a convex inner approximation of (1) that can be used to
determine the nodal reactive capacities.

###  III-B Optimizing wind farm nodal reactive capacity

The bounds from (11) and (12) allow us to omit (1d) entirely and replace the
original variables $P$, $Q$, and $V$ with their corresponding upper and lower
bounds $(.)^{+}$ and $(.)^{-}$ in (4). Since $(.)^{+}$ and $(.)^{-}$ are outer
approximations, using them in an OPF formulation results in a feasible set
that is contained in the original, non-convex AC OPF, which means that (P1)
and (P2) below represent convex inner approximations and can be used to
determine the wind farm nodal reactive capacity:

$\displaystyle\text{(P1)}\quad q^{+}=\arg\min_{q_{i}}\
-Q_{\text{10}}^{-}(q)+\sum_{i=1}^{N}f_{i}(q_{i})$ (13)
$\displaystyle\text{s.t.}\quad\eqref{eq:P_relation_1}-\eqref{eq:V_relation_2},\eqref{eq:l_upper},\eqref{eq:l_lower}$
(14) $\displaystyle\underline{V}\leq V^{-}(q)\quad V^{+}(q)\leq$
$\displaystyle\overline{V}$ (15) $\displaystyle
l_{\text{ub}}\leq\overline{l}\quad\underline{q}\leq q\leq\overline{q}$ (16)
$\displaystyle\text{(P2)}\quad q^{-}=\arg\min_{q_{i}}\
Q_{\text{10}}^{+}(q)+\sum_{i=1}^{N}f_{i}(q_{i})$ (17)
$\displaystyle\text{s.t.}\quad\eqref{eq:P_relation_1}-\eqref{eq:V_relation_2},\eqref{eq:l_upper},\eqref{eq:l_lower}$
(18) $\displaystyle\underline{V}\leq V^{-}(q)\quad V^{+}(q)\leq$
$\displaystyle\overline{V}$ (19) $\displaystyle
l_{\text{ub}}\leq\overline{l}\quad\underline{q}\leq q\leq\overline{q}$ (20)

where $Q_{10}^{-}$ and $Q_{10}^{+}$ are the lower and upper bound of the
reactive power flow in line connecting nodes $1$ and $0$ (also called
$Q_{\text{head}}$), (15) and (16) ,(19) and (20), ensure that any feasible
dispatch $q$ from (P1) and (P2) satisfies nodal voltages and branch flows in
the original AC OPF based on (1). To determine the nodal reactive capacity, we
must solve (P1) for the upper capacity $q^{+}$ and (P2) for the lower capacity
$q^{-}$. Thus, the objective function components, $f_{i}(q_{i})$, must be
designed to engender $q_{i}^{-}$ and $q_{i}^{+}$. For example, when computing
$q_{i}^{-}$, we can choose $f_{i}(q_{i}):=\alpha_{i}q_{i}\,$ and, for
$q_{i}^{+}$, we can designate $f_{i}(q_{i}):=-\alpha_{i}q_{i}$, where
$\alpha_{i}$ is the relative priority of nodal reactive capacity at node $i$.
Clearly, the objective function determines how nodal reactive capacities are
allocated over the network, but objective function design will be explored in
future work and is outside the scope of this paper.

######  Remark.

In this work, we use simple box constraints for the nodal reactive power
limits. However, other convex constraints, such as quadratic reactive
capability constraints (e..g,
$p_{i}^{2}+q_{i}^{2}\leq(S_{i}^{\text{max}})^{2}$ for apparent power limit,
$S_{i}^{\text{max}}$) can readily be included in the formulation. For example,
an extensive comparison of different active and reactive power inverter
schemes (and resulting constraints) is presented within the CIA formulation in
[13].

The optimization problem in (P1) and (P2) is applied to the 3-node example
shown in Fig. 2 to determine the inner convex set. The results for this
example network are shown in Fig. 4, where the green rectangular set is the
inner approximation obtained through (P1) and (P2). In this example, we also
adapt the approach in [13] to reactive power to iteratively expand the nodal
reactive capacities as indicated by the red dots in Fig. 4. A detailed
description of the iterative approach can be found in [13] and is beyond the
scope of this paper. The convex set in Fig. 4 allows for dispatching reactive
power resources without need for coordination among them, while at the same
time guaranteeing satisfaction of network constraints. For comparison we also
apply the method from [5] to this example network. Points resulting from this
method are shown by the cyan colored stars in Fig. 4.

Figure 4:  Algorithm is adapted to increase the admissible region via
iterations (red dots), where the inner (green) set’s boundary defines the
nodal reactive capacities. Also, solutions obtained by the method in [5] are
indicated by the cyan stars.

###  III-C Simulation resulting on nodal reactive capacity

To showcase the effectiveness of the proposed convex inner approximation
method, we conduct simulation-based analysis on the 19-turbine wind farm in
Fig. 5. We determine the reactive power capability of the wind farm and the
nodal reactive capacities for each turbine. We consider three scenarios
similar to those in [5], determine the resulting wind farm reactive power
capabilities and compare the results with [5]. These three scenarios compare
the reactive power capability of the wind-farm under different active power
conditions. For reference, the method in [5] broadcasts a reactive power
reference signal to each wind turbine. The local control scheme of each
turbine seeks to maintain the reactive power output at that broadcast
reference value. However, turbine protection will override reactive power
control, if necessary, to ensure the turbine’s terminal voltage does not
deviate beyond its limits.

We will also compare the results obtained through convex inner approximation
with those obtained using a convex relaxation method (outer approximation) and
by solving the full non-linear model (local solution). Each of the 19 turbines
in the network is rated at 1.65 MW and can operate in a voltage range of 0.9
to 1.1 pu. Each of the turbines has reactive power capability of $[-0.5,0.5]$
MVAr. The scenario descriptions are listed below:

  * •

Scenario 1: All wind turbines are operating at their active power generation
capacity of $p_{j}=1.65$ MW.

  * •

Scenario 2: The active power generation is half the nameplate capacity of the
wind farm, with the turbines at the ends of branches operating at full
capacity $p_{j}=1.65$ MW, and the others producing zero active power.

  * •

Scenario 3: The active power generation is half the nameplate capacity, with
the turbines operating at full and zero capacity swapped from scenario 2.

Figure 5: Layout of the 19 turbine wind farm.

Applying (P1) and (P2) to scenarios 1, 2, and 3 gives the nodal reactive
capacities shown in Figs. 7a, 7b and 7c, respectively. Specifically, Figs. 7a
and 7b show that the reactive power capacity is limited in scenarios 1 and 2.
This is due to the upper nodal voltage limits at the turbines, which constrain
the amount of reactive power support that can be provided by the entire wind
farm. In contrast, no voltage limits are binding in scenario 3, allowing full
reactive power support. To provide insights into the conservativeness of the
CIA method, we compare the total reactive power capability of the wind farm
network with results from three other approaches and formulations, the method
presented in [5], a convex relaxation (from [10]), and the NLP based on (1).
The comparison is provided in Table I and indicates that the proposed CIA
method can compute practically relevant capacities (i.e., CIA does not result
in overly conservative bounds) across three different active power (wind
generation) scenarios and against three different approaches/formulations.
Specifically, despite being an inner approximation (i.e., inherently
conservative), the CIA method’s iterative approach results in reactive
capacities (i.e., $Q_{\text{head}}$) that compare well overall against both a
convex relaxation method, which is an outer approximation (i.e., overly
optimistic), and the non-convex NLP-based method.

To further highlight that the CIA method is not overly conservative, Fig. 6
compares the accuracy of the second-order approximation in (5) with the non-
linear expression from (1d) for the 19-turbine wind farm in Fig. 5. Note that,
the worst-case approximation error for $l_{ij}$ is less than $6.1\times
10^{-4}$pu over the wide range of net injections $[-3000,3000]$ kVAR.

Figure 6:  Comparison of the second-order approximation from (5) with the
original (nonlinear) expression of $l$ in (1d) for the 19-turbine wind farm.
The maximum absolute approximation error is found at branch $l_{12}$ (head
branch) when the reactive net-injection at node $2$ ($q_{\text{g},2}$) is
$-3000$kVAR and is $6.1\times 10^{-4}$pu and the relative error is always
within $0.8\%$.

######  Remark.

The method in [5] achieves a larger reactive power capacity $Q_{\text{head}}$
than the CIA method in part because it does not restrict the reactive
injections to be positive at all nodes. (This requirement is enforced in the
CIA approach to obtain nodal capacities, $q_{i}^{+}$.) This additional degree
of freedom enlarges the operating range. From a practical perspective, the two
methods have different implementations. While the CIA method provides a priori
predictions of $Q_{\text{head}}$ and corresponding decoupled, network-aware
operating ranges, $[q_{i}^{-},q_{i}^{+}]$, it requires full knowledge of the
wind farm (network parameters and up-to-date active power generation) and
centralized computing to determine and broadcast local reactive power nodal
capacities. On the other hand, the method in [5] requires only local sensing
and control at each wind turbine, but cannot predict the wind farm’s total
$Q_{\text{head}}$ capability in advance.

(a)

(b)

(c)

Figure 7: Nodal reactive capacities for (a) scenario 1, (b) scenario 2 and (c) scenario 3. TABLE I: Comparison of $Q_{\text{head}}$ for different schemes Scheme | Scenario 1 | Scenario 2 | Scenario 3
---|---|---|---
CIA-based | [-9.9,7.0] | [-9.8,7.9] | [-9.6,9.4]
Nonlinear | [-9.9,7] | [-9.8,7.9] | [-9.6,9.4]
Relaxation | [-9.9,9.1] | [-9.8,9.3] | [-9.6,9.4]
From [5] | $\leq 7.3$ | $\leq 8.0$ | $\leq 9.4$

These scenarios indicate that the reactive power capacity of the wind farm
changes as active power generation changes. Specifically, there are two
constraints that are responsible for limiting the reactive power capacity: 1)
the (local) voltage constraints, and 2) the (local) turbine reactive power
limit. To understand the effects of these constraints on the aggregate wind
farm reactive power capacity, Fig. 8 shows the relationship between active
power generation $p_{i}$ (with all turbines producing the same
$p_{i}\in[0,1.65]$ MW) and the corresponding reactive power capacity of the
entire wind farm at the head-node, $Q_{\text{head}}$. As the active power
generation increases from $0$ (i.e., no turbines producing active power), the
wind farm’s reactive power capacity decreases slowly due to increasing losses.
In this initial phase, the reactive power is limited only by the upper
reactive power limits $\overline{q}_{i}$ of the turbines. However, at around
$p_{i}=1.2$ MW, some of the network and turbine voltage constraints become
active causing the reactive power capacity to reduce sharply.

Figure 8: Change in wind farm reactive power capacity with change in active
power generation.

##  IV Real-time voltage regulation algorithm

Based on the computed nodal reactive capacities, a real-time control algorithm
can be developed to dispatch reactive-power set-points to each turbine in the
wind farm in order to provide voltage support at the PCC. We again consider
the 19-turbine wind farm network shown in Fig. 5, where all the wind turbines
are operating at full capacity of 1.65 MW. It is assumed that the wind farm is
connected to a large power system at the PCC and the high-voltage transmission
system is represented by a Thévenin equivalent circuit with a voltage source
denoted by the voltage $V_{\text{grid}}$. The aim of this controller is to
maintain the wind farm PCC voltage $V_{\text{PCC}}$ at $1.06$ pu in the
presence of power system disturbances. It is assumed that the grid experiences
a fault, which causes $V_{\text{grid}}$ to suddenly drop. The grid’s voltage
then recovers slowly as part of a recovery event, which is modeled as a ramp.

The feedback control scheme is shown in Fig. 9. A PI controller is used to
regulate the reactive power dispatch based on the deviation of
$V_{\text{PCC}}$ from the reference value $V_{\text{ref}}$. The saturation
block restricts the total wind farm reactive power reference
$Q_{\text{TG}}^{\text{ref}}$ to within the pre-calculated maximum and minimum
wind turbine reactive power limits. A standard anti-windup mechanism is
implemented to ensures that if $Q_{\text{TG}}^{\text{ref}}$ saturates then the
PI controller’s integrator does not wind up. The ‘Disagg’ block disaggregates
the reactive power reference $Q_{\text{TG}}^{\text{ref}}[k]$ at time-step $k$
amongst the individual turbines relative to their nodal reactive capacities
(i.e., $q_{i}^{-}$ and $q_{i}^{+}$) as

$\displaystyle
q_{i}[k]=\begin{cases}\frac{q_{i}^{+}}{\sum_{i}q_{i}^{+}}Q_{\text{TG}}^{\text{ref}}[k]\qquad
Q_{\text{TG}}^{\text{ref}}[k]>0\\\
\frac{q_{i}^{-}}{\sum_{i}q_{i}^{-}}Q_{\text{TG}}^{\text{ref}}[k]\qquad
Q_{\text{TG}}^{\text{ref}}[k]<0.\end{cases}$ (21)

The PCC voltage $V_{\text{PCC}}$ is measured and fed back to achieve closed-
loop tracking of the voltage reference $V_{\text{ref}}$.

We compare this disaggregation scheme with a grid-agnostic scheme and contrast
the results of the real-time controller under these two approaches. The
disaggregation schemes can be summarized as:

  * •

CIA-based disaggregation: Disaggregation is proportional to the nodal reactive
capacities ($q^{-},q^{+}$) and the saturation block uses the computed reactive
power capacity limits for the wind farm.

  * •

Grid-agnostic disaggregation: Disaggregation is proportional to the turbine
reactive power capacities ($\underline{q},\overline{q}$) and the saturation
block uses the sum of those turbine capacity values. This scheme ignores the
collector network constraints.

The different disaggregation schemes are compared in Fig. 10. It is assumed
that all the wind turbines are operating at their rated active power, i.e., we
utilize scenario 1. Hence, for the method based on CIA, the nodal reactive
power capacity is provided in Fig. 7a. For the grid-agnostic approach, the
network is ignored so the only constraints are the turbine reactive power
limits ([-0.5,0.5] MVAr in this case). Hence, for this method, the nodal
reactive power capacity is the same as depicted in Fig. 7c. The two methods
for allocating reactive power across the wind turbines (grid-agnostic and CIA-
based) result in the PCC voltages $V_{\text{PCC}}$ shown in Figs. 10a and 10d.
Since the grid-agnostic approach employs the full range of nodal reactive
capacities, the RMSE error in tracking $V_{\text{ref}}$ for the CIA-based
approach is $\approx 2$ times that of the grid-agnostic approach. A comparison
of the reactive power dispatches $Q_{\text{TG}}^{\text{ref}}$ (control signal)
and $Q_{\text{head}}$ (physical quantity) is shown in Figs. 10b and 10e. From
the figures it can be seen that the grid-agnostic approach provides much
larger reactive power which results in the better voltage tracking
performance. The CIA-based method provides reactive power up to the pre-
calculated wind farm capacity (labelled “WF Capacity” in Fig. 10e), which is
obtained from the solution of (P1) and (P2). This results in the moderate
voltage tracking performance. The difference between the dispatch control
signal $Q_{\text{TG}}^{\text{ref}}$ and $Q_{\text{head}}$ is due to the losses
in the network. The impact of the reactive power dispatch on the wind farm’s
nodal voltages is depicted in Figs. 10c and 10f. It can be seen from Fig. 10c
that the large reactive power output results in voltage violations across the
wind farm. This is a consequence of the grid-agnostic approach not taking into
account the wind farm network when dispatching wind turbine reactive power. As
a result, it overestimates the network’s reactive power capacity, resulting in
voltage violations. In contrast, Fig. 10f shows that wind farm voltages are
within limits, except for violations during transients. This is not unexpected
as the CIA-based method only guarantees steady-state operating conditions.
Extending the CIA-based method to capture transient behaviour is an
interesting avenue for future work. This example illustrates the usefulness of
the CIA-based method for enabling wind farms to provide reactive power support
to transmission systems while ensuring reliability of the wind farm collector
network.

Figure 9:  Proposed control scheme for real-time disaggregation and grid
voltage support.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10: Comparing the methods of disaggregating a desired reactive power
set-point among the 19 wind turbines. (a,d): $V_{\text{PCC}}$ tracking under
disturbance from grid voltage $V_{\text{grid}}$ for (a) grid-agnostic scheme
(RMSE$=0.0055$ pu), and (d) the proposed CIA scheme (RMSE$=0.0113$ pu). (b,e):
Head node physical reactive power $Q_{\text{head}}$ and the reference control
signal for (b) grid-agnostic scheme, and (e) the proposed CIA scheme. (c,f):
wind farm network nodal voltages under (c) grid-agnostic scheme and (f) the
proposed CIA scheme.

##  V Conclusions and Future Work

The paper has considered the application of a convex inner approximation
method for determining the reactive power support that can be provided by a
wind farm. This method determines the nodal reactive power capacities that
guarantee satisfaction of network constraints. These nodal capacities form the
basis for a feedback control algorithm that provides voltage support to the
grid by dispatching the reactive power of wind turbines. Through simulation of
wind farm networks, we have shown the effectiveness of this approach.

Future work will consider systematic design of the objective function in (P1)
and (P2), e.g., maximization of $Q_{\text{head}}$. We also seek to extend this
work to incorporate system dynamic behaviour into the formulation of the CIA
to guarantee transient operation of networks. Future work will also extend the
method in [5] to address lower reactive power limits and integrate that
feedback methodology with the CIA-based method to improve performance for
practical wind farms. Finally, we will seek to collaborate with wind industry
experts to validate performance on high-fidelity models and support practical
applications.

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