# Model-Free Optimal Control of Inverter for Dynamic Voltage Support

Yifei Guo, Bikash C. Pal, Rabih A. Jabr This work was supported by the
Resilient Operation of Sustainable Energy Systems (ROSES) UK-China (EPSRC-
NSFC) Programme on Sustainable Energy Supply (EP/T021713/1, NSFC-52061635102).
(_Corresponding author: Bikash C. Pal_)Yifei Guo and Bikash C. Pal are with
the Electrical and Electronic Engineering Department, Imperial College London,
London SW7 2AZ, U.K. (e-mail<EMAIL_ADDRESS>b.pal@imperial.ac.uk)Rabih A. Jabr is with the Department of Electrical and
Computer Engineering, American University of Beirut, Beirut 1107 2020, Lebanon
(e-mail<EMAIL_ADDRESS>
###### Abstract

Inverter-based resources (IBRs) are required to provide dynamic voltage
support (DVS) during voltage dips to enhance the low-voltage ride-through
capability. In this paper, we develop a model-free control method to achieve
the optimal DVS (ODVS) without relying on the knowledge of grid parameters.
Delving into the optimum trajectory of the ODVS problem, it is found that
either the current constraint and the maximum active power constraint of IBRs
are binding or one of the constraints is binding. This inspires us to search
for the optimum in a closed-loop way by a perturb-and-observe (P&O)-based
optimum seeking (OS) controller with either the power factor angle or the
reactive current being the manipulated (perturbed) variable. The system is
guaranteed to converge asymptotically to the optimum provided the stepsize
sequence is diminishing and non-summable. The proposed model-free optimal
control is finally implemented within a single-stage photovoltaic (PV) system,
where dynamic simulations demonstrate the optimal and fast DVS performance.

###### Index Terms:

Dynamic voltage support (DVS), inverter-based resources (IBRs), low-voltage
ride-through, model-free control, optimum seeking (OS), perturb-and-observe
(P&O).

##  I Introduction

THERE has been a broad consensus among worldwide grid codes on the operational
requirements for inverter-based resources (IBRs) in response to voltage
disturbances—IBRs should provide dynamic voltage support (DVS); for example, a
certain amount of reactive current provision in proportion to the retained
voltage at the point of connection (PoC) is usually mandatory during grid
voltage dips [1, 2, 3, 4, 5, 6, 7].

Although there is little doubt that the reactive current injection can help
support the voltage, prioritizing reactive current or even purely injecting
reactive current does not maximize the DVS. Furthermore, with the increasing
integration of IBRs and the retirement of synchronous generators in power
systems, IBRs play an increasingly vital role in improving system immunity
against disturbances. So, it is imperative to explore a better way for IBRs to
provide DVS, as simple heuristic rules usually cannot unlock the maximum DVS
capability of IBRs and may even cause instability [8, 9].

Recent years have seen a tremendous surge of interest in developing advanced
DVS strategies of IBRs based on optimization, specifically for boosting the
positive-sequence voltage magnitude [10, 11, 12, 13, 14, 15]. The work in [10]
formulates an optimization model for maximizing the PoC voltage where the
maximum fault current injection is imposed, based on a sensitivity-based
linearized relationship between the PoC voltage and active and reactive
currents of IBRs. Rather than relying on the linear approximation, the studies
in [11, 12, 13] present the optimization models for maximizing the PoC voltage
subject to the maximum current limit, based on the entire nonlinear voltage-
current relationship. The work in [14] further takes into account a load
impedance at the PoC. A thorough optimality analysis of the optimal DVS (ODVS)
problem is provided in [15], considering the current, active power, and
synchronization stability constraints of IBRs.

It has been extensively verified in previous studies [10, 11, 12, 13, 14, 15]
that the current injection of IBRs that yields the maximum (optimal) DVS
depends not only on the current and power limits of IBRs, but also on the
Thévenin equivalent of the grid seen from the PoC; the underlying mathematical
relationship has been revealed [15]. Most of those studies assume the
availability of the grid model (grid voltage and/or impedance). Indeed, it is
straightforward to implement the ODVS based on the knowledge of grid
parameters. However, as there are typically stringent requirements on the
response time of DVS111For example, solar and wind farms shall provide DVS
within 30 and 60 ms since a voltage dip occurs, respectively, in China [5,
6]., the time window available for grid estimation may be too short even for
most state-of-the-art techniques [16, 17, 18, 19] to work effectively and
reliably, especially during dramatic voltage disturbances222For example, the
online event-based grid impedance estimation technique for IBRs requires 750
ms to update the result [19].. This reveals the fact that it is very
challenging in practice to achieve the ODVS based on the full knowledge of
grid parameters; note that using erroneous grid parameters could not only
result in loss of optimality, but may even cause instability, e.g., loss of
synchronism (LoS) or power imbalance between ac and dc sides. The authors in
[10] realize the difficulty in computing the sensitivity coefficients without
knowing external grid conditions and thus propose a compromise solution:
outputting the maximum and equal active and reactive currents regardless of
grid conditions. Such a strategy eliminates the need for grid estimation;
however, it may not only sacrifice the optimality but also induce instability
under certain conditions.

To overcome these challenges, this work explores the ODVS control of IBRs that
does not rely on a feedforward grid parameter estimation. Without the
availability of grid parameters, the ODVS becomes a gray-box optimization
problem333Note that the model structure is imposed, i.e., the Thévenin
equivalent still holds, but the model parameters are no longer assumed to be
known.. In this context, we propose a model-free control method to seek the
optimum of a gray-box ODVS problem in an online fashion. Revisiting the
optimum trajectory of the ODVS problem derived in our earlier work [15], it is
found that the current constraint (CC) and/or the maximum active power
constraint (MAPC) of the IBR is binding; that is, the IBR should either output
the maximum current or maximum active power to achieve ODVS. This indicates
that one can search for the optimum along the feasible space boundary on which
the optimum is instead of exhaustively searching over the whole feasible
space. To do so, we first discover the functional relationship between the PoC
voltage and power factor angle (reactive current) based on the condition that
the IBR operates along the boundary of CC (MAPC). It is proven that the
functions have a unique local maximum within a specific domain, which exactly
corresponds to the optimum of the ODVS problem. In this context, solving the
two-dimensional (2-D) ODVS problem can be reduced to a one-dimensional (1-D)
search. This inspires a perturb-and-observe (P&O)-based optimum seeking (OS)
with either the power factor angle or the reactive current being the
manipulated (perturbed) variable. The criteria associated with the stepsize
rule design are given, guaranteeing that the P&O-based OS can converge to the
optimum without steady-state oscillations. Finally, we detail the
implementation method of the model-free optimal DVS control and embed it into
a single-stage photovoltaic (PV) generation system. Its performance is tested
and compared with other existing practices under various conditions.

The rest of this paper is organized as follows. Section II revisits the
problem statement and the optimum trajectory. Section III introduces the
dimensionality reduction approach. In Section IV, we develop the P&O-based OS
algorithm. Section V presents the implementation method of the model-free DVS
control embedded within a single-stage PV generation system. In Section VI,
dynamic simulations are conducted, followed by conclusions. The proofs are
collected in the Appendix.

##  II Preliminaries

###  II-A Problem Formulation

Figure 1: Schematic diagram of a three-phase grid-connected IBR.

The schematic diagram of a three-phase grid-connected IBR is shown in Fig. 1,
where $V_{g}$ and $R+jX$ are the positive-sequence grid voltage and impedance.
The three-phase inverter employs a typical cascaded control loops, i.e., an
outer loop that regulates the power/voltage, and an inner loop that controls
the current. The standard dual current control strategy under the dq-frame
that allows the decoupled control of positive-and negative-sequence currents
is adopted. The system modeling has been detailed in [15], so is omitted
here.444Since we focus only on the positive-sequence voltage support in this
work, the negative-sequence current is kept to zero. But note that advanced
negative-sequence control is also compatible, as long as it can be well
coordinated with the positive-sequence control to respect all the physical
limits.

To reduce the risk of being tripped by the under-voltage protection, IBRs
shall provide DVS during voltage dips. So, the ODVS seeks to maximize the
positive-sequence PoC voltage magnitude ($V$) by optimizing the IBR’s active
and reactive current injections ($I_{\rm d}$ and $I_{\rm q}$) while respecting
all the physical constraints. It is formulated as

$\displaystyle\underset{I_{\rm d},I_{\rm q}}{\rm maximize}\hskip 8.53581pt$
$\displaystyle V(I_{\rm d},I_{{\rm q}})$ (1a) $\displaystyle{\rm
subject\,to}\hskip 8.53581pt$ $\displaystyle g_{c}(I_{\rm d},I_{\rm q})\leq
0\hskip 14.22636pt{\rm Current\,Constraint}$ (1b) $\displaystyle g_{p}(I_{\rm
d},I_{\rm q})\leq 0\hskip 14.22636pt{\rm Max.\,Power\,Constraint}$ (1c)

where

$\displaystyle V(I_{\rm d},I_{{\rm q}})$
$\displaystyle:=\sqrt{V_{g}^{2}-(RI_{\rm q}+XI_{\rm d})^{2}}+RI_{\rm
d}-XI_{\rm q}$ (2) $\displaystyle g_{c}(I_{\rm d},I_{\rm q})$
$\displaystyle:=I_{\rm d}^{2}+I_{\rm q}^{2}-I_{\rm max}^{2}$ (3)
$\displaystyle g_{p}(I_{\rm d},I_{\rm q})$ $\displaystyle:=\frac{3}{2}V(I_{\rm
d},I_{\rm q})I_{\rm d}-P_{\rm max}.$ (4)

$I_{\rm max}>0$ and $P_{\rm max}>0$ denote the IBR’s maximum current limit and
the maximum power limit. The maximum available power $P_{\rm max}$ depends on
the dc-side energy source, e.g., wind, solar, battery storage, etc., and may
thus vary over a wide range. Note that, the synchronization stability
constraint (SSC)

$\displaystyle g_{s}(I_{\rm d},I_{\rm q})$ $\displaystyle:=|RI_{\rm q}+XI_{\rm
d}|-V_{g}\leq 0$ (5)

is hidden in (1). If this constraint is violated, the IBR will lose
synchronism with the grid.

Compared with the original formulation presented in [15], we relax all the
constraints that are definitely non-binding (inactive) under different model
parameters ($V_{g},R,X,I_{\rm max}$ and $P_{\rm max}$). Clearly, this does not
affect the solution.

###  II-B Optimum Trajectory

If all the model parameters ($V_{g},R,X,I_{\rm max},$ and $P_{\rm max}$) are
available, the ODVS problem (1) can easily be solved according to the
optimality analysis conducted in [15]. Not surprisingly, the optimum is
strongly related to the model parameters. Let $P_{\rm max}$ decrease from a
sufficiently large value to zero and fix other parameters. The optimum of (1)
may experience three stages: $S_{1}\rightarrow S_{2}\rightarrow S_{3}$, and
correspondingly, there exists two boundary conditions, C1 and C2, that
determine which stage is valid, which is detailed as follows:

  * •

Stage 1 (if C1 holds):

$\displaystyle S_{1}:\left\\{\begin{array}[]{l}I_{\rm
d}^{\star}=\dfrac{R}{Z}{I}_{\rm max},\\\\[5.69054pt] I_{\rm
q}^{\star}=-\dfrac{X}{Z}{I}_{\rm max}\end{array}\right.$ (8)

where $Z:=\sqrt{R^{2}+X^{2}}$.

  * •

Stage 2 (if neither C1 nor C2 holds): $S_{2}$ is the unique solution
satisfying the following conditions:

$\displaystyle S_{2}:\left\\{\begin{array}[]{l}g_{c}(I_{\rm d}^{\star},I_{\rm
q}^{\star})=0\\\\[5.69054pt] g_{p}(I_{\rm d}^{\star},I_{\rm
q}^{\star})=0\\\\[5.69054pt] I_{\rm d}^{\star}\geq 0,I_{\rm q}^{\star}\leq
0.\end{array}\right.$ (12)

  * •

Stage 3 (if C2 holds):

$\displaystyle S_{3}:\left\\{\begin{array}[]{l}I_{\rm
d}^{\star}=-\dfrac{1}{2Z}\left(V_{g}-\sqrt{V_{g}^{2}+\dfrac{8R{P}_{\rm
max}}{3}}\right)\\\\[14.22636pt] I_{\rm
q}^{\star}=-\dfrac{X}{2RZ}\left(V_{g}+\sqrt{V_{g}^{2}+\dfrac{8R{P}_{\rm
max}}{3}}\right).\end{array}\right.$ (15)

The boundary conditions C1 and C2 are

$\displaystyle{\rm C1:}$ $\displaystyle\hskip 2.84526pt{P}_{\rm max}\geq
P_{b}$ (16) $\displaystyle{\rm C2}:$ $\displaystyle\hskip 2.84526ptI_{\rm
max}\geq I_{b}$ (17)

where $P_{b}$ and $I_{b}$ are two specific parameters associated with the grid
parameters.

The optimum trajectory under varying $P_{\rm max}$ is illustrated in Fig. 2,
where the color bar represents the resultant maximum PoC voltage; the dotted
arc is the maximum current boundary; ${L}_{1}$ and ${L}_{2}$ are two parallel
straight lines: $RI_{\rm q}+XI_{\rm d}=0$ and $RI_{\rm q}+XI_{\rm
d}+V_{g}X/Z=0$, respectively. In geometry, $S_{1}$ is the intersection of the
maximum current boundary and the straight line ${L}_{1}$, $S_{2}$ is the
intersection of the maximum current boundary and the maximum power boundary,
and $S_{3}$ is the intersection of the maximum power boundary and the straight
line ${L}_{2}$. Keep in mind that if the grid is strong, and/or the voltage
dip is mild (so that $L_{2}$ does not intersect with the maximum current
boundary in the fourth quadrant), $S_{3}$ may not exist no matter how $P_{\rm
max}$ varies. More details of formulation and optimality analysis can be
referred to [15].

Figure 2: Optimum trajectory of optimal DVS problem under varying $P_{\rm
max}$.

###  II-C Gray-Box Optimization

It is clear that the optimum is strongly related to the grid parameters
($R,X$, and $V_{g}$). However, as mentioned earlier, accurate grid parameters
are often unavailable for DVS in practice, indicating that the above
analytical solution cannot be straightforwardly used. In this context,
optimization problem (1) becomes a gray-box optimization problem since some
model parameters are unknown. Solving such a gray-box optimization program is
typically very challenging for many conventional numerical algorithms, in the
sense that 1) the gradient information is unavailable, 2) the objective
function cannot be directly evaluated, and 3) the constraint violations are
hard to foresee. In addition, solution efficiency is also a big concern, as
the DVS from IBRs should be fast and agile to comply with modern grid codes.
So, in what follows, we will initiate a new way of solving the gray-box
counterpart of (1).

##  III Dimensionality Reduction

No gradient information makes it nontrivial to solve for the optimum
efficiently, in the sense that the best search direction among all the
candidates is unknown. However, one exception could be solving a one-
dimensional problem, where there are only two potential search directions,
“positive ($+1$)” and “negative ($-1$)”. This inspires us to explore a
dimensionality reduction of the search, which provides the foundation for the
later OS.

###  III-A Reduced Solution Space

Looking into the optimum trajectory again, it can be found that CC is
_binding_ if C1 holds, i.e., $g_{c}(I_{\rm d}^{\star},I_{\rm q}^{\star})=0$;
MAPC is _binding_ if C1 does not hold, i.e., $g_{p}(I_{\rm d}^{\star},I_{\rm
q}^{\star})=0$. This implies that one can search for the optimum either along
the maximum current boundary

$\displaystyle B_{c}:=\left\\{(I_{\rm d},I_{\rm q}):g_{c}(I_{\rm d},I_{\rm
q})=0\right\\},$ (18)

or along the maximum power boundary

$\displaystyle B_{p}:=\left\\{(I_{\rm d},I_{\rm q}):g_{p}(I_{\rm d},I_{\rm
q})=0\right\\},$ (19)

instead of exhaustively searching over the whole 2-D feasible domain.
Furthermore, over ${B}_{c}$ or ${B}_{p}$, $I_{\rm d}$ and $I_{\rm q}$ are no
longer mutually independent, enabling the dimensionality reduction.

###  III-B Dimensionality Reduction Based on $B_{c}$

Let us introduce the power factor angle $\phi:={\rm atan2}(I_{\rm q},I_{\rm
d})$. Then, we have $B_{c}=\big{\\{}(I_{\rm max}\cos\phi,I_{\rm
max}\sin\phi)|-180^{\circ}\leq\phi\leq 180^{\circ}\big{\\}}$. Correspondingly,
the objective function $V$ over $B_{c}$ can be re-expressed as a function of
$\phi$, i.e.,555For simplicity of exposition, we abuse the notation of
$V(\cdot)$ depending on the context. This also applies to $g_{c}(\cdot)$,
$g_{p}(\cdot)$, and $g_{s}(\cdot)$.

$\displaystyle V(\phi)=V(I_{\rm max}\cos\phi,I_{\rm max}\sin\phi).$ (20)

We find that if C1 holds, $V$ strictly increases over
$[\underline{\phi},\phi^{\star}]$ and strictly decreases over
$[\phi^{\star},\overline{\phi}]$, where

$\displaystyle\phi^{\star}$ $\displaystyle:={\rm atan2}(-X,R)$ (21)
$\displaystyle[\underline{\phi},\overline{\phi}]$
$\displaystyle:=\left\\{\phi\in[\underline{\phi_{s}},\overline{\phi_{s}}]:g_{p}(\varphi)\leq
0,\forall\varphi\in[\underline{\phi_{s}},\phi]\right\\}$ (22)
$\displaystyle[\underline{\phi_{s}},\overline{\phi_{s}}]$
$\displaystyle:=\left\\{\phi\in[-90^{\circ},0^{\circ}]:g_{s}(\phi)\leq
0\right\\}.$ (23)

This suggests that there exists an interval
$[\underline{\phi},\overline{\phi}]$, over which $V$ has a unique local
maximizer $\phi^{\star}$, which exactly corresponds to $S_{1}$. So, solving
(1) can be achieved by searching for $\phi^{\star}$ over
$[\underline{\phi},\overline{\phi}]$ if C1 holds. Fig. 3 gives two
illustrative examples of dimensionality reduction under different grid
conditions where we assume that $P_{\rm max}$ is sufficiently large. It shows
that if the voltage dip is mild and/or the grid is strong,
$[\underline{\phi},\overline{\phi}]$ will be $[-90^{\circ},0^{\circ}]$ (see
the top example); if the voltage dip is sufficient and/or the grid is weak,
$[\underline{\phi},\overline{\phi}]$ may shrink subject to SSC (see the bottom
example where $\underline{\phi}=-45^{\circ}$, and
$\overline{\phi}=-8.13^{\circ}$). If $P_{\rm max}$ takes a value such that
$P_{\rm max}\geq P_{b}$ and $g_{p}(\phi)>0$ for some
$\phi\in[\underline{\phi_{s}},\overline{\phi_{s}}]$, $\Phi$ will further
shrink subject to MAPC.

Figure 3: Dimensionality reduction based on $B_{c}$. Top: $V_{g}=0.5$ pu,
$Z=0.1$ pu, $R/X=2$; bottom: $V_{g}=0.1$ pu, $Z=0.2$ pu, $R/X=2$.

###  III-C Dimensionality Reduction Based on $B_{p}$

For the solutions $(I_{\rm d},I_{\rm q})\in B_{p}$, there exists an implicit
functional relationship between $I_{\rm d}$ and $I_{\rm q}$, i.e., $I_{\rm
d}=\Psi(I_{\rm q})$ or $I_{\rm q}=\Psi^{-1}(I_{\rm d})$; consequently, the
objective function $V$ over $B_{p}$ can be reduced to a function of solely
$I_{\rm q}$, that is,

$\displaystyle V(I_{\rm q}):=V\left(\Psi(I_{\rm q}),I_{\rm q}\right).$ (24)

Then, it is found that if neither C1 nor C2 holds, $V$ strictly decreases over
$[\underline{I_{\rm q}},\overline{I_{\rm q}}]$, where the upper bound is

$\displaystyle\overline{I_{\rm
q}}=\dfrac{X}{2Z^{2}}\left(V_{g}-\sqrt{V_{g}^{2}+\frac{8Z^{2}P_{\rm
max}}{3R}}\right)$ (25)

and the lower bound $\underline{I_{\rm q}}$ uniquely satisfies

$\displaystyle\left\\{\begin{array}[]{l}g_{p}\Big{(}\sqrt{I^{2}_{\rm
max}-\underline{I_{\rm q}^{2}}},\underline{I_{\rm q}}\Big{)}=0\\\\[5.69054pt]
\underline{I_{\rm q}}\leq 0.\end{array}\right.$ (28)

It is not hard to see that $\underline{I_{\rm q}}$ is also the unique local
maximizer $I_{\rm q}^{\star}$ over $[\underline{I_{\rm q}},\overline{I_{\rm
q}}]$, which exactly corresponds to $S_{2}$.

If C2 holds, $V$ is strictly increasing over $[\underline{I_{\rm q}},I_{\rm
q}^{\star}]$ and strictly decreasing over $[I_{\rm q}^{\star},\overline{I_{\rm
q}}]$, where $I_{\rm q}^{\star}$ is the same as in (15), i.e., $S_{3}$;
$\overline{I_{\rm q}}$ is the same as in (25); $\underline{I_{\rm q}}$ is the
same as in (28) if

$\displaystyle I_{\rm
max}^{2}\geq\frac{R^{2}+Z^{2}}{2R^{2}Z^{2}}V_{g}^{2}+\frac{2P_{\rm
max}}{3R}+\frac{X^{2}V_{g}}{2R^{2}Z^{2}}\sqrt{V_{g}^{2}+\frac{8Z^{2}RP_{\rm
max}}{3X^{2}}}$ (29)

otherwise,

$\displaystyle\underline{I_{\rm
q}}=-\frac{R^{2}+Z^{2}}{2RZ^{2}}V_{g}-\frac{X^{2}}{2RZ^{2}}\sqrt{V_{g}^{2}+\frac{8Z^{2}RP_{\rm
max}}{3X^{2}}}.$ (30)

This reveals that there is a unique maximizer $I_{\rm q}^{\star}$ of $V$ over
a specific interval $[\underline{I_{\rm q}},\overline{I_{\rm q}}]$, which
corresponds to $S_{3}$ if C2 holds or $S_{2}$ otherwise. The upper bound
$\overline{I_{\rm q}}$ is given in (25), which in geometry corresponds to the
unique intersection point between ${L}_{1}$ and $B_{p}$ in the fourth
quadrant. It should be clarified that $\overline{I_{\rm q}}$ is somewhat
conservative, i.e., $V$ is expected to decrease over a wider range (see
examples later in Fig. 4). The lower bound $\underline{I_{\rm q}}$ is
determined by CC or SSC. If the grid is weak and/or the voltage dip is severe
[so that the condition in (29) does not hold], $\underline{I_{\rm q}}$ is
determined by SSC; otherwise, it is determined by CC. Therefore, searching for
$S_{2}$ or $S_{3}$ can be achieved by searching for $I_{\rm q}^{\star}$ over
$[\underline{I_{\rm q}},\overline{I_{\rm q}}]$ if C1 does not hold.

Fig. 4 gives two illustrative examples of dimensionality reduction associated
with $B_{p}$. In the first (top) case where neither C1 nor C2 holds, $V$
strictly decreases within $[-1.318,-0.374]$ pu and has a unique local
maximizer $I_{\rm q}^{\star}=-0.374$ pu, which exactly corresponds to $S_{2}$
computed by (12); in the second (bottom) case where C2 holds, $V$ strictly
increases within $[-1.316,-0.836]$ pu and strictly decreases within
$[-0.836,-0.356]$ pu. It has a unique local maximizer, $I_{\rm
q}^{\star}=-0.836$ pu, which exactly corresponds to $S_{3}$ computed by (15).

Figure 4: Dimensionality reduction based on $B_{p}$. Top ($S_{2}$):
$V_{g}=0.5$ pu, $Z=0.1$ pu, $R/X=2$, $P_{\rm max}=0.436$ pu; bottom ($S_{3}$):
$V_{g}=0.1$ pu, $Z=0.1$ pu, $R/X=2$, $P_{\rm max}=0.126$ pu.

##  IV Optimum Seeking

Thanks to the dimensionality reduction, solving the ODVS problem (1) can be
reduced to a 1-D search no matter the value of $P_{\rm max}$. More precisely,
one can search for the maximizer $x^{\star}$ of a 1-D _unimodal_ function
$V(x)$ over a specific interval $[\underline{x},\overline{x}]$, where $x$
denotes either $\phi$ or $I_{\rm q}$. Note that we have only shown the
existence of such a 1-D functional relationship, but its explicit expression
is, in fact, not available due to the lack of grid parameters. So, an
immediate question of interest is: _how to search for $x^{\star}$ efficiently
without the need to know the explicit expression of $V(x)$? _

Perturb-and-observe is considered an effective and easy-to-implement method
for 1-D search, provided the objective function value is available.
Fortunately, the PoC voltage, i.e., the objective function, is usually
measurable in practice. This enables us to develop a P&O-based OS algorithm,
of which the updated rule at the $k$th step is given as

$\displaystyle x_{k+1}$ $\displaystyle\leftarrow\left[x_{k}+\alpha_{k}\cdot
d_{k}\right]_{\underline{x}}^{\overline{x}}$ (31) $\displaystyle d_{k+1}$
$\displaystyle\leftarrow{\rm sgn}\big{(}V(x_{k})-V(x_{k-1})\big{)}\cdot{\rm
sgn}\big{(}d_{k}\big{)}$ (32)

where $\alpha_{k}$ is the perturbation stepsize; $d_{k}$ is the search
direction; $[\cdot]_{\underline{x}}^{\overline{x}}$ denotes the projection
operator onto the set $[\underline{x},\overline{x}]$; ${\rm sgn}(u)=1$ for
$u\geq 0$ and ${\rm sgn}(u)=-1$ for $u<0$.

Of great importance will be the design of stepsize rule $\\{\alpha_{k}\\}$,
which dictates the convergence performance. Conventionally, the stepsize can
be fixed, but its selection is very tricky—a large stepsize could lead to
significant steady-state oscillations, whereas a small stepsize could slow
down the OS. In light of this, we provide the following rigorous criteria and
practical examples for stepsize rule design to enhance convergence.

_Theorem 1:_ The sequence $\\{x_{k}\\}$ generated by (31)–(32) will converge
to the optimum $x^{\star}$, if $0<\alpha_{k}<\infty$ for all $k$, and

$\displaystyle\lim_{k\rightarrow\infty}\alpha_{k}=0,\hskip
8.53581pt\sum_{k=0}^{\infty}\alpha_{k}=\infty.$ (33)

This suggests that the sequence $\\{\alpha_{k}\\}$ should be _diminishing_ and
_non-summable_ , as $k\rightarrow\infty$. The property of “ _diminishing_ ”
guarantees zero steady-state oscillation around the limit point of
$\\{x_{k}\\}$, while the property of “ _non-summable_ ” guarantees the
optimality of the limit point. There are indeed numerous available stepsize
rules meeting the above criteria. Below, we provide an option based on the
“$p$-series.”

_Corollary 1:_ Suppose $0<p\leq 1$ and $\lambda>0$. By taking the stepsize
rule:

$\displaystyle\alpha_{k}=\frac{\lambda}{k^{p}}$ (34)

the sequence $\\{x_{k}\\}$ generated by (31)–(32) will converge to the optimum
$x^{\star}$.

$\lambda$ is a scale factor that dictates the _average_ convergence
performance throughout the whole process. A larger $\lambda$ can accelerate
the convergence but may, in turn, cause oscillations. $p$ determines the
diminishing rate of stepsize and thus dictates the convergence performance at
different stages of the process. A larger $p$ can alleviate large oscillations
but possibly slow down the convergence. So, good convergence dynamics should
be established on a suitable combination of $\lambda$ and $p$.

Keep in mind that the convergence performance should definitely be related to
the unknown grid parameters, in the sense that these parameters dictate the
shape of $V(x)$ and the value of $[\underline{x},\overline{x}]$; the same
stepsize rule may work out differently under different grid conditions.
Therefore, thorough offline trials and analyses under different grid
conditions are advocated to ensure the robustness of the stepsize rule.

##  V Implementation

So far in this paper, we have elaborated in theory on how to achieve the
dimensionality reduction and how to perform the OS algorithm thereafter. With
those fundamentals in place, in this section, we will present the
implementation method of the model-free ODVS, which will be embedded within a
single-stage PV system.

###  V-A Perturbed Variable and OS Mode

In Section IV, we base the OS on the prior knowledge of the perturbed variable
($\phi$ or $I_{\rm q}$); in other words, the valid stage of optimum ($S_{1}$
or $S_{2}/S_{3}$) should have been known before OS. It should be clarified
that there is no need to further distinguish between $S_{2}$ and $S_{3}$ as
they share the same perturbed variable and the corresponding OS algorithm.
Accordingly, there will be two OS modes: OS-a for $S_{1}$ and OS-b for
$S_{2}/S_{3}$.

Recall in Section II that C1 gives a theoretical criterion for OS mode
selection. Unfortunately, it cannot be straightforwardly used since the grid
parameters are unavailable. However, the rationale behind C1—whether or not
the IBR can provide adequate active power in favor of $S_{1}$—is clear and
useful. Inspired by this, we propose the following strategy to handle the mode
selection issue. _The IBR operates with OS-a by default; if the system cannot
afford the required active power corresponding to $S_{1}$, then switch the OS
to OS-b._

Whether $S_{1}$ is affordable can be reflected by the dc voltage behavior of
the IBR, provided the primary dc-side source injects the maximum available
power; more precisely, the power mismatch between the dc and ac sides will
induce significant dc voltage drops once the dc-side source cannot offer the
required power corresponding to $S_{1}$. So, an easy-to-implement criterion
for mode switching is thus given by

$\displaystyle\widetilde{V}_{\rm dc}\leq\rho V_{\rm dc}^{\ast}$ (35)

where ${V}_{\rm dc}^{\ast}$ is the dc voltage reference of the IBR immediately
before the PoC voltage drops, which is held constant over the whole period of
DVS; $\widetilde{V}_{\rm dc}$ is the notch-filtered dc voltage; the
coefficient $\rho$ ($0<\rho<1$) determines the threshold.

###  V-B Enforcement of $B_{c}$ and $B_{p}$

A prerequisite for the dimensionality reduction (as well as the OS) is that
the IBR shall always operate along the maximum current boundary $B_{c}$ (for
OS-a) or the maximum power boundary $B_{p}$ (for OS-b). The former can be
achieved in a feedforward manner, as shown in Fig. 5. The latter can be
achieved by combining the dc-side maximum power control and the dc voltage
control. For instance, consider a single-stage PV system in this work. The dc
voltage of the PV inverter is controlled (by regulating $I_{\rm d}$) to track
the dc voltage reference yielding the maximum power point of PV panels.

###  V-C Feasibility of Perturbed Variable

Recall in Section IV that $x_{k}$ should be restricted within a specific
interval $[{\underline{x}},{\overline{x}}]$ at each iteration, so that the
convergence of OS can be guaranteed. Unfortunately, the lack of grid
parameters makes it impossible to directly compute $\underline{x}$ and
$\overline{x}$, as they are related to the grid parameters. So instead, we
will leverage the physical essence of such bounds to avoid the violation.

_OS-a:_ According to the analysis in Section III-B,
$[\underline{\phi},\overline{\phi}]$ is determined by SSC and MAPC.
$[\underline{\phi_{s}},\overline{\phi_{s}}]$ corresponds to SSC, which is not
violated provided the IBR maintains synchronism with the grid; we will discuss
how to physically achieve this shortly. Theoretically, it is hard to guarantee
that OS-a does not violate MAPC due to the lack of grid parameters. However,
the violation of MAPC is very unlikely to occur in practice if C1
holds.666This is because if the violation of MAPC occurs, $P_{b}\leq P_{\rm
max}$ and $g_{p}(\phi)>0$ for some
$\phi\in[\underline{\phi_{s}},\overline{\phi_{s}}]$ should be satisfied,
simultaneously. Let us take the examples in Fig. 3 again. Accordingly, $P_{\rm
max}$ should exactly fall within $[0.7375,0.8012]$ and $[0.5367,0.5525]$ pu,
respectively, of which the probabilities are very low. Note that this is only
a necessary condition for the violation of MAPC. That is, even though $P_{\rm
max}$ exactly falls within that specific interval, it does not mean that the
violation will definitely happen; it should also depend on the design of the
OS controller. For example, a proper stepsize rule or initialization may help
avoid such violation. Even though the violation of MAPC does occur in
practice, according to Section V-A, the OS will be switched from OS-a to OS-b
once a significant dc voltage is detected. This will cause the system to reach
a near-optimal solution, instead of becoming unstable.

_OS-b:_ According to the analysis in Section III-C, $\underline{I_{\rm q}}$
corresponds to either CC or SSC. This implies that it is not violated,
provided neither of those two constraints is violated. As depicted in Fig. 5,
two additional current limiters can accomplish the current limiting. As
mentioned above, SSC is not violated, provided the IBR can remain synchronized
with the grid. In contrast, $\overline{I_{\rm q}}$ does not directly
correspond to any physical constraint of an IBR; hence, theoretically, it is
unlikely to know whether such bound is violated or not. But as mentioned
earlier, $\overline{I_{\rm q}}$ is rather conservative; thus, in reality, a
moderate violation does not undermine the convergence of OS777It is expected
that the monotonicity of objective function (strictly decreasing) is preserved
for $\overline{I_{\rm q}}\leq I_{\rm q}\leq 0$, though it is hard to
rigorously prove this result; see the numerical examples in Fig. 4.. Since the
optimum must be in the fourth quadrant, a conservative but effective upper
bound, $I_{\rm q}\leq 0$, can be imposed. Besides, as a remedy, we can
carefully initialize the OS to prevent it from reaching the upper bound888If
the R/X ratio of the main grid is roughly known, one can obtain a crude
estimate of $[\underline{I_{\rm q}},\overline{I_{\rm q}}]$ and then initialize
the OS by letting $x_{0}=(\underline{I_{\rm q}}+\overline{I_{\rm q}})/2$;
otherwise, just simply, let $x_{0}=-I_{\rm max}/2$..

Figure 5: Model-free DVS controller design.

_Synchronization Stability:_ The remaining issue is to guarantee that the IBR
maintains synchronism with the grid during the OS. As known, when LoS occurs,
the frequency of the PoC voltage calculated by the PLL ($f_{PLL}$)
significantly deviates from the nominal frequency $f_{N}$ [8], which can be
considered a sign of the violation of SSC. LoS usually results in
significantly distorted PoC voltage, under which the OS may fail. The
instability is even exacerbated due to the interaction between the OS and the
system’s transient dynamics immediately after the voltage drops. To avoid
this, we propose the following “freezing strategy.” Once the PLL frequency
deviation exceeds a given tolerance $\Delta f$, i.e.,

$\displaystyle|f_{PLL}-f_{N}|\geq\Delta f,$ (36)

the OS will be frozen with a fixed output $\phi=-45^{\circ}$ (for OS-a) or
$I_{\rm q}=-I_{\rm max}/4$ (for OS-b). When the PLL frequency returns within
the deadband $|f_{PLL}-f_{N}|<\Delta f$, the OS will be unfrozen.

###  V-D Implementation Within A Single-Stage PV System

Based on the above discussions, the overall controller design of model-free
DVS is shown in Fig. 5, which is embedded within a single-stage PV system. We
do not modify the control scheme of the PV system under normal operating
conditions. The whole procedure of DVS in response to a voltage dip is further
detailed below.

  * •

During normal operating conditions, the single-stage PV system adopts the MPPT
strategy and operates with a unity power factor (steady-state reactive power
support from the PV inverter is also allowed if necessary).

  * •

If the positive-sequence PoC voltage drops below a given threshold (e.g.,
$0.9$ pu), the MPPT control is immediately frozen, and the generated dc
voltage reference is held constant afterwards999Here, an implicit assumption
is that the solar irradiance does not significantly change during the DVS.
This is widely believed to hold in practice since the duration of DVS is
typically very short (several tens of milliseconds to several seconds)..
Meanwhile, the DVS is triggered.

  * •

During the DVS, OS-a is first carried out by default, i.e., $\phi$ is
perturbed to seek the maximum voltage support based on the iterative algorithm
(31)–(32).

  * •

If the condition (35) is satisfied, the control mode will be switched from
OS-a to OS-b, where $I_{\rm q}$ becomes the perturbed variable; otherwise,
OS-a is kept activated.

  * •

The DVS controller generates the (positive-sequence) current references
$I_{\rm d}^{\ast}$ and $I_{\rm q}^{\ast}$ for the inner current control of the
inverter.

TABLE I: System Parameters Description |  Value
---|---
Power rating of PV |  250 kW (1 pu)
Nominal ac voltage |  250V/25kV (1 pu)
Nominal frequency |  60 Hz
Pre- and post-fault SCRs |  20, 10
R/X ratio |  2
Maximum current limit |  1.5 pu
Sampling rate of OS |  30 Hz
$\rho$ |  0.95
$\Delta f$ |  0.3 Hz
$x_{0}$ |  $-45^{\circ}$ (OS-a), $-0.75$ pu (OS-b)
$d_{0}$ |  $-1$ (OS-a/b)
$\lambda$ |  15 (OS-a), 0.2(OS-b)
$p$ |  1 (OS-a/b)

##  VI Simulation Results

We test the model-free ODVS control with a grid-connected PV system via
dynamic simulations in the MATLAB/Simulink R2021a environment. The main
specifications of the test system are given in Table I. Four test cases
(A,B,C,D) are considered to examine the DVS control performance under
different operating conditions. We compare the model-free ODVS method with the
model-based ODVS method [15] and the widely-used droop-based DVS strategy [7].
In the model-based ODVS, the accurate grid parameters are assumed to be
available at the very beginning of DVS; hence, the ODVS can be performed
without any delay101010Here, the implementation method of the model-based ODVS
is the same as in [15], except that it no longer requires a three-cycle delay
to accomplish grid estimation. Though it is somewhat unrealistic for real-
world applications, it can be used as the benchmark to validate the
effectiveness of the model-free ODVS.. In the droop-based DVS strategy, the
reactive current injection is controlled as per the piece-wise linear rule
[7]:

$\displaystyle I_{\rm q}=\left\\{\begin{array}[]{cl}-{I}_{\rm max},&{\rm
if}\,\,V\leq 0.5\\\\[5.69054pt] \dfrac{0.9-V}{0.5-0.9}\cdot{I}_{\rm max},&{\rm
if}\,\,0.5<V<0.9\\\\[8.53581pt] 0,&{\rm if}\,\,V\geq 0.9.\end{array}\right.$
(40)

The active current reference is generated by the dc voltage PI controller
(same as in OS-b) but note that the reactive current is prioritized (different
with OS-b) in the droop-based strategy.

###  VI-A Search for $S_{1}$

Figure 6: Simulation results—Case A. From top to bottom: 1) three-phase grid
voltage, 2) three-phase PoC voltage, 3) three-phase inverter current
injection, 4) power factor angle, 5) dc voltage, 6) OS mode selection, and 7)
positive-sequence PoC voltage under different control strategies.  Figure 7:
Simulation results—Case B. From top to bottom: 1) three-phase grid voltage, 2)
three-phase PoC voltage, 3) three-phase inverter current injection, 4) active
and reactive currents, 5) dc voltage, 6) OS mode selection, and 7) PoC voltage
under different control strategies.

In Case A, an unbalance grid voltage dip occurs at $t=0$s, where the positive-
sequence and negative-sequence components are $0.4\angle 0^{\circ}$pu and
$0.3\angle 50^{\circ}$pu, respectively. The solar irradiance is $1000\rm
W/m^{2}$, and the cell temperature is $25^{\circ}\rm C$. In this context, the
IBR shall operate with $S_{1}$ to achieve the maximum DVS. It can be seen from
Fig. 6 that the model-free ODVS control is triggered immediately after the
voltage dip happens. It constantly operates with OS-a because the PV
generation can afford the required power corresponding to $S_{1}$ ($184$ kW).
Due to the self-protection capability of a single-stage PV generation system,
the dc voltage is boosted to a new equilibrium ($\approx 560$V); the mode
switching criterion in (35) is never met. The perturbed variable $\phi$ of OS
quickly converges to the optimum $\phi^{\star}={\rm
atan2}(-X,R)=-26.57^{\circ}$ within a few iterations ($\leq 5$ iterations),
and therefore, the PoC voltage is recovered rapidly to $0.55$pu—the maximum
DVS. It achieves almost the same DVS performance as the model-based ODVS,
where the optimal active and reactive current set-points are directly given
and outperforms the droop-based strategy, by which the PoC voltage can only be
recovered to $0.44$pu.

###  VI-B Search for $S_{2}$

The simulation settings in Case B are the same as in Case A, except for the
solar irradiance, which is set as $\rm 400W/m^{2}$ here. So, the IBR shall
operate with $S_{2}$ to achieve the maximum DVS. Fig. 7 showcases the
simulation results. After the DVS control is triggered, OS-a is first carried
out by default. It is observed that the dc voltage rapidly drops below the
threshold $95\%V^{\ast}_{\rm dc}$ ($\approx 480$V) since the PV generation
cannot provide sufficient power to support $S_{1}$ anymore under such a solar
irradiance condition, after which the OS switches from OS-a to OS-b (at
$t=0.0112$s). Then, the dc voltage is recovered quickly to the reference value
($\approx 505$V) within $30$ms. Meanwhile, the system rapidly converges to the
optimum S2 within $40\sim 50$ms through the OS, and correspondingly, the PoC
voltage is recovered to $0.515$pu—the maximum DVS. The model-free ODVS
achieves the similar DVS performance as the model-based ODVS (both of them
converge to the optimum). However, the dynamics are slightly different, and it
performs better than the droop-based strategy, by which the PoC voltage is
recovered only to $0.44$pu.

###  VI-C Search for $S_{3}$

Figure 8: Simulation results—Case C. From top to bottom: 1) three-phase grid
voltage, 2) three-phase PoC voltage, 3) three-phase IBR current injection, 4)
active and reactive currents, 5) dc voltage, 6) OS mode selection, and 7) PoC
voltage under different control strategies.

In Case C, a balanced voltage dip ($V_{g}=0.1\angle 0^{\circ}$pu) is simulated
at $t=0$s; the solar irradiance is set as $100\text{W}/\text{m}^{2}$. So, the
IBR shall operate with $S_{3}$ for the maximum DVS. The simulation results are
depicted in Fig. 8. Similar to Case B, OS-a is first enabled by default, then
switched to OS-b as the dc voltage drops below $95\%V_{\rm dc}^{\ast}$
($\approx 460$V). The model-free ODVS drives the IBR system towards the
optimum $S_{3}$, where the PoC voltage is recovered to $0.169$pu—the maximum
DVS. The model-free and model-based ODVS methods ultimately achieve a similar
DVS performance after the system settles down. In contrast, the droop-based
strategy results in significantly distorted PoC voltage because it drives the
system towards an infeasible operating point that potentially violates SSC and
thus induces the LoS.

###  VI-D Test of “Freezing Strategy”

Many grid codes require the connectivity of IBRs for a certain length of time
(e.g., 150 ms [6]) even when the voltage drops below 10% of the nominal
voltage, whereby LoS is very likely to happen. So, in case D, we consider a
very severe voltage dip by setting $V_{g}=0.05$ pu; other simulation settings
are the same as in Case C. As shown in Fig. 9, without the freezing strategy,
the system loses synchronism quickly with the main grid, which will result in
disconnection of the IBR in practice. In contrast, the model-free ODVS with
the freezing strategy promptly stabilizes the system, reflected by the fact
that the PLL frequency is quickly corrected close to the nominal value.
Consequently, the system not only avoids LoS but also converges to the optimal
operating point corresponding to the maximum DVS (similar to the model-based
ODVS). This demonstrates the necessity of such a freezing strategy, especially
under a severe voltage dip.

Figure 9: Simulation results—Case D. PLL frequency (top) and positive-sequence
PoC voltage (bottom) under different control strategies.

##  VII Conclusions

This paper proposes a model-free ODVS control method for maximizing the DVS
provided by IBRs without relying on the knowledge of grid parameters. We first
determine the potential binding constraints of the ODVS problem, CC, and MAPC.
Then, we develop a P&O-based OS algorithm to search for the optimum along the
boundary of CC and MAPC. The algorithm is provably convergent to the optimum
provided the stepsize sequence is diminishing and non-summable. It is
demonstrated that the proposed model-free ODVS responds to voltage dips
promptly and achieves similar DVS performance as the model-based ODVS does
(the optimality is guaranteed). Moreover, it has several other advantages,
such as inherent robustness against model mismatch, low computational burden,
and dynamic responsiveness.

##  Appendix A Proof of Theorem 1

###  A-A Proof of Theorem 1

_Proof of OS-a:_ We first show that there does _not_ exist a positive integer
$m$, such that for all $k\geq m$, $\phi_{k}>\phi^{\star}$ or
$\phi_{k}<\phi^{\star}$. Assume the contrary, that is, there exists some $m$,
such that,

$\displaystyle(\text{Case a}):\hskip 8.53581pt\phi_{k}$
$\displaystyle<\phi^{\star},\hskip 8.53581pt\forall k\geq m\hskip
8.53581pt{\rm or}$ $\displaystyle(\text{Case b}):\hskip 8.53581pt\phi_{k}$
$\displaystyle>\phi^{\star},\hskip 8.53581pt\forall k\geq m.$

Accordingly, due to the property of function $V(\phi)$ and the rule of OS
algorithm, there must exist some $i\geq m$, such that,

$\displaystyle{V}(\phi_{i})<{V}(\phi_{i+1})<{V}(\phi_{i+2})<\cdots$
$\displaystyle d_{i}=d_{i+1}=d_{i+2}=\cdots=\left\\{\begin{matrix}1,\text{Case
a}\\\ -1,\text{Case b}\end{matrix}\right.$

and thus,

$\phi_{i+1}=\big{[}\phi_{i}+\alpha_{i}d_{i}\big{]}^{\overline{\phi}}_{\underline{\phi}}=\phi_{i}+\alpha_{i}d_{i}.$

Since the stepsize sequence is non-summable, it follows that,

$\displaystyle\lim_{k\rightarrow\infty}\phi_{k}=\phi_{i}+\lim_{k\rightarrow\infty}\sum_{j=i}^{k-1}\alpha_{j}=\infty>\phi^{\star},\,\,\,(\text{Case
a})$
$\displaystyle\lim_{k\rightarrow\infty}\phi_{k}=\phi_{i}-\lim_{k\rightarrow\infty}\sum_{j=i}^{k-1}\alpha_{j}=-\infty<\phi^{\star},\,\,\,(\text{Case
b})$

which contradicts our earlier assumption.

So, this means that $\phi_{k}$ will always jump around the optimum
$\phi^{\star}$ as $k\rightarrow\infty$; so, there exists some positive integer
$m$, such that,

$\displaystyle\phi_{m-1}$
$\displaystyle<\varphi^{\star}\leq\phi_{m},\,\,\,\text{or}$
$\displaystyle\phi_{m-1}$ $\displaystyle>\varphi^{\star}\geq\phi_{m},$

which yields,

$\displaystyle\left\|\phi_{m}-\phi^{\star}\right\|<\left\|\phi_{m}-\phi_{m-1}\right\|=\alpha_{m-1}.$

Then, it follows that

$\displaystyle\left\|\phi_{m+1}-\phi^{\star}\right\|$
$\displaystyle=\left\|\big{[}\phi_{m}+\alpha_{m}d_{m}\big{]}_{\underline{\phi}}^{\overline{\phi}}-\phi^{\star}\right\|$
$\displaystyle\leq\left\|\phi_{m}+\alpha_{m}d_{m}-\phi^{\star}\right\|$
$\displaystyle\leq\left\|\phi_{m}-\phi^{\star}\right\|+\left\|\alpha_{m}\right\|$
$\displaystyle<\left\|\alpha_{m-1}\right\|+\left\|\alpha_{m}\right\|.$

Since the iteration will always switch between $\phi_{k}\leq\phi^{\star}$ and
$\phi_{k}\geq\phi^{\star}$ as $k\rightarrow\infty$. Define a _cycle_ as the
procedure that starts from the $m$th iteration where
$\phi_{m-1}<\phi^{\star}\leq\phi_{m}$ and ends up with the $n$th iteration
where $n$ is the minimum number satisfying
$\phi_{n}<\phi^{\star}\leq\phi_{n+1}$; let
$\mathcal{K}(c):=\\{m,m+1,m+2,...,n\\}$ be the set of indexes associated with
the $c$th cycle and
$D(c):=\max_{k\in\mathcal{K}(c)}\left\|\phi_{k}-\phi^{\star}\right\|.$ Then,
we have

$0\leq D(c)<\left\|\alpha_{m-1}\right\|+\left\|\alpha_{m}\right\|$

Given that the stepsize sequence is diminishing, we have (by the Squeeze
Theorem)

$\displaystyle\lim_{c\rightarrow\infty}D(c)=0$

which indicates that

$\displaystyle\lim_{k\rightarrow\infty}\left\|\phi_{k}-\phi^{\star}\right\|=0.$

This completes the proof of OS-a. ∎

_Proof of OS-b:_ If $S_{3}$ is valid, the proof is similar to the proof of
OS-a, which is thus omitted for brevity. Let us consider the case where
$S_{2}$ is valid, that is, ${V}(I_{\rm q})$ is strictly decreasing over
$[\underline{I_{\rm q}},\overline{I_{\rm q}}]$.

We first show that there must exist some $k$ such that $I_{{\rm
q},k}=\underline{I_{\rm q}}=I_{\rm q}^{\star}$. Similarly, assume the
contrary, that is, $I_{{\rm q},k}>\underline{I_{\rm q}}$ for any $k$. Then, as
per the OS algorithm and the monotonicity of ${V}(I_{\rm q})$, we have for any
$k\geq 1$

$\displaystyle I_{{\rm q},k+1}$ $\displaystyle=\Big{[}I_{{\rm
q},k}+\alpha_{k}d_{k}\Big{]}_{\underline{I_{\rm q}}}^{\overline{I_{\rm
q}}}=I_{{\rm q},k}-\alpha_{k}.$

Given that the stepsize sequence is non-summable, we have

$\displaystyle\lim_{k\rightarrow\infty}I_{{\rm q},k}$ $\displaystyle=I_{{\rm
q},1}-\lim_{k\rightarrow\infty}\big{(}\alpha_{1}+\cdots+\alpha_{k-1}\big{)}=-\infty<\underline{I_{\rm
q}}$

which contradicts our earlier assumption.

So, there must exist some $k$, such that $I_{{\rm q},k}=\underline{I_{\rm
q}}=I_{\rm q}^{\star}$. Then, due to the iteration rule of OS, it follows
that,

$\displaystyle I_{{\rm q},m}=I_{{\rm q},k}=\underline{I_{\rm q}}=I_{\rm
q}^{\star},\,\,\,\forall m\geq k.$

In other words, once the OS reaches the optimum, it will be stuck there. This
completes the proof of OS-b. ∎

##  Appendix B Proof of Corollary 1

It is not hard to see that

$\displaystyle 0<\frac{\lambda}{k^{p}}$ $\displaystyle<\infty,\,\,\text{for
all}\,\,\,k$ $\displaystyle\lim_{k\rightarrow\infty}\frac{\lambda}{k^{p}}$
$\displaystyle=0,$ $\displaystyle\sum_{k=0}^{\infty}\frac{\lambda}{k^{p}}$
$\displaystyle=\infty,$

for $0<p\leq 1$ and $\lambda>0$. Then, according to Theorem 1, the convergence
of OS is guaranteed. ∎

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