11institutetext: Mohamed bin Zayed University of Artificial Intelligence, Abu
Dhabi, UAE
11email<EMAIL_ADDRESS>

# FissionFusion: Fast Geometric Generation and Hierarchical Souping for
Medical Image Analysis

Santosh Sanjeev  Nuren Zhaksylyk  Ibrahim Almakky  Anees Ur Rehman Hashmi
Mohammad Areeb Qazi  Mohammad Yaqub

###### Abstract

The scarcity of well-annotated medical datasets requires leveraging transfer
learning from broader datasets like ImageNet or pre-trained models like CLIP.
Model soups averages multiple fine-tuned models aiming to improve performance
on In-Domain (ID) tasks and enhance robustness against Out-of-Distribution
(OOD) datasets. However, applying these methods to the medical imaging domain
faces challenges and results in suboptimal performance. This is primarily due
to differences in error surface characteristics that stem from data
complexities such as heterogeneity, domain shift, class imbalance, and
distributional shifts between training and testing phases. To address this
issue, we propose a hierarchical merging approach that involves local and
global aggregation of models at various levels based on models’ hyperparameter
configurations. Furthermore, to alleviate the need for training a large number
of models in the hyperparameter search, we introduce a computationally
efficient method using a cyclical learning rate scheduler to produce multiple
models for aggregation in the weight space. Our method demonstrates
significant improvements over the model souping approach across multiple
datasets (around 6% gain in HAM10000 and CheXpert datasets) while maintaining
low computational costs for model generation and selection. Moreover, we
achieve better results on OOD datasets than model soups. The code is available
at https://github.com/BioMedIA-MBZUAI/FissionFusion.

###### Keywords:

Model Soups Medical Image Analysis Model Merging Transfer Learning

##  1 Introduction

Deep learning (DL) has emerged as the de facto standard for various computer
vision tasks, consistently achieving state-of-the-art performance. A pivotal
contributor to this success lies in the availability of pre-trained models. In
recent years, a well-established paradigm has emerged: pre-training models on
large-scale data, such as ImageNet [5] followed by fine-tuning on target tasks
with limited training data [20, 22]. This strategy has proven particularly
effective in domains with constrained data availability, such as medical
imaging. The scarcity of annotated medical datasets, coupled with the
challenges of data acquisition and ethical considerations, highlights the
importance of transfer learning from large-scale datasets. Although public
data in medical imaging is increasing, it is usually smaller in size compared
to natural image databasets, which has led to the widespread adoption of
transfer learning from ImageNet [5], to improve performance on medical tasks
[21, 23, 28].

Figure 1: This figure illustrates the validation error on a two-dimensional
slice of the error landscapes for various natural and medical domain datasets
following an approach outlined in [7]. (a) CIFAR-10 [14] (b) CIFAR-100 [14]
(c) FGVC-Aircrafts [17] (d) RSNA Pneumonia [27] (e) APTOS [13] (f) HAM10000
[31]. We employ the 3 best-performing models from the validation set, with the
best model serving as the reference (origin).

The common practice in transfer learning is to adapt a pre-trained model to a
downstream task by fine-tuning. This involves conducting a grid search to
explore various hyperparameter combinations and selecting the model that
performs best on the validation set. Another approach is employing ensemble
techniques [6], where multiple models are utilized simultaneously, albeit at
the expense of increased computational and memory requirements, especially at
inference time. Recent research by [25] has noted that fine-tuned models
optimized independently from the same pre-trained initialization often
converge to similar error basins. A previous work [10] has demonstrated that
averaging weights along a single training trajectory can enhance model
performance in non-transfer settings. Motivated by these observations, [33]
proposed model averaging - Model Soups (MS) as an alternative approach to
ensembles, aiming to derive a single model that achieves a good performance in
terms of accuracy and inference time on In-Domain (ID) as well as Out-of-
Distribution (OOD) datasets. They show that averaging several fine-tuned
models trained with different hyperparameter settings results in a better
model. This method is particularly effective in scenarios with natural imaging
datasets or those without significant domain shifts. Several other works have
improved upon the model merging concept by adopting Fisher-based weight
averaging [19], Task Arithmetic [8], Pruning [36], Gradient-based Matching
[3], Tie-Merging [34], and FedSoups [1] for Federated Learning setting.

However, complexities arise when dealing with medical datasets. As illustrated
in Fig. 1, a significant contrast arises between the error surfaces of natural
imaging datasets (a-c), characterized by smoothness and convexity, and those
of medical datasets (d-f), exhibiting pronounced roughness and the presence of
multiple local optima within the global minima. This roughness is attributed
to various intricacies inherent in medical data, including data heterogeneity,
domain shift from pre-trained (ImageNet) networks, class imbalance, and
distribution shift between the training and testing phases. Such complexities
often result in challenging optimization landscapes, where models are prone to
getting stuck in suboptimal minima. Consequently, the effectiveness of model
averaging is compromised, often leading to subpar performance outcomes. Few
works have adopted model souping for medical datasets [29, 35, 16, 18]. Most
of these studies have merely applied uniform or greedy souping to few models,
and the majority have not explored or analyzed model souping from the
perspective of error surfaces. Additionally, the process of model averaging
typically involves training multiple models with different hyperparameter
settings, which can be computationally intensive. In contrast to previous
research, our work explores computationally efficient model generation and
averaging in a transfer learning context, particularly to domains with
significant domain shifts.

In Model Soups [33], the process of fine-tuning consists of two main steps:
(1) Model generation - fine-tuning models with various hyperparameter
configurations, and (2) Model selection - selecting the model with the highest
accuracy on the held-out validation set and then performing either uniform or
greedy souping. In our study, we explore the challenges associated with both
steps in the medical image analysis domain. Drawing from insights in [25] and
[7], we propose a Fast Geometric Generation (FGG) approach to efficiently
generate models with minimal computational overhead. Additionally, we address
the model selection process by introducing Hierarchical Souping (HS), which is
better suited for medical data because of the aforementioned complexities. Our
key contributions are as follows:

  * •

We propose Fast Geometric Generation (FGG) approach, which uses cyclical
learning rate scheduler in the weight space for efficient model generation.
This approach achieves superior results compared to model soups at a lower
computational cost.

  * •

We introduce a novel selection mechanism - Hierarchical Souping (HS), tailored
specifically for the medical image analysis domain, which performs model
averaging at different levels.

  * •

We perform a comprehensive analysis of model souping across various datasets
in both natural and medical domains. We demonstrate that combination of FGG
and HS for selection significantly enhance results, improve robustness and
increase generalization to out-of-distribution datasets.

Figure 2: An illustration of (a) Loss landscape of fine-tuned models (b) Fast
Geometric Generation(FGG) approach using cyclical learning rate scheduler (c)
FGG and the Hierarchical Souping (HS) approach.

##  2 Methodology

Let $\mathcal{D}_{\text{train}}=\\{(x_{i},y_{i})\\}_{i=1}^{N}$ denote the
training dataset where $x_{i}\in\mathbb{R}^{d}$ represents the input data and
$y_{i}\in\\{1,2,\ldots,C\\}$ denotes the corresponding label from a set of C
classes. Similarly, let
$\mathcal{D}_{\text{val}}=\\{(x_{j},y_{j})\\}_{j=1}^{M}$ be the validation
dataset. We adapt a pre-trained model to our task by performing linear probing
for a few epochs as a warm-up step, where we only update the weights of the
last layer while freezing the rest of the model. Let $\theta_{\text{0}}$
denote the parameters of the linear probed model. As mentioned above, the
souping procedure involves two main steps (1) Generation of the models by
fine-tuning (2) Selection of models that perform best on the
$\mathcal{D}_{\text{val}}$. The illustration of our proposed approaches (FGG
and HS) can be seen in Fig. 2.

Fast Geometric Generation (FGG). We design FGG to generate a range of diverse
neural networks while navigating through the weight space taking small steps.
The primary objective is to explore variations in network weights without
straying into regions likely to lead to low test performance. Instead of
employing the conventional hyperparameter grid search, we opt for a more
focused strategy. In FGG, we only iterate through a single hyperparameter i.e.
(learning rate), and the rest of the hyperparameters are kept constant. This
helps reduce the hyperparameter search space and the computational cost.

In FGG, we start with $\theta_{\text{0}}$ as an initialization and fully train
the model using $n$ different learning rates, resulting in a set of parameters
$\Theta=\\{\theta_{1},\dots,\theta_{n}\\}$. After obtaining $\Theta$, we
implement the Fission process. We initialize the weights
$\theta_{t}\in\Theta$, and carry out a second training process for a set of
iterations $\textit{I}=\\{1,\dots,k\\}$ to generate a set of parameters
$\hat{\Theta}=\\{\hat{\theta}^{1},\dots,\hat{\theta}^{m}\\}$. During this, we
employ a learning rate scheduler to cyclically change the learning rate every
cycle $c$, where $c$ is defined as a set of number of iterations and is an
even number. This encourages $\theta_{t}$ to diverge from its initialization,
while maintaining validation accuracy. We use a function $\alpha(i)$ to
control the learning rate at iteration $i\in\textit{I}$ as follows:

$\alpha(i)=\begin{cases}\alpha_{2}.(2t(i))+\alpha_{1}(1-2t(i))&\text{if
}0<t(i)\leq 0.5,\\\ \alpha_{1}.(2t(i)-1)+\alpha_{2}(2-2t(i))&\text{if
}0.5<t(i)\leq 1,\end{cases}$

where $t(i)$$=$$\frac{1}{c}(\text{mod}(i-1,c)+1)$, $\alpha_{2}$ and
$\alpha_{1}$ ($\alpha_{2}$ $\leq$ $\alpha_{1}$) are hyperparameters used to
control the minimum and maximum learning rates, respectively. Using
$\alpha(i)$, we train the network $\theta_{t}$ to form the new set of
parameters $\hat{\Theta}$, where each $\hat{\theta}\in\hat{\Theta}$ is the
model parameters when the learning rate reaches its minimum value,
$\alpha(i)=\alpha_{2}$. This occurs at the point where
$\text{mod}(i-1,c)+1=\frac{c}{2}$ and $t(i)=\frac{1}{2}$. We follow this
training process for every $\theta_{t}\in\Theta$, which results in a
$\\{\hat{\Theta}_{1},\dots,\hat{\Theta}_{n}\\}$ .

During high learning rate intervals (close to $\alpha_{1}$), the weight
$\theta_{t}$ traverses the weight space with larger strides, potentially
leading to higher test error. Conversely, in low learning rate episodes,
$\theta_{t}$ transitions to smaller steps, reducing test error. This mechanism
facilitates incremental movements in weight space, aiding models in evading
local minima. Additionally, it gathers diverse models for averaging, reducing
the necessity for an extensive grid search.

Hierarchical Souping (HS). In [33], the greedy souping approach outperforms
uniform averaging by sequentially adding models to the soup if they improve
accuracy on $\mathcal{D}_{\text{val}}$. This approach can yield suboptimal
results when the best model is stuck in a local optimum because of the uneven
error surface caused due to domain shift. To solve this, we propose
Hierarchical Souping, where models are merged at different levels. Starting
from the parameter sets $\Theta$ and $\hat{\Theta}$ acquired during the FGG
phase, we adopt a local souping approach where the generated models
$\\{\hat{\Theta}_{1},\dots,\hat{\Theta}_{n}\\}$ are averaged along with the
corresponding initialization $\theta_{t}$ (greedy or uniform i.e.
${\hat{\theta}}_{\text{sp}_{t}}=\frac{1}{m+1}[\sum_{i=1}^{m}\hat{\theta}_{t}^{i}+\theta_{t}]$
) at different levels resulting in
$\\{\hat{\theta}_{{sp}_{1}},\hat{\theta}_{{sp}_{2}},\dots,\hat{\theta}_{{sp}_{n}}\\}$
and a greedy averaging technique is used at the top level giving
${\theta}_{\text{sp}}$ (GoG refers to Greedy at all levels whereas GoU refers
to Greedy at the top level and Uniform at the lower levels). This approach
enables networks to escape local minima at the lower levels by local
aggregation, giving a good subset of generalizable models that can be averaged
in a greedy manner at the top level.

##  3 Experiments

Implementation Details. Our research investigates two model architectures:
DeiT-S (Transformer) and ResNet50 (CNN), both pre-trained on ImageNet. Before
full fine-tuning, we conduct a 10-epoch warmup where the whole network except
the last layer is frozen ($\theta_{\text{0}}$). We then unfreeze the entire
model and perform a full fine-tuning known as the LP-FT approach [15]. We use
AdamW optimizer with a cosine annealing scheduler for 50 epochs, batch size of
128, and an image resolution of 224$\times$224.

For the model soups experiments, we perform a grid search over learning rate,
seeds, and augmentations similar to [33] for the general grid search (GS)
experiments. Learning rates (LR) include (1e-3, 5e-4, 1e-4, 5e-5, 1e-5, 5e-6,
1e-6, 1e-7) and Augmentations include minimal (random crop covering 90-100% of
image), medium (default timm library settings [32]), and heavy (RandAugment
with $N=2$, $M=15$) [2]. Each hyperparameter configuration is run with two
seeds, resulting in 48 models in total for baseline results [33].

For our method, we focuses solely on the learning rate as the varying
hyperparameter, where we fine-tune eight models initially. Each model is
trained with a different learning rate, employing heavy augmentation and a
fixed seed. According to [33, 24], the learning rate emerges as the most
influential parameter, positioning the model in the loss surface. All
experiments use the AdamW optimizer with a cyclical learning rate scheduler,
where the learning rate ranges between $\alpha_{1}=1e-5$ and
$\alpha_{2}=1e-8$. From each base model, we generate five models using FGG,
resulting in a total of 48 models. We train the eight initial models for 50
epochs and then we perform the second stage training process (FGG) for 17 (4
epochs per cycle) epochs with cyclical learning rate collecting 5 models per
base model, at the end of the training process. We also provide an analysis
and justification for choosing learning rate as a top-level hyperparameter
using Linear Mode Connectivity (LMC) in Fig. 1 in B1 in the Appendix.
Datasets. For the primary experiments, we consider two natural imaging domain
datasets and five medical domain datasets. We use CIFAR10 [14] and CIFAR100
[14], partitioning the training dataset into train/validation sets in a
90%:10% ratio. We utilize the official test split provided. For the CheXpert
[9] and HAM10000 [31] datasets, we adhere to the official
train/validation/test splits. For APTOS [13], EyePACs [12], and RSNA-Pneumonia
[27] datasets, we split the data into train/validation/test sets in an
80%:10%:10% ratio, given that only the training dataset was publicly
available. It is noteworthy that all datasets are multiclass, except CheXpert,
which is a multilabel dataset. For the OOD experiments, we consider the
CIFAR10.1 [26, 30] having 2000 test samples from the natural imaging domain.
For the medical imaging domain, we use the Messidor and Messidor-2 [4]
datasets, sampling 10% of the data for the test set. We also use the MIMIC-CXR
[11] dataset and follow its official test split.

##  4 Results and Discussion

Table 1: Performance comparison of different methods. (GS (best) - best model
on the validation set from Grid Search, FGG(best) - best model on the
validation set from Fast Geometric Generation, GoU - Greedy of Uniform, GoG -
Greedy of Greedy), Bold numbers mean best and underlined are the second best

| Model
---
Method |  | CIFAR10
---
(Acc.)
| CIFAR100
---
(Acc.)
| APTOS
---
(F1)
| HAM10000
---
(Recall)
| RSNA
---
(F1)
| CheXpert
---
(AUC)
| EyePACs
---
(F1)
ResNet50 | GS(best) | 0.9769 | 0.8380 | 0.7086 | 0.6074 | 0.9444 | 0.8444 | 0.4750
Uniform Soup | 0.8703 | 0.7652 | 0.5509 | 0.5698 | 0.9171 | 0.5752 | 0.1738
Greedy Soup | 0.9769 | 0.8401 | 0.7247 | 0.6074 | 0.9444 | 0.8444 | 0.4750
FGG(best) | 0.9783 | 0.8464 | 0.7172 | 0.6614 | 0.9518 | 0.8434 | 0.4874
GoU (Ours) | 0.9785 | 0.8457 | 0.6909 | 0.6818 | 0.9545 | 0.8351 | 0.4905
GoG (Ours) | 0.9785 | 0.8477 | 0.7172 | 0.6614 | 0.9518 | 0.8488 | 0.4900
DeiT-S | GS(best) | 0.9871 | 0.8919 | 0.6903 | 0.6487 | 0.9503 | 0.8143 | 0.4807
Uniform Soup | 0.9386 | 0.8551 | 0.1637 | 0.1429 | 0.4147 | 0.7177 | 0.1697
Greedy Soup | 0.9892 | 0.8968 | 0.6785 | 0.6487 | 0.9503 | 0.8068 | 0.4865
FGG(best) | 0.9876 | 0.8963 | 0.7011 | 0.6393 | 0.9529 | 0.8619 | 0.5029
GoU (Ours) | 0.9899 | 0.8968 | 0.6976 | 0.6495 | 0.9579 | 0.7609 | 0.4903
GoG (Ours) | 0.9901 | 0.8987 | 0.7003 | 0.6393 | 0.9529 | 0.8644 | 0.4940

Figure 3: OOD analysis for different architectures on various datasets (a)
CIFAR10 v/s CIFAR10.1 - ResNet50 (b) CheXpert v/s MIMIC - ResNet50 (c) APTOS
v/s (EyePacs, Messidor, Messidorv2) - ResNet50 (d) CIFAR10 v/s CIFAR10.1 -
DeiT-S (e) CheXpert v/s MIMIC - DeiT-S (f) APTOS v/s (EyePacs, Messidor,
Messidorv2) - DeiT-S. We do not plot the results of Uniform Soups as it
performs poorly.

As seen in Table 1, we observe that our method achieves better results both in
the natural as well as medical imaging domains. In CIFAR datasets, we observe
that GoU and GoG achieve better results compared to model soups, though it is
not significant due to the smooth convex error surface. However, when testing
on medical datasets, we observe around 6% improvement in Recall (GoG) for the
HAM10000 dataset (ResNet50) as well as around 6% gain (GoG) in AUC(DeiT-S) for
the CheXpert dataset. Both the GoU as well as GoG approaches achieve similar
results except in the case of CheXpert (DeiT-S), where GoU significantly lags
behind GoG. The GoG is always better than the Greedy Soup approach in almost
all cases and is achieved at a significantly lower computational cost, as we
do not have to perform the full grid search. Greedy soups does not perform as
expected when fine-tuning on medical datasets as the averaging starts from the
best model, which can be an issue due to uneven error surfaces. For example,
for the DeiT-S on the CheXpert dataset, we observe that the best model has a
high validation score, but a low test score indicating its not generalizable
as it might have been stuck in a local minima. The FGG process overcomes this
issue through its ability to escape numerous local minima, while the acquired
models cluster around the same area in the error surface, facilitating
smoother averaging. The local averaging in HS approach allows smoother
averaging at local surfaces and greater diversity at the top level. We further
conduct an ablation study in Table 1 in Section A1 in the Appendix, exploring
greedy souping on the FGG models, and HS on the grid-search generated models.

OOD Analysis. We also conduct an analysis on OOD scenarios for both natural
and medical domain datasets. Performance comparison of souped models on OOD
data is important as averaged models should demonstrate robustness to
distribution shifts [33]. In Fig. 3, we compare the performance of various
approaches in both ID and OOD datasets, where the top five models are selected
based on validation set results. In almost all the cases, either GoU or GoG
yields similar or better results compared to greedy soups in both ID and OOD
tasks at a much lower computational cost. Our approach results in higher ID
and OOD performance gains, particularly for medical imaging datasets, which
can be attributed to the FGG approach aiding models in escaping local minima.
HS also contributes to this, by facilitating easier averaging between models
in the error surface.

##  5 Conclusion

This work investigates challenges associated with model averaging in transfer
learning settings, particularly in medical imaging domain where significant
domain shifts can occur. To address this, we introduce Fast Geometric
Generation (FGG) leveraging hyperparameter significance and a cyclical
learning rate scheduler. Moreover, we propose a Hierarchical Souping
mechanism, which involves averaging models at different levels based on the
smoothness of the error surface and hyperparameter significance. The proposed
generation and selection methodologies yield notable performance enhancements
compared to the traditional model souping approach. While our work achieves
improved results, we observe instances where models from grid search
occasionally outperform averaged models due to very rough error landscapes.
This suggests a potential improvement for enhancing generalizability by
smoothing the error surface, an aspect we plan to focus on in future
endeavors.

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## Appendix

##  Appendix A1 Ablation study for different generation and selection
techniques

Table 1: Ablation study exploring the Performance comparison of different
combinations. (GS - Grid search for model souping, GoU - Greedy of Uniform ,
GoG - Greedy of Greedy),Bold numbers mean best and underlined are the second
best

| Model
---
Method |  | CIFAR10
---
(Acc.)↑
| CIFAR100
---
(Acc.)↑
| APTOS
---
(F1)↑
| HAM10000
---
(Recall)↑
| RSNA
---
(F1)↑
| CheXpert
---
(AUC)↑
ResNet50 | GS+HS(GoU) | 0.9777 | 0.8434 | 0.6553 | 0.6589 | 0.9474 | 0.8194
GS+HS(GoG) | 0.9771 | 0.8409 | 0.7045 | 0.6071 | 0.9444 | 0.8444
FGG+ UniformSoup | 0.9218 | 0.8076 | 0.5912 | 0.6070 | 0.9104 | 0.7313
FGG + GreedySoup | 0.9776 | 0.8466 | 0.7172 | 0.6614 | 0.9518 | 0.8488
FGG+HS(Aug@top) | 0.9769 | 0.8410 | 0.7118 | 0.6317 | 0.9444 | 0.8444
DeiT-S | GS+HS(GoU) | 0.9887 | 0.8975 | 0.6797 | 0.6713 | 0.9499 | 0.8609
GS+HS(GoG) | 0.9886 | 0.8957 | 0.6990 | 0.6487 | 0.9530 | 0.8707
FGG+ UniformSoup | 0.9309 | 0.8509 | 0.3067 | 0.1429 | 0.3781 | 0.7199
FGG + GreedySoup | 0.9895 | 0.9003 | 0.6828 | 0.6833 | 0.9529 | 0.8653
FGG+HS(Aug@top) | 0.9892 | 0.8946 | 0.6741 | 0.6487 | 0.9503 | 0.8691

##  Appendix B1 Linear Mode Connectivity

Figure 1: Hyperparameter Analysis using Linear Mode Connectivity (LMC)
($\theta=\lambda\cdot\theta_{A}+(1-\lambda)\cdot\theta_{B}$), where
$\theta_{A}$ and $\theta_{B}$ differ only in one hyperparameter. (a) and (d)
LMC between models varying only in seed. (b) and (e) LMC between models
varying only in augmentation. (c) and (f) LMC between models varying only in
learning rate.


# Hierarchical Frequency Control of Hybrid Power Plants Using Frequency
Response Observer

Qian Long, Kaushik Das, and Poul Sørensen

###### Abstract

Frequency control (FC) enables utility-scale grid-connected hybrid power
plants (HPPs) to operate in compliance with grid code requirements while to
capture value streams from provision of frequency control services (FCSs). In
this paper, a novel hierarchical FC approach is proposed to allow HPPs to
provide three types of FCSs, namely fast frequency response (FFR), frequency
containment response (FCR) and frequency restoration response (FRR). To
accommodate state-of-the-art fast FC, controllers for fast FCSs, such as FFR
and FCR, are implemented at asset controllers, while controllers for slow FCSs
like FRR are implemented at plant controllers or the HPP controller (HPPC).
Control counteraction issue, which arises across control hierarchy, is then
discussed. To solve this issue, an innovative frequency response observer
(FROB) is proposed. Inspired by the concept of disturbance observer (DOB),
FROB at plant controllers and the HPPC accurately estimates frequency response
initiated at asset controllers, and the obtained estimation is used for
control compensation at plant controllers and the HPPC to avoid control
counteraction. This scheme achieves robust performance even when there are
system uncertainties existing in HPPs, such as parameter uncertainty, unknown
control malfunction, and time-varying communication delays. The proposed
approach is implemented in a power system dynamic model in MATLAB/Simulink to
highlight its effectiveness and robustness.

###### Index Terms:

hierarchical frequency control, hybrid power plants, fast frequency control,
disturbance observers, frequency response observers.

##  I Introduction

As frequency stability has been challenged by the increasing share of
inverter-based resources in modern power systems, the requirement for
frequency control (FC) becomes demanding, posing challenges for the existing
FC methods for power plants [1]. Utility-scale hybrid power plants (HPPs) have
received global attention due to enhanced controllability and efficient
utilization of electrical infrastructure [2]. FC methodologies that enable
HPPs to provide enhanced frequency response need to be further investigated,
especially since there are additional number of assets and controllers in an
HPP as compared to single-technology power plants [3]. However, there is
little literature addressing FC of HPPs [4, 5].

Figure 1: An example of control counteraction during an under-frequency event.

Hierarchical FC for single-technology power plants, which consists of asset
controllers and a plant control, has been well established. A conventional
approach is to introduce auxiliary FC loop for active power control at the
plant controller, which generates time-varying active power setpoints for
asset controllers when FC is activated [6, 7, 8, 9]. While this approach
enables power plants to provide slow frequency control services (FCSs) such as
FRR, factors such as communication delays and limited control bandwidth could
be the bottleneck for plant controllers to provide fast FCSs. To date, fast FC
have been mostly implemented at asset controllers for wind turbines (WTs),
photovoltaics (PVs) and energy storage (ESs) [10, 11, 12, 13, 14, 15, 16].
Nevertheless, without any coordination strategy, when frequency response is
activated at asset controllers, against which the plant controller will start
to counteract, during a frequency event. Fig. 1 shows an example of such
control counteraction. Therefore, a new hierarchical FC approach that supports
fast FC at asset controllers and coordinates between asset controllers and
plant controllers during the period of FCS provision is in need. Considering
the HPP controller (HPPC) as an extra control level that provides control and
supervision for wind power plants (WPPs), solar power plants (SPPs) and energy
storage systems (ESSs), the design of the coordination strategy across the
whole control hierarchy becomes even more complicated.

The main challenge of the coordination arises from frequency response being an
unknown control action (as seen by the HPPC and plant controllers) from asset
controllers. A possible way of addressing control counteraction could be
estimating the total frequency response and compensating as the feedback at
plant controllers and the HPPC. However, uncertainties existing in HPPs bring
extra challenges into accurate estimation of frequency response. This leads to
one of the innovations in the paper, which is to develop an observer that can
accurately estimate the total frequency response activated at asset
controllers given uncertainties in HPPs, for the purpose of avoiding
counteraction across control hierarchy.

Disturbance observers (DOBs) algorithms for estimating system uncertainties
and disturbances, have been used extensively in various power-related
industrial applications. Time-domain DOBs, such as extended-state observer,
have been utilized in the field of load frequency control to estimate total
disturbance from loads and renewable generation [17, 18]. In [19], similar DOB
techniques have been applied to charging and discharging control of a flywheel
ESS. Another category of DOBs, that is, frequency-domain DOBs, have been
proposed and adopted in applications like load frequency control [20],
converter control [21] and PV generation [22]. The above-mentioned DOBs help
generate a robust control system that achieves good performance in disturbance
rejection and handles system uncertainties at the same time. This paper
focuses on frequency-domain DOBs originally proposed in [23] due to its simple
and straightforward nature. However, it should be emphasized that frequency-
domain DOBs can not be directly applied as a solution to the counteraction
issue in hierarchical FC. It is because the design purpose of frequency-domain
DOBs is to reject external disturbance, while frequency response from asset
controllers is supposed to be accepted rather than rejected by plant
controllers and the HPPC. Therefore, a new observer called frequency response
observer (FROB), following the idea of frequency-domain DOBs, estimates the
total frequency response from asset controllers as ”the disturbance”, and
modification is introduced such that the closed-loop characteristic of FROB
accepts frequency response. This modification also leads to the need to
establish new design guidelines.

In this paper, a novel hierarchical FC approach using FROB is proposed to
enable HPPs to provide coordinated FCSs. The main contribution of this
approach is threefold: i) the proposed approach allows coordinated
participation in frequency response from multiple technology power plants,
such as WPPs, SPPs and ESSs; ii) the new control architecture allows HPPs to
provide three types of FCSs including FFR, FCR and FRR, where controllers for
fast FCSs like FFR and FCR are implemented at asset controllers and
controllers for slow FCS like FRR is implemented at plant controllers and the
HPPC. Under this architecture, state-of-the-art fast FC that is mainly
developed at asset level is easy to apply; iii) an innovative FROB is
introduced at plant controllers and the HPPC to avoid control counteraction
against asset controllers. The proposed FROB achieves robust performance on
estimating frequency response even when there are system uncertainties in
HPPs.

Next section introduces dynamic modeling of HPPs with the proposed
hierarchical FC approach. Section III starts with a brief introduction of the
DOB and then focuses on FROB design and analysis. In Section IV, simulations
are conducted and the effectiveness of the proposed control is illustrated.
Section V concludes the paper.

##  II Dynamic Modeling and Control of HPP

###  II-A HPP Component Model

Figure 2: WT dynamic model. Figure 3: PV dynamic model. Figure 4: ES dynamic
model.

The dynamic models for WTs, PVs and ESs are shown in Fig. 2, Fig. 3 and Fig.
4, respectively, with converter models and converter control models being
developed. Since this paper is only concerned with FC, only active power
control module is considered in converter control. Reactive power control
module is beyond the scope. Modeling details of WTs, PVs and ESs for frequency
studies can be found in [24, 25, 26, 9].

The WT dynamic model is built based on a variable speed Type 4 WT
configuration, where the aerodynamic rotor of WT is directly coupled to the
permanent magnet synchronous generator through a gearless drive train as
described in IEC 61400-27-1 [27]. For turbine model, aerodynamic model is
included and a two-mass model is used for representing the mechanical shaft
system of the WT. Turbine control model consists of two parts, torque control
and pitch angle control. In order to control generator rotor speed, turbine
control sends either power (or torque) command to converter control or pitch
angle command to pitch actuator. Active power control module receives power
(or torque) command from turbine control as the input, and generates active
current command for WT converter model.

The PV dynamic model is built based on a generic WECC model for large-scale
SPPs [28]. The PV module takes weather conditions, such as ambient temperature
and solar irradiance, as the input. Maximum power point tracking (MPPT)
controller seeks for maximum power point of the PV module given weather
conditions, PV terminal voltage and current. Active power control module
receives DC voltage measurement, which is regulated by converter control, and
generates active current command for PV converter model.

The ES dynamic model is built based on a generic WECC ES model [29]. The
battery module takes ambient temperature as the input. The battery management
system (BMS) has been developed in order to estimate state of charge and to
determine maximum discharging and charging current for ES converter control
model. The input of the BMS includes ES terminal voltage and current. Active
power control module takes external power command as the input, and generates
active current command for ES converter model.

###  II-B Hierarchical Frequency Control

Figure 5: The proposed hierarchical FC architecture.

In the existing approaches, controllers for FFR and FCR are integrated into
plant controllers by adding auxiliary FC loops to active power control [4, 8].
However, these approaches could lead to a degraded frequency response due to
several factors, such as communication delays or failures, control bandwidth
and ramp rate limits. The above-mentioned factors restrict the speed of
frequency response. The proposed hierarchical FC architecture is shown in Fig.
5, consisting of three control levels, asset controllers, plant controllers
and the HPPC. The main innovation of the proposed approach is that controllers
for FFR and FCR are implemented at asset controllers. This approach allows
fast FCSs, such as FFR and FCR, to be triggered at asset controllers by local
frequency measurements. Therefore, no communication is involved during FCS
provision. Meanwhile, control bandwidth of asset controllers is large enough
to support fast FC, and control signal for fast FC is unaffected by ramp rate
limits. FROB is implemented at both plant controllers and the HPPC to avoid
control counteraction with asset controllers.

The references sent from the HPPC to plant controllers are defined as follows:

$\displaystyle\begin{split}\boldsymbol{u_{WPP}}=&[P^{*}_{WPP},\boldsymbol{P^{*}_{WPP+}},\boldsymbol{P^{*}_{WPP-}},\boldsymbol{T_{FFR}},\\\
&DB_{FFR},DB_{FCR},R_{FCR}]\\\ \end{split}$ (1)

$\displaystyle\begin{split}\boldsymbol{u_{SPP}}=&[P^{*}_{SPP},\boldsymbol{P^{*}_{SPP+}},\boldsymbol{P^{*}_{SPP-}},\boldsymbol{T_{FFR}},\\\
&DB_{FFR},DB_{FCR},R_{FCR}]\\\ \end{split}$ (2)

$\displaystyle\begin{split}\boldsymbol{u_{ESS}}=&[P^{*}_{ESS},\boldsymbol{P^{*}_{ESS+}},\boldsymbol{P^{*}_{ESS-}},\boldsymbol{T_{FFR}},\\\
&DB_{FFR},DB_{FCR},R_{FCR}]\\\ \end{split}$ (3)

where $P^{*}_{WPP}$, $P^{*}_{SPP}$, and $P^{*}_{ESS}$ are power references for
WPP, SPP and ESS, respectively. $\mathbf{P^{*}_{WPP+}}$,
$\mathbf{P^{*}_{WPP-}}$, $\mathbf{P^{*}_{SPP+}}$, $\mathbf{P^{*}_{SPP-}}$,
$\mathbf{P^{*}_{ESS+}}$ and $\mathbf{P^{*}_{ESS-}}$ are upwards and downwards
frequency reserve vectors for WPP, SPP and ESS, respectively. Each frequency
reserve vector contains three elements representing the amount of FFR, FCR and
FRR, respectively. $\mathbf{T_{FFR}}$ is a time vector that defines rising
time, duration and falling time of FFR. $DB_{FFR}$, $DB_{FCR}$ and $R_{FCR}$
are frequency deadband of FFR, frequency deadband and droop coefficients of
FCR, respectively. The references sent from plant controllers to asset
controllers are defined as follows:

$\displaystyle\begin{split}\boldsymbol{u_{WTi}}=&[P^{*}_{WTi},\boldsymbol{P^{*}_{WT+}},\boldsymbol{P^{*}_{WT-}},\boldsymbol{T_{FFR}},\\\
&DB_{FCR},R_{FCR}]\\\ \end{split}$ (4)

$\displaystyle\begin{split}\boldsymbol{u_{PVi}}=&[P^{*}_{PVi},\boldsymbol{P^{*}_{PV+}},\boldsymbol{P^{*}_{PV-}},\boldsymbol{T_{FFR}},\\\
&DB_{FCR},R_{FCR}]\\\ \end{split}$ (5)

$\displaystyle\begin{split}\boldsymbol{u_{ESi}}=&[P^{*}_{ESi},\boldsymbol{P^{*}_{ES+}},\boldsymbol{P^{*}_{ES-}},\boldsymbol{T_{FFR}},\\\
&DB_{FCR},R_{FCR}]\\\ \end{split}$ (6)

where $P^{*}_{WTi}$, $P^{*}_{PVi}$, and $P^{*}_{ESi}$ are power references for
the ith WT, the ith PV and the ith ES, respectively. $\mathbf{P^{*}_{WT+}}$,
$\mathbf{P^{*}_{WT-}}$, $\mathbf{P^{*}_{PV+}}$, $\mathbf{P^{*}_{PV-}}$,
$\mathbf{P^{*}_{ES+}}$ and $\mathbf{P^{*}_{ES-}}$ are upwards and downwards
frequency reserve vectors for the ith WT, the ith PV and the ith ES,
respectively. $\mathbf{T_{FFR}}$, $DB_{FFR}$, $DB_{FCR}$ and $R_{FCR}$ are the
same FC settings as the ones passed from the HPPC. Note that these settings
are rarely updated once fixed.

Figure 6: Asset controller design. Figure 7: Plant controller design. Figure
8: The HPPC design.

As shown in Fig. 6, a parallel control loop for FFR and FCR is implemented at
asset controllers. Controller for FCR is implemented using a droop control
with frequency deadband, while controller for FFR is implemented using an
open-loop pre-defined shape with a FFR activation block. The output of this
control loop is added to active power control of asset controllers [25]. It is
worth pointing out that the proposed asset controller design is only an
example and there are alternatives for implementing controllers for FFR and
FCR with comparable performance [13]. To develop the optimal FC strategy from
the assets is beyond the focus of the paper. Fig. 7 shows the diagram of a
plant controller, whose structure is universally applicable to the WPP
controller, the SPP controller and the ESS controller. While no FC is
implemented at the plant controller, a FROB is added to estimate the total
frequency response from asset controllers. By subtracting the FROB output from
active power measurement at plant point of connection (PoC), the compensated
feedback only contains active power measurement irrelevant to FC. Another FROB
is also added to the HPPC for the same purpose, as shown in Fig. 8, allowing
the HPPC to provide a stacked functionality of energy trade and FCS provision.
Controller for FRR is implemented at the HPPC given that the response of FRR
is slow, in the range of tens of minutes. An extra control command
$P_{FRR\\_set}$ is added to HPP active power controller to control the
activation of FRR. The design details of plant controller and the HPPC could
be found in the companion paper [30].

##  III Frequency Response Observer

The idea of FROB is inspired by frequency-domain DOBs which accurately track
uncertainties and unknown disturbances. From the perspective of plant
controllers and the HPPC, frequency responses activated at asset controllers
can be considered as ”unknown disturbances with uncertainties”. The difference
that distinguishes FROB from frequency-domain DOBs is that the total frequency
response should be accepted rather than rejected at plant controllers and the
HPPC. This section briefly reviews classical frequency-domain DOBs and then
gives design and analysis for FROB.

###  III-A Overview of DOB

Figure 9: A generic DOB structure.

Frequency-domain DOBs were initially proposed in [23] to improve the
robustness of the controllers for DC servo motor systems, given disturbances
and uncertainties in the system. As shown in Fig. 9, the lumped disturbance
$d_{l}(s)$ is written as [31]:

$\displaystyle\begin{split}d_{l}(s)=&[G_{n}(s)^{-1}-G(s)^{-1}]y(s)+d(s)+G_{n}(s)^{-1}n(s)\\\
\end{split}$ (7)

where $d_{l}(s)$ captures all the disturbance and the uncertainties in the
system. $G(s)$ and $G_{n}(s)$ are the physical system and the nominal model,
respectively. $y(s)$ is the system output. $d(s)$ is the external disturbance.
$n(s)$ is the measurement noise. The filter $Q(s)$ is used to estimate the
lumped disturbance. The estimated lump disturbance $\hat{d}_{l}(s)$ is given
by:

$\displaystyle\begin{split}\hat{d}_{l}(s)=Q(s)d_{l}(s)\\\ \end{split}$ (8)

As the estimated lump disturbance is fed back to compensate the influence of
the disturbance, the output of the equivalent system considering the DOB is
expressed as:

$\displaystyle\begin{split}y(s)=G_{cy}(s)c(s)+G_{dy}(s)d(s)+G_{ny}(s)n(s)\\\
\end{split}$ (9)

where $G_{cy}(s)$, $G_{dy}(s)$ and $G_{ny}(s)$ are transfer functions from
$c(s)$, $d(s)$ and $n(s)$ to $y(s)$, respectively. In the frequency range
where $Q(s)\approx 1$, (9) reduces to [31]:

$\displaystyle\begin{split}y(j\omega)\approx
G_{n}(j\omega)c(j\omega)-n(j\omega)\\\ \end{split}$ (10)

The above representation implies the real physical system is forced to behave
like the nominal system without disturbance given $Q(s)\approx 1$.

###  III-B FROB

The main idea of FROB is to estimate frequency response provided by asset
controllers, and then add the estimation to plant-level and HPP-level
measurements as a compensation term, in order to avoid control counteraction.
The FROB uses signals only from its own control level, and therefore
communication is avoided during FCS provision. The FROB shown in Fig. 10 is
different from the original frequency-domain DOB design shown in Fig. 9. The
DOB design adds the estimated disturbance directly to the output of the
controller $C(s)$ to suppress disturbance and uncertainty. Here in the
proposed FROB design, the estimated disturbance is added to the input of the
controller $C_{p}(s)$ in order to compensate the influence of frequency
response.

Figure 10: FROB design for plant controller.

Fig. 10 shows the diagram of FROB for plant controllers. The FROB diagram is
generic to any type of power plant like WPP, SPP and ESS. Asset level includes
controller before FC $C_{A1}$, controller after FC $C_{A2}$, asset dynamics
$G_{A}$ and asset FC $G_{d}$. The input of $G_{d}$ is the measured frequency
deviation $f_{d}$ and the output of $G_{d}$ is the FC command $P_{d}$. Output
at asset level is the total power output $y$, and $n$ represents measurement
noise. Plant level includes plant controller $C_{p}$, dispatch function $D$,
and the FROB that consists of the nominal model $G_{n}$, and the FROB filter
$Q$. The model $G_{n}$ is the nominal representation of $G$, where
$G=DC_{A1}C_{A2}G_{A}$. The combined frequency response of FFR and FCR is
estimated by taking the difference between $y_{m}$ and $G_{n}u$. Ideally, it
is desired that

$\displaystyle\begin{split}\hat{y}_{d}&=Q(y_{m}-G_{n}u)\approx y_{d}\approx
P_{d}C_{A2}G_{A}\\\ \end{split}$ (11)

Figure 11: Linearized unity feedback system.

Based on Fig. 10, plant controller with FROB is restructured as a linearized
unity feedback system, shown in Fig. 11. Plant controller with FROB is
equivalent to controller $C_{p}(s)$ reshaped by $Q(s)$ and $G_{n}(s)$, with a
prefilter $\frac{1}{1-Q(s)}$ added to control reference $y_{r}$. The transfer
function from control reference to output $G_{y_{r}y}(s)$ is defined as (12).
The transfer function from FC command to output $G_{P_{d}y}(s)$ is defined as
(13). The transfer function from measurement noise to output $G_{ny}(s)$ is
defined as (14). When the filter $Q(s)\approx 1$, the transfer functions are
simplified into:

$\displaystyle G_{y_{r}y}(s)$
$\displaystyle=\frac{C_{p}(s)G(s)}{1+Q(s)C_{p}(s)G_{n}(s)-Q(s)C_{p}(s)G(s)+C_{p}(s)G(s)}$
(12) $\displaystyle G_{P_{d}y}(s)$
$\displaystyle=C_{A2}(s)G_{A}(s)\frac{1+Q(s)C_{p}(s)G_{n}(s)}{1+Q(s)C_{p}(s)G_{n}(s)-Q(s)C_{p}(s)G(s)+C_{p}(s)G(s)}$
(13) $\displaystyle G_{ny}(s)$
$\displaystyle=-\frac{(1-Q(s))C_{p}(s)G(s)}{1+Q(s)C_{p}(s)G_{n}(s)-Q(s)C_{p}(s)G(s)+C_{p}(s)G(s)}$
(14)

$\displaystyle G_{y_{r}y}|_{Q\approx 1}$
$\displaystyle\approx\frac{C_{p}G}{1+C_{p}G_{n}}$ (15) $\displaystyle
G_{P_{d}y}|_{Q\approx 1}$ $\displaystyle\approx C_{A2}G_{A}$ (16)
$\displaystyle G_{ny}|_{Q\approx 1}$ $\displaystyle\approx 0$ (17)

It is seen from (16) and (17) that, when $Q(s)$ is close to 1, frequency
response is fully accepted in the closed-loop system, and measurement noise is
completely attenuated. Although the unity might seem a perfect design for
$Q(s)$, it would lead to zero loop transfer function, and modified
characteristics of reference command tracking, seen from Fig. 11 and (15).
Only when the nominal model is correct ($G_{n}=G$) does $G_{y_{r}y}(s)$ stay
unchanged, and plant controller $C_{p}$ and the FROB filter can be designed
independently. Therefore, low-pass filters have been used for the design of
$Q(s)$.

Figure 12: FROB design for HPPC.

FROB for the HPPC is similar to FROB for plant controllers, as shown in Fig.
12. The difference lies mainly in the model being used in FROB to represent
the nominal model of the HPP, $\mathbf{G_{n}}$, which is a multiple-input
single-output model. The actual HPP dynamics are presented using the closed-
loop transfer functions including both plant level and asset level. These
closed-loop transfer functions for WPPs, SPPs and ESSs are obtained using
(12), (13) and (14). A similar linearized unity feedback system to Fig. 11 can
be easily derived including HPP level, plant level and asset level. To avoid
redundancy, only plant level and asset level are considered as the example in
the following design guideline and robustness analysis. Nevertheless, they can
readily be applied to the system including all three levels.

###  III-C Design Guidelines

Algorithm 1 $Q(s)$ Selection
---
1: Initialize the algorithm by setting up a general structure for $Q(s)$:
$Q(s)=\frac{1+\sum_{m=1}^{n-1}f_{m}s^{m}}{1+\sum_{m=1}^{n}f_{m}s^{m}}$
2: Start the filter degree $n=1$ and choose $\omega_{c}$ to be equal to or
larger than the frequency of measurement noises.
3: Perform bode analysis of $G_{P_{d}y}$ and $G_{ny}$, and fine tune
$\omega_{c}$, such that $G_{P_{d}y}$ has unity gain around the frequency range
of frequency response and $G_{ny}$ has small gain around the frequency range
of measurement noises.
4: Increase the filter degree by 1 and choose the butterworth filter
coefficients while keeping the same $\omega_{c}$. Check if $G_{P_{d}y}$ has
unity gain for a wider frequency range, that is, a preferable performance on
frequency response acceptance, and $G_{ny}$ has a smaller gain for a wider
frequency range, that is, a preferable performance on measurement noise
attenuation.
5: If $G_{P_{d}y}$ and $G_{ny}$ show a better performance with the increased
filter degree
6: Repeat Step 4.
7: Else
8: Finalize the design of $Q(s)$ with the original degree.

The design of the filter $Q(s)$ plays a significant role in the FROB. The main
requirement is that $Q(s)$ should be a low-pass filter with unity gain for low
frequency to accept frequency response and zero gain for high frequency to
attenuate measurement noise. Unlike the DOB filter design [32, 33], no degree
requirements exist for the FROB filter since there is no inverse of the model.
There is an amount of freedom to select $Q(s)$ as long as $Q(j\omega)\approx
1$ in the frequency range covering frequency response and measurement noise.
The main parameters to be selected for $Q(s)$ are the cut-off frequency
$\omega_{c}$ and the filter degree $n$. A straightforward algorithm is
developed to select the filter $Q(s)$, shown as Algorithm 1.

Figure 13: Bode plots of transfer functions - $G_{P_{d}y}$ and $G_{ny}$.

Fig. 13 presents bode plots of $G_{P_{d}y}(s)$ and $G_{ny}(s)$. Without FROB
being included, $G_{P_{d}y}(s)$ shows the rejection of frequency response in
low frequency range, which explains the importance of the FROB in hierarchical
FC. With the filter degree equal to 1, 2 and 3, $G_{P_{d}y}(s)$ shows the
characteristics of frequency response acceptance. It is also seen that without
FROB, $G_{ny}(s)$ shows a limited performance of measurement noise
attenuation. With a higher filter degree, the performance of measurement noise
attenuation has been improved. Therefore, the degree of 3 is preferable for
the design of $Q(s)$.

PI gains of plant controllers are selected based on the consideration of
coordination with other control levels, and performance requirements of
command reference tracking. The bandwidth of the closed-loop system including
plant level and asset level is selected to be 0.25 Hz and the phase margin is
selected to be 150∘.

###  III-D Robustness Analysis

In this subsection, robustness analysis under two scenarios are considered for
the FROB: robustness in the presence of uncertain parameters and time-varying
communication delays. When uncertainties are considered, it implies that
$G_{n}(s)=G(s)$ is invalid.

In the first scenario, assume that asset dynamics has changed to a different
system $G^{\prime}_{A}$, leading to the actual physical system $G_{m1}$
different from $G_{n}$, which is written as:

$\displaystyle G_{m1}$ $\displaystyle=DC_{A1}C_{A2}G^{\prime}_{A}$ (18)

The communication delay exists between asset controllers and plant
controllers. In the second scenario, a time-varying communication delay is
introduced as $e^{-Ts}$ as shown in Fig. 10, and $T$ is uncertain but is known
to lie in the range [$T_{1}$, $T_{2}$]. The integration of communication
delays leads to another physical system $G_{m2}$, which is written as:

$\displaystyle G_{m2}$ $\displaystyle=e^{-Ts}G=e^{-Ts}DC_{A1}C_{A2}G_{A}$ (19)

Based on the perturbation multiplicative model [34], the magnitude bound for
system stability in these two scenarios is written as:

$\displaystyle M_{1}(s)$ $\displaystyle=\frac{G^{\prime}_{A}(s)}{G_{A}(s)}-1$
(20) $\displaystyle M_{2}(s)$ $\displaystyle=e^{-Ts}-1$ (21)

The loop transfer function of the FROB including an asset controller and a
plant controller is derived from Fig. 11:

$\displaystyle L(s)$
$\displaystyle=\frac{(1-Q(s))C_{p}(s)G(s)}{1+Q(s)G_{n}(s)C_{p}(s)}$ (22)

To check the stability of the closed-loop system in the presence of uncertain
parameters, it should be guaranteed that

$\displaystyle|M_{1}(j\omega)|$
$\displaystyle<|1+\frac{1}{L(j\omega)}|\textrm{ for all }\omega\textrm{ and
}a_{min}\leq a\leq a_{max}$ (23) $\displaystyle|M_{2}(j\omega)|$
$\displaystyle<|1+\frac{1}{L(j\omega)}|\textrm{ for all }\omega\textrm{ and
}T_{1}\leq T\leq T_{2}$ (24)

Since $a$ and $T$ are uncertain, the magnitude of the LHS of (23) and (24) are
not known. As long as functions $W_{1}(s)$ and $W_{2}(s)$ are found, such that

$\displaystyle|\frac{G^{\prime}_{A}(j\omega)}{G_{A}(j\omega)}-1|$
$\displaystyle<|W_{1}(j\omega)|\textrm{ for all }\omega\textrm{ and
}a_{min}\leq a\leq a_{max}$ (25) $\displaystyle|e^{-j\omega T}-1|$
$\displaystyle<|W_{2}(j\omega)|\textrm{ for all }\omega\textrm{ and }T_{1}\leq
T\leq T_{2}$ (26)

the robust stability conditions with uncertain parameters and time-varying
communication delay are satisfied by

$\displaystyle|W_{1}(j\omega)|<|1+\frac{1}{L(j\omega)}|\textrm{ for all
}\omega$ (27) $\displaystyle|W_{2}(j\omega)|<|1+\frac{1}{L(j\omega)}|\textrm{
for all }\omega$ (28)

Fig. 14 shows magnitude plots for $|M(j\omega)|$, $|W(j\omega)|$ and
$|1+\frac{1}{L(j\omega)}|$ with two types of uncertainties. In Fig. 14(a), it
can be seen that, when the actual dynamics vary around the nominal model, a
function $W_{1}(s)$ can be obtained as the upper bound of
$|\frac{G^{\prime}_{A}(j\omega)}{G_{A}(j\omega)}-1|$. Likewise, based on
industrial practice, communication delay between plant controllers and asset
controllers ranges from 0 to 2 sec. The function $W_{2}(s)$ is obtained as the
upper bound of $|e^{-j\omega T}-1|$. It can be seen that the robust stability
condition is met for both scenarios. Also a high magnitude of
$|1+\frac{1}{L(j\omega)}|$ indicates the FROB results in a very robust system.

(a) $M_{1}(s)=\frac{G^{\prime}_{A}}{G_{A}}-1$

(b) $M_{2}(s)=e^{-Ts}-1$

Figure 14: Magnitude plot of $|M(j\omega)|$, $|W(j\omega)|$ and
$|1+\frac{1}{L(j\omega)}|$.

##  IV Simulation Results

In order to validate the proposed hierarchical FC approach as well as the
relevant analytical results, three case studies are carried out using a
single-bus power system model with system parameters shown in [24]. This power
system model, along with a detailed HPP model including the WPP, the SPP and
the ESS and the corresponding controls, is built in MATLAB/Simulink™. The
first case study is the benchmark case showing the performance of the proposed
FC on the coordination of three types of FCSs. The comparison with the
centralized implementation is presented in the second case study. The focus of
the third case study is on the robustness of the FROB when different types of
uncertainties are included in the system.

###  IV-A Benchmark Studies

(a) Only ESS participates in FCS provision with HPP power reference equal to
0.9 p.u. and HPP power reserve reference equal to 0.05 p.u.

(b) Both WPP and ESS participate in FCS provision with HPP power reference
equal to 0.84 p.u. and HPP power reserve reference equal to 0.10 p.u.

Figure 15: FCS provision of an HPP during an under-frequency event.

The under-frequency event is triggered at t = 150 sec by a heavy load
switching-in within the power system. A secondary FC is initiated by system
operator at t = 400 sec to bring system frequency back to nominal. Fig. 15
shows frequency response provided by the HPP while the HPP tracks power
setpoint decided by the HPP EMS. Fig. 15(a) presents the case when all the
FCSs including FFR, FCR and FRR are fulfilled by the ESS. At t = 150 sec when
the frequency dip occurs, the ESS provides a combined frequency response
including FFR and FCR. When the system frequency reaches steady-state, only
FCR is still activated and the charging power of the ESS achieves a new
steady-state value. At t = 400 sec, the FRR setpoint is sent to the HPPC by
system operator, with the frequency restored back to 50 Hz due to the effect
of system-level secondary frequency response. Meanwhile, FCR provided by the
HPP dies away when the frequency deviation reduces to zero.

Fig. 15(b) presents another case when FCSs are fulfilled by both the ESS and
the WPP. According to the active power reserve dispatch algorithm in [30], the
capacity of the ESS is reserved to provide FFR and FCR, and the WPP runs at
the derated operation to provide FCR and FRR. When the frequency dip occurs at
t = 150 sec, the ESS provides a combined frequency response including FFR and
FCR while the WPP provides FCR. At t = 400 sec, the FCR dies away due to
frequency restoration, so it can be seen that the charging power of the ESS
returns to the pre-event value. Meanwhile, the WPP participates in secondary
frequency response initiated by system operator via the contribution of FRR.

The above results show that the proposed hierarchical FC is effective in
coordinating three types of FCSs, namely FFR, FCR and FRR, during frequency
events. Besides, the approach works in harmony with conventional active power
control. Thanks to FROB, no counteraction is identified. More thorough
operating conditions will be considered to validate the proposed approach in
the future work.

###  IV-B Comparison with Centralized Approach

Figure 16: Centralized implementation of hierarchical FC.

(a) $T_{cd}=0.0s$.

(b) $T_{cd}=0.1s$.

(c) $T_{cd}=1.0s$.

(d) $T_{cd}=2.0s$.

Figure 17: Performance comparison of FCS provision between the proposed
approach and the centralized approach considering communication delays.

Instead of at each asset, FC can be implemented at the HPPC [4]. A simplified
diagram of architecture design with centralized implementation of FC is shown
in Fig. 16. The main difference between the proposed approach and the
centralized implementation is that the controller for FFR and the FCR is
located at the HPPC and activated by the frequency measured at the HPP PoC.
The active power setpoints obtained from the HPPC are passed and added to the
active power control at each plant controller. In this case, FROB is still
necessary at the HPPC to distinguish frequency response from active power
tracking, but is unnecessary at plant controllers.

TABLE I: Impact of Communication Delay on Frequency Response Provision

Comm Delay $T_{cd}$ (sec) | Rise Time (sec)
---|---
Distributed | Centralized
0.0 | 1.9 | 2.9
0.1 | 1.9 | 2.9
1.0 | 1.9 | 3.4
2.0 | 1.9 | 4.6

Fig. 17 presents the performance of FFR and FCR provision considering that
communication delay exists between asset controllers and plant controllers.
The communication delay $T_{cd}$ is assumed to lie in the interval between 0
sec and 2 sec, and values including 0.0 sec, 0.1 sec, 1.0 sec and 2.0 sec are
selected. This assumption is realistic for a large renewable power plant [35].
It is obviously seen that with no or small communication delay, frequency
response via the proposed approach and the centralized approach are
comparable. However, for larger communication delays like 1 sec or 2 sec, the
centralized approach provides a delayed frequency response that doesn’t meet
technical requirement of FFR and FCR [1]. As shown in Table I, even if there
is negligible communication delay, time constant of frequency response via the
centralized approach is still larger than the proposed approach. It is because
for the centralized approach, how fast frequency response is activated is
subject to control bandwidth of plant controller. On the contrary, the
proposed approach provides the same level of frequency response regardless of
the communication delay.

###  IV-C Study on FROB Robustness

In order to investigate the robustness of FROB, three types of uncertainties
are considered in this subsection. The first type is uncertainty in system
parameters. The second type is a random malfunction, which is unknown to the
plant controller, of FC at asset level. The third type refers to time-varying
communication delay between asset controllers and plant controllers. Either
type of uncertainty leads to a nominal model $G_{n}(s)$ that doesn’t represent
the actual system $G(s)$. For simplicity purpose, only asset controllers and
plant controllers are included in this study. However, the conclusion also
applies to the case when the HPPC is considered.

To evaluate the performance of FROB, ideal responses of FFR and FCR are
defined by the performance obtained from an open-loop plant controller. When
the plant controller is open-loop, there is no counteraction between asset
controllers and plant controllers during the frequency event. Another
coordination strategy to avoid counteraction is to estimate the total
frequency response from asset controllers using FC settings that are made
available to plant controllers, plus the frequency measured at the plant PoC.
In the following studies, the above-mentioned strategies are named Strategy #1
and Strategy #2, respectively. The proposed approach is named Strategy #3.

####  IV-C1 Impact of uncertain system parameters

(a) upper bound uncertainties

(b) lower bound uncertainties

Figure 18: FROB robustness considering uncertain system parameters.

The HPP might have several parameters that are constants but uncertain within
a range. When actual parameters deviate from the nominal values used in the
FROB design, the performance of FROB should be investigated. For example,
control parameters of an asset converter vary around a nominal value that is
used to construct the nominal WPP model for the FROB. Here, uncertainties in
parameters are considered as:

$\begin{split}\delta 1&=T_{cc\\_MSC}\in[0.001,0.1],\delta
2=K_{p\\_pc\\_MSC}\in[0.05,0.15],\\\ \delta
3&=K_{i\\_pc\\_MSC}\in[5,15],\delta 4=T_{cc\\_GSC}\in[0.001,0.1],\\\ \delta
5&=K_{p\\_vc\\_GSC}\in[7.5,22.5],\delta
6=K_{i\\_vc\\_GSC}\in[75,225].\end{split}$ (29)

where $T_{cc\\_MSC}$ is time constant of current control of machine-side
converter, $K_{p\\_pc\\_MSC}$ and $K_{i\\_pc\\_MSC}$ are proportional and
integral gains of power control of machine-side converter, $T_{cc\\_GSC}$ is
time constant of current control of grid-side converter, $K_{p\\_vc\\_GSC}$
and $K_{i\\_vc\\_GSC}$ are proportional and integral gains of DC-link voltage
control of grid-side converter. Fig. 18(a) and 18(b) shows frequency responses
for upper and lower bound uncertain systems. Strategy #3 always provides
frequency response identical to the ideal response (Strategy #1). On the other
hand, the deviation of system parameters causes the failure of estimating
frequency responses from asset controllers using Strategy #2. Thus, FROB shows
robustness over uncertain systems.

####  IV-C2 Impact of unknown FC malfunction

Figure 19: FROB robustness considering unknown FC malfunction.

It happens when FC at some assets fail to respond to frequency dips due to
device malfunction. When the failure is unknown to plant controller, whether
frequency response is properly coordinated across control hierarchy needs to
be investigated. Fig. 19 shows that frequency response from Strategy #3
adaptively coincides with that from Strategy #1. It implies that FROB is
robust in estimating the total frequency response, regardless of what
percentage of assets experience FC failure. Meanwhile, frequency response from
Strategy #2 indicates overcompensation from plant controller, because the
total frequency response from all the assets is overestimated with plant
controller not being notified of those malfunctioning units.

####  IV-C3 Impact of time-varying communication delay

(a) $T_{cd}$ = 0.0 s

(b) $T_{cd}$ = 0.1 s

(c) $T_{cd}$ = 1.0 s

(d) $T_{cd}$ = 2.0 s

Figure 20: FROB robustness considering time-varying communication delay.

The exact value of communication delay between asset controllers and plant
controllers is uncertain, and the interval $0\leq T_{cd}\leq 2s$ is used. Fig.
20 shows frequency response considering $T_{cd}$ equal to 0.0 sec, 0.1 sec,
1.0 sec and 2.0 sec. While Strategy #2 experiences instability issues with
large communication delay, there is no stability issue observed from Strategy
#3, which in all cases provides the identical frequency response to that of
Strategy #1. Although tuning control gains could be an option of resolving
instability for Strategy #2, it could still be challenging to obtain a set of
control gains accommodating time-varying communication delays. With the FROB
integrated into plant controllers, control gains can be kept the same as those
designed for cases without communication latency. This characteristic shows
the superior performance of the FROB regarding robustness against time-varying
communication delay.

##  V Conclusion

This paper proposes an innovative hierarchical FC approach that enables HPPs
to provide FCSs. The proposed approach coordinates not only among multiple
technology power plants but also across control hierarchy. Fast FCSs like FFR
and FCR are implemented at asset level while slow FCS such as FRR is
implmented at plant level and HPP level. A novel observer called FROB, which
is inspired by frequency-domain DOBs, is developed to estimate the total
frequency responses activated by asset controllers, for the purpose of
avoiding control counteraction. In this way, the proposed approach
accommodates state-of-the-art fast FC designed at asset level, coping with the
increasingly demanding technical requirements, even when there are system
uncertainties, such as uncertain control parameters, unknown malfunction or
time-varying communication delays.

## Acknowledgment

The work is done as a part of Indo-Danish HYBRIDize project funded by
Innovation Fund Denmark (IFD) with grant number 8127-00015B.

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