A Corrective Controller for Improved Ratio-Based Frequency Support through Multiterminal High-Voltage Direct Current Grids

Abstract

A recently proposed droop controller design achieved approximate ratio-based frequency support through Multiterminal High-Voltage Direct Current (MTDC) grids connecting asynchronous AC areas. The design was performed via a reduced-order model, which neglects system losses. In this paper, to achieve improved tracking, a model-reference-estimation-based corrective control approach is presented, which estimates the differences between the reduced and actual models and compensates for the same. The sufficient condition for perfect ratio tracking by the proposed controller in the presence of modeling uncertainty is established. It is shown that a continuous sliding-mode controller that is robust to bounded modeling uncertainty takes the form of the PI compensator in the proposed corrective control. The region of stability of the proposed controller is established through root locus and numerical eigenvalue analyses. Finally, the effectiveness of this corrective strategy is illustrated through time-domain simulations performed in a full-order model, which is a detailed phasor-based differential-algebraic representation of the AC-MTDC grid.

1. Introduction

In view of the envisioned projects that aim to connect multiple asynchronous AC grids in Europe around the North Sea to harness offshore wind energy [1], there is a need to share frequency support resources through the interconnecting Multiterminal High-Voltage Direct Current (MTDC) grid. Moreover, as the generation mix shifts towards renewable energy resources, including wind and solar, the decrease in the inertia of an individual area would have to be compensated for by inertial and primary frequency support provided from other areas in the interconnection. In the literature, (i) droop control-based methods and (ii) distributed/centralized/coordinated methods have been proposed for frequency support.

References [2,3,4,5,6,7,8] propose strategies that fall into the first category. References [2,3] propose two similar consensus-based controllers. Building on these concepts, Refs. [4,5] propose a distributed proportional–integral (PI) controller and distributed model predictive control (MPC), respectively. Papers [6,7] propose distributed controllers for secondary frequency support. Notably, in all these works [2,3,4,5,6,7], small-signal stability is analytically proven, at least for some special cases. On the other hand, Ref. [8] proposes MPC for secondary frequency control, and designed parameters through trial and error to obtain a stable and damped response.

The concept of providing frequency support through droop control of MTDC converter stations is first proposed in [9], following which multiple works on this topic have used droop in some form. Reference [10] is the first work to present communication-free droop control to extract support from offshore wind farms (OWFs). Related works on frequency support from OWFs include [11,12]. A few papers [13,14] analyze the effect of coupling between voltage and frequency droop in a steady state. To decouple the droop control loops, Ref. [15] proposes an integration method, [16,17] present receding horizon control, and Ref. [18] proposes an MPC-based approach. Some works, including [19,20], have proposed an adaptive frequency support mechanism. Reference [19] proposes the emulation of a virtual synchronous generator through voltage source converter (VSC) controls to extract inertial and primary frequency support. In [20], voltage-droop coefficients are adaptively varied based on the frequency deviation.

To the best of our knowledge, none of the papers among [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] have introduced a way to quantify the amount of support provided by each area towards the interconnection. Moreover, most of these works are focused on achieving certain performance objectives, and do not consider any robustness analysis due to parametric uncertainty.

Our prior works [21,22,23,24] propose a ratio-based approach to quantify the support provided by a single AC area towards the interconnection. In those works, the droop coefficients are analytically designed to meet the prescribed ratios based on a reduced-order model, which neglects various dynamics and does not account for losses in the AC-MTDC system. Consequently, the controller proposed in these prior works fails to accurately meet the prescribed support. Moreover, it is dependent on the accurate knowledge of model parameters, as rightly pointed out in [25]. Reference [25] aimed to achieve the objective of ratio-based frequency support (the ratios were not plotted over time) by explicitly communicating the system frequencies without any time delay, which is not realistic, even with a high bandwidth. Also, in the test system of [25], none of the AC areas is a multimachine area, which is impractical. In reality, considering communication latency and multiple machines in one area can challenge the design of the controller. Importantly, the proposed scheme in this work is not shown to be based upon a mathematical foundation, unlike [21,22]. Moreover, ref. [25] neither established any analytical guarantees for maintaining the ratios nor provided sufficient conditions for system stability—only numerical root locus analysis was presented.

In this paper, we aim to address the aforementioned issues of [21,22,23,24], namely (a) approximate ratio-based tracking, and (b) dependence on knowledge of model parameters by using a model-reference-estimation-based corrective controller. The contributions of this paper, including the improvements achieved by the proposed controller, are summarized below.

• Analytical characterization of system losses in ratio-tracking issue: We analytically establish the condition for ratio-based tracking in the presence of both AC and DC losses. This shows what role system losses play in approximate ratio tracking in [21,22,23,24].
• Model reference estimation for improved ratio tracking: Building on the first contribution, a model-reference approach for loss estimation is proposed, which ensures that compared to the approximate tracking of previous works, ratios are now accurately met.
• Fast corrective control: We propose a further improvement in the model-reference estimation approach by separating the faster and slower transients in system losses, which leads to more accurate tracking during the inertial phase.
• Removing need for knowledge of system parameters: In our proposed control method, accurate knowledge of system parameters is not a necessity to achieve the desired performance.
• Robustness analysis: The sufficient condition for perfect ratio tracking in the presence of modeling uncertainty is established. It is shown that a sliding-mode controller that is robust to modeling uncertainty takes the form of the PI compensator in the proposed corrective control.

This paper is organized as follows. For the sake of completeness, the objectives of recently proposed ratio-based frequency support are explained first. Next, the limitations of droop-based design proposed in [21,22] are illustrated analytically and through simulation studies. In order to solve these issues, a model-reference-estimation-based corrective control strategy is proposed. This is followed by a robustness analysis of the proposed approach under modeling uncertainty. Finally, the results of time-domain simulations are illustrated using a detailed model called the ‘full-order model’ that represents the AC-MTDC grid using phasor-based differential-algebraic equations. Unlike [25], realistic latency in communication channels is considered and the proposed controllers are shown to work with at least one AC area with multiple synchronous generators, loads, and transmission lines.

2. Ratio-Based Selective Frequency Support

This section is a review of the ratio-based selective frequency support mechanism that was proposed in [21,22]. First, let us consider a system, shown in Figure 1, with four asynchronous AC areas connected through a five-terminal bipolar MTDC network with metallic return. AC Area #1 is a modified four-machine system [26], while each of the remaining AC areas is represented by a single aggregated generator. We will use this system for explaining the ratio-based selective frequency support concept and also use it as our study system in this paper.

2.1. Motivation and Objectives

Consider that there is a sudden reduction in generation in one of the relatively smaller AC areas, e.g., Area $\#4$. Due to low inertia, this will lead to a steep rate of change of frequency (RoCof) and a deep nadir, which might result in underfrequency load shedding (UFLS). To avoid this, the MTDC grid can derive frequency support from other AC areas. However, not all asynchronous AC areas in the interconnection might be willing to provide frequency support due to various politico-economic or reliability issues. Therefore, the areas in the MTDC interconnection are divided into participating and non-participating areas. However, note that depending on the type and location of contingency, an AC area’s designation as participating or non-participating might change. Thus motivated, the first objective of such a frequency support mechanism is to have the ‘selectivity’ property.

Another important aspect in this process is to quantify the contribution of each participating AC area. In [21], a ratio-based metric of primary frequency support contribution was proposed. In this metric, a prescribed ratio has to be achieved among the participating areas’ frequency deviations under a steady state. In [22], the ratio-based metric was further expanded to quantify the inertial support contribution of AC areas by imposing the desired ratio among the frequency deviations of the participating areas for the entire time after the contingency. This ratio can be translated into the change in real power import/export from each area, which can be used for economic analysis and market participation. Thus, the second objective of the frequency support mechanism is ‘quantifiability’.

Note that our proposed approach is geared towards a future ancillary services market mechanism where different areas will participate. Most notably, with the appropriate droop control design for ratio-based control, the MTDC grid just acts as a conduit for support. The individual areas need to procure the resources (for inertial and primary frequency support) to be able to maintain the pre-determined ratios.

2.2. Mathematical Representation of Objectives

Let $f_i$ and $f_0$ denote the center of interia (COI) frequency of the ith AC area and the nominal frequency, respectively. The deviation in COI frequency in the ith AC area is given by $\Delta f_i=f_0-f_i$. Also, suppose the number of participating areas among N asynchronous AC areas is equal to $N_r$, and without loss of generality, assume that all the non-participating areas are concatenated to the end. With these assumptions and a prescribed ratio of {$r_1:\cdots :r_{N_r}$}, both objectives can be translated to the equation shown below:

$$\begin{array}{c}\hfill \Delta f_i\left(t\right)=\left\{\begin{array}{cc}0,\forall t>t_0\hfill & \mathrm{if}i>N_r\hfill \\ r_i\Delta f_{i^{*}}\left(t\right)/r_{i^{*}},\forall t>t_0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

(1)

where $i^{*}\in \{1,\cdots ,N_r\}$ is the disturbed area number, and $t_0$ denotes the time at which contingency occurred. To clarify, $r_i^{*}$ is the prescribed ratio of frequency deviation of the disturbed area, which is one of the numbers in {$r_1:\cdots :r_{N_r}$}. Therefore, (1) shows that if Area #i is a participating area, then the ratio of frequency deviation of the ith area and the disturbed area will be ${\displaystyle \frac{\Delta f_i\left(t\right)}{\Delta f_{i^{*}}\left(t\right)}}={\displaystyle \frac{r_i}{r_{i^{*}}}}$.

2.3. Overall Control Architecture

In this section, we describe the overall control architecture of the MTDC system to achieve the two objectives mentioned above. Figure 2 shows the power–voltage-droop control along with the frequency support reference generator for the aggregated g-pole converter station connected to the ith AC area. This is the so-called outer loop that feeds the inner current control loop of the converter station operating under the traditional vector control approach.

Assume that both the frequency support reference generator and voltage droop are turned on, and that the PI controller in Figure 2 is perfectly tracking its reference. Then, the net power injected into the ith AC area from the aggregated g-pole, $P_{Cgi}$ is given by

$$\begin{array}{c}\hfill P_{Cgi}=P_{refgi}+k_{vgi}(V_{DCgcom}^2-V_{DCgref}^2)/4+\Delta P_{fgi}\end{array}$$

(2)

where $g=p$ and n represent positive and negative pole variables, respectively. The variable $k_{vgi}$ denotes the voltage-droop constant; $V_{DCgref}$ and $V_{DCgcom}$ are the reference and measured common DC voltages from the DC network belonging to g-pole, respectively. This voltage is measured from a common node in the DC grid (for the study system in Figure 1, the DC bus of station#1 is assumed to be the common node) and communicated to all the converters for autonomous power sharing [27]. Note that $\Delta P_{fgi}$ is the output of the frequency support reference generator, which is added to the power reference $P_{refgi}$.

2.3.1. Achieving Selectivity

In a nominal operating condition, frequency support is switched off, i.e., $\Delta P_{fgi}$ is set to zero, and all the common DC voltage-droop controllers are active. Following an AC-side contingency, as illustrated in Figure 2, a distress signal is sent from the converter in the disturbed AC area, which triggers the activation of frequency control and the deactivation of voltage droop in participating and non-participating AC areas, respectively. Therefore, in effect, the non-participating areas hold their power constant and their frequency remains unchanged, i.e., $\Delta f_i=0$, ∀$i>N_r$, as required by the first part in Equation (1). Note that for all simulation studies of the realistic model, a 100 ms delay is assumed in communicating the distress signal.

2.3.2. Achieving Quantifiability

Different techniques for frequency support generation can be used to achieve this objective. In [21,22,23], inertial and primary frequency droop are used and their droop coefficients are designed to meet the ratio-based support. In the next section, we summarize the overall system modeling followed by the droop coefficient design process.

3. Modeling and Droop Coefficient Design for Ratio-Based Frequency Support

3.1. Full-Order Model

As the name suggests, the full-order model is a detailed representation of the AC-MTDC interconnection obtained through rigorous nonlinear differential–algebraic equations. Within a single AC area, it allows the inclusion of multiple sixth-order subtransient models of generators equipped with IEEE-type DC-1A excitation systems and non-reheat turbine-governorsconnected through AC transmission lines and transformers. Within the MTDC grid, a bipolar DC network with metallic return and converter stations, represented through averaged models of VSCs with the standard vector control strategy, are considered. The DC links are modeled using a fitted $\pi$ cable model [28]. Further details of generic full-order system modeling can be found in [29].

The full-order model is employed when there is a need for a realistic representation of the study system. For example, in order to validate the performance of the existing droop control and the proposed corrective strategies via time-domain simulations, we considered the full-order model of the test system shown in Figure 1.

3.2. Reduced Nth-Order Model for Droop Controller Design

In order to analytically design droop coefficients and derive the conditions for stability, a reduced-order model is essential. Therefore, a reduced Nth-order model is derived from the full-order model with inertial and frequency droop controllers based on some approximations, including:

• Neglecting DC lines’ losses and inductive and capacitive dynamics, and only considering the algebraic power sharing effect of voltage-droop control;
• Assuming instantaneous tracking for controllers in the converter stations;
• Aggregating each asynchronous AC area into a single synchronous generator and only considering the swing dynamics of that generator.

The aggregate generator in the ith AC area has an inertial constant of $H_{gi}$ and an inverse governor droop of $k_{gi}$. For an AC-MTDC interconnection with N asynchronous areas, this reduced-order model has N dynamic states, where the ith state corresponds to the COI frequency of the ith AC area. When inertial and primary frequency droop controllers are activated in the participating areas, the net frequency support reference of the ith AC area, $\Delta P_{fi}$ is given by

$$\begin{array}{c}\hfill \Delta P_{fi}=2\Delta P_{fpi}=2\Delta P_{fni}=k_{fi}\Delta f_i-2h_i{f}_i{\dot{f}}_i\end{array}$$

(3)

where $h_i$ and $k_{fi}$ denote the inertial and primary frequency droop coefficients, respectively. Also, from (2), assuming identical positive and negative poles, we can write

$$P_{Ci}=P_{cpi}+P_{cni}=P_{refi}+k_{vi}\left({\mathrm{V}}_{\mathrm{DCcom}}^2-V_{DCref}^2\right)/4+\Delta P_{fi}$$

(4)

We can write the aggregated swing equation of all the synchronous generators in the ith AC area as

$$\begin{array}{c}2H_{gi}{f}_i{\dot{f}}_i=P_{Mi}+k_{gi}\Delta f_i-P_{Li}+P_{Ci}\hfill \end{array}$$

(5)

We perform algebraic manipulations with (4) in (5) by replacing $P_{Ci}$, which leads to the a reduced model describing the frequency dynamics as follows:

$$\begin{array}{c}2[H_{gi}+(1-{\overline{k}}_{vi})h_i]f_0{\dot{f}}_i-{\overline{k}}_{vi}({\displaystyle \sum _{k\ne i}^N}2h_k{f}_0{\dot{f}}_k)\hfill \\ =[k_{gi}+(1-{\overline{k}}_{vi})k_{fi}]\Delta f_i-{\overline{k}}_{vi}({\displaystyle \sum _{k\ne i}^N}{k}_{fk}\Delta f_k)+\Delta P_i\hfill \end{array}$$

(6)

where $i=1,\cdots ,N$; ${\overline{k}}_{vi}=k_{vi}/{\sum }_k^N{k}_{vk}$ is the normalized voltage-droop coefficient; and $\Delta P_i$ is given by

$$\begin{array}{c}\Delta P_i=P_{Mi}-P_{Li}+P_{refi}-{\overline{k}}_{vi}({\displaystyle \sum _k^N}{P}_{refk})\hfill \end{array}$$

(7)

From (6), the Nth-order model can be written as follows (note that $\Delta f_i=f_0-f_i$ implies that ${\dot{f}}_i=-\Delta {\dot{f}}_i$):

$$\begin{array}{c}\hfill 2f_0{\mathbf{H}}_{\mathbf{N}}\Delta \dot{\mathbf{f}}=-{\mathbf{K}}_{\mathbf{N}}\Delta \mathbf{f}-\Delta \mathbf{P}\end{array}$$

(8)

Here, the $i^{th}$ elements of vectors $\Delta \mathbf{P}\in {\mathbb{R}}^N$ and $\Delta \mathbf{f}\in {\mathbb{R}}^{\mathbf{N}}$ are $\Delta P_i$ and $\Delta f_i$, respectively. The DC- and/or AC-side disturbances in the system are denoted by $\Delta \mathbf{P}$, whose pre-disturbance steady-state value is $\mathbf{0}$. The matrices ${\mathbf{H}}_{\mathbf{N}}\in {\mathbb{R}}^{N\times N}$ and ${\mathbf{K}}_{\mathbf{N}}\in {\mathbb{R}}^{N\times N}$ are structurally similar, and their elements can be represented as follows:

$$\begin{array}{c}{H}_N(i,j)=\left\{\begin{array}{c}{H}_{gi}+(1-{\overline{k}}_{vi})h_i\mathrm{if}i=j\\ -{\overline{k}}_{vi}{h}_j\mathrm{if}i\ne j\end{array}\right.\\ \\ K_N(i,j)=\left\{\begin{array}{c}{k}_{gi}+(1-{\overline{k}}_{vi})k_{fi}\mathrm{if}i=j\\ -{\overline{k}}_{vi}{k}_{fj}\mathrm{if}i\ne j\end{array}\right.\end{array}$$

(9)

The final form of the model that is sufficient to approximately capture frequency dynamics in $N_r$ participating AC areas following an AC-side contingency can be compactly represented as

$$\begin{array}{c}\hfill 2f_0{\mathbf{H}}_{\mathbf{Nr}}\Delta \dot{\mathbf{f}}=-{\mathbf{K}}_{\mathbf{Nr}}\Delta \mathbf{f}-\Delta \mathbf{P}\end{array}$$

(10)

$$\begin{array}{c}{\mathbf{X}}_{\mathbf{Nr}}=\left[\begin{array}{cccc}{C}_1+(1-{\overline{k}}_{v1})x_1& -{\overline{k}}_{v1}{x}_2& \cdots & -{\overline{k}}_{v1}{x}_{Nr}\\ -{\overline{k}}_{v2}{x}_1& C_2+(1-{\overline{k}}_{v2})x_2& \cdots & -{\overline{k}}_{v2}{x}_{Nr}\\ ⋮& ⋮& \ddots & ⋮\\ -{\overline{k}}_{vNr}{x}_1& -{\overline{k}}_{vNr}{x}_2& \cdots & C_{Nr}+(1-{\overline{k}}_{vNr})x_{Nr}\end{array}\right]=\left\{\begin{array}{c}\frac{{\mathbf{H}}_{\mathbf{Nr}}\begin{array}{c}\mathrm{if},(x_i,C_i)\\ =(h_i,H_{gi})\end{array}}{{\mathbf{K}}_{\mathbf{Nr}}\begin{array}{c}\mathrm{if},(x_i,C_i)\\ =(k_{fi},k_{gi})\end{array}}\end{array}\right.\hfill \end{array}$$

(11)

Here, $\Delta \mathbf{f}$ and $\Delta \mathbf{P}$ are vectors of length $N_r$, with their ith entry corresponding to frequency deviation and power imbalance in the ith AC area, respectively. The matrices ${\mathbf{H}}_{\mathbf{Nr}}$ and ${\mathbf{K}}_{\mathbf{Nr}}$ have similar structures, and are expanded in Equation (11). Parameter ${\overline{k}}_{vi}$ in (11) is given by ${\overline{k}}_{vi}=k_{vi}/{\sum }_{j=1}^{N_r}{k}_{vj}$.

The next step is to review the design process of droop control coefficients to achieve the objective of quantifiability.

3.3. Droop Coefficient Design for Achieving Quantifiability

We review two key theorems presented and proven in [23,24]. The first one provides analytical constraints on the droop coefficients to guarantee stability of the $N_r$th-order system, while the second one specifies further constraints to achieve quantifiability.

Theorem 1.

The $N_r^{th}$-order system (10) is small-signal-stable, if the inertial and frequency droop coefficients satisfy $h_i>-H_{gi}$ and $k_{fi}>-k_{gi}$.

Theorem 2.

Given that disturbance occurred in Area#1 at $t=t_0$, if matrices ${\mathbf{K}}_{\mathbf{Nr}}$ and ${\mathbf{H}}_{\mathbf{Nr}}$ satisfy the stability constraints in Theorem 1 and the performance constraints

$$\begin{array}{c}\hfill {\mathbf{K}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}{r}_1& \cdots & r_{N_r}\end{array}\right]}^T=(\sum _k^{N_r}{r}_k{k}_{gk}){\widehat{\mathbf{e}}}_{\mathbf{1}}\end{array}$$

(12a)

$$\begin{array}{c}\hfill {\mathbf{H}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}{r}_1& \cdots & r_{N_r}\end{array}\right]}^T=(\sum _k^{N_r}{r}_k{H}_{gk}){\widehat{\mathbf{e}}}_{\mathbf{1}}\end{array}$$

(12b)

then the frequency deviation will satisfy (1) in the Nth-order model. Here, ${\widehat{\mathbf{e}}}_{\mathbf{1}}$ is the standard notation for the unit vector along the first dimension.

We are not repeating the proof of Theorem 1 presented in [24]. However, we will review the proof of Theorem 2, since it helps determine the reason behind the failure of the controller proposed in [21,22] in accurately tracking the ratios.

Proof of Theorem 2. 

Consider the following first-order system obtained by adding the equations in (10) when the desired ratios are maintained,

$$\begin{array}{c}\hfill 2f_0(\sum _k^{N_r}{r}_k{H}_{gk})\Delta {\dot{f}}^{*}=-(\sum _k^{N_r}{r}_k{k}_{gk})\Delta f^{*}-\Delta P_1\end{array}$$

(13)

Clearly, this first-order system is stable, since the summations are positive on both sides of the equality. If (12a) and (12b) are satisfied, then, by multiplying each side of (13) by ${\widehat{\mathbf{e}}}_{\mathbf{1}}$, we obtain,

$$\begin{array}{c}\hfill 2f_0{\mathbf{H}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}{r}_1& \cdots & r_{Nr}\end{array}\right]}^T\Delta {\dot{f}}^{*}=\\ \hfill -{\mathbf{K}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}{r}_1& \cdots & r_{Nr}\end{array}\right]}^T\Delta f^{*}-\Delta P_1{\widehat{\mathbf{e}}}_{\mathbf{1}}\end{array}$$

(14)

Following the disturbance in Area#1, according to the Nth-order model (8), the frequency dynamics can be described by

$$\begin{array}{c}\hfill 2f_0{\mathbf{H}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}\Delta {\dot{f}}_1& \cdots & \Delta {\dot{f}}_{Nr}\end{array}\right]}^T=\\ \hfill -{\mathbf{K}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}\Delta f_1& \cdots & \Delta f_{Nr}\end{array}\right]}^T-\Delta P_1{\widehat{\mathbf{e}}}_{\mathbf{1}}\end{array}$$

(15)

where $\Delta P_1$ denotes the power imbalance in Area$\#1$ due to the disturbance. Since all losses are neglected in the Nth-order model, this leads to $\Delta P_i=0,i\ne 1$. After subtracting (14) from (15), we obtain

$$\begin{array}{c}\hfill 2f_0{\mathbf{H}}_{\mathbf{Nr}}\Delta \dot{\mathbf{F}}=-{\mathbf{K}}_{\mathbf{Nr}}\Delta \mathbf{F}\end{array}$$

(16)

where $\Delta f_i-r_i\Delta f^{*}=\Delta F_i$, and $\Delta \mathbf{F}$ is a vector with $\Delta F_i$ as its ith entry. In pre-disturbance a steady state, we have $\Delta F_i\left(t\right)=0\forall i,t\le t_0$. Following the disturbance at $t_0$, if the stability constraint Theorem 1 is satisfied, then $\Delta F_i\left(t\right)$ will remain at zero $\forall i,t>t_0$. By definition, this implies that $\Delta f_i\left(t\right)=r_i\Delta f^{*}\left(t\right)\forall i,t>t_0$. □

It can be shown that the generalized solution to (12a) and (12b) is of the form

$$\begin{array}{c}\hfill x_1=(\sum _{j=2}^{N_r}{r}_j{C}_j)/r_1+p({\overline{k}}_{v1}/r_1)\\ \hfill x_i=-C_i+p({\overline{k}}_{vi}/r_i)\forall i=2\mathrm{to}{N}_r\end{array}$$

(17)

with $p>0$ and $p\in \mathbb{R}$ ensuring stability. Here, $x_i=k_{fi}$ and $C_i=k_{gi}$ for (12a), whereas $x_i=h_i$ and $C_i=H_{gi}$ for (12b). Please see Appendix A for the derivation. Although p can be different for inertial and primary frequency droop coefficient designs, the same value of p is used in this paper for both designs.

3.4. Tracking Performance of Existing Droop Control [21,22]

Figure 3 shows the frequency dynamics simulated in the full-order nonlinear model in response to a 10% step reduction in the power output of $G7$ belonging to AC Area #4 (see Figure 1). The design of the frequency droop coefficients aims to achieve a desired ratio of 1:1.5:NP:2 among the frequency deviations of the four AC areas during the entire period following the disturbance. Here, NP stands for non-participating. From Figure 3b, it is clear that the true ratios did not accurately track the desired set points. Thus motivated, we next propose a model-reference-estimation-based corrective control strategy, which improves the tracking of the prescribed ratios (see (1)) and is also capable of dealing with uncertainty in the system parameters.

4. Model-Reference-Estimation-Based Corrective Controller

The approach proposed in this section attempts to address the drawbacks of the droop controller presented in [21,22,23], whose coefficients were designed based on the lossless assumption of the Nth-order model. As mentioned in the proof of Theorem-II, the AC- and DC-side losses had to be neglected to derive the droop coefficients in (17). This shortcoming of the design process negatively affects the tracking performance (see Figure 3).

In our new approach, we build on the same design methodology—albeit in such a way that the error stemming from system losses is compensated. As a first step in that direction, we represent the change in power loss in the ith participating AC area as an unknown variable $\Delta {\tilde{P}}_{aci}$. We also represent the change in the total DC loss shared by that AC area as $\Delta {\tilde{P}}_{dci}$. This is followed by the inclusion of $\Delta {\tilde{P}}_{aci}$ and $\Delta {\tilde{P}}_{dci}$ in Theorem 2, which leads to Corollary 1.

4.1. Inclusion of Changes in AC and DC Losses in Theorem 2

With certain assumptions, it is possible to show that Theorem 2 is also valid in the presence of system losses. We present this as a corollary of Theorem 2.

Corollary 1.

If the change in loss in each participating AC area and the corresponding change in DC system loss contributed by each area are known (or can be accurately estimated), then Theorem 2 holds when (a) these changes are replenished through the disturbed AC area (Area$\#1$) and (b) the effects of the changes are nullified in other areas.

Proof of Corollary 1. 

With the realistic effect of losses considered, the frequency dynamics of the Nth-order model following a disturbance in Area$\#1$ is given by

$$\begin{array}{c}2f_0{\mathbf{H}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}\Delta {\dot{f}}_1& \cdots & \Delta {\dot{f}}_{Nr}\end{array}\right]}^T=\hfill \\ -{\mathbf{K}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}\Delta f_1& \cdots & \Delta f_{Nr}\end{array}\right]}^T-{\left[\begin{array}{cccc}\Delta P_1& \Delta {\tilde{P}}_2& \cdots & \Delta {\tilde{P}}_{Nr}\end{array}\right]}^T\hfill \end{array}$$

(18)

Here, $\Delta {\tilde{P}}_i=\Delta {\tilde{P}}_{aci}+\Delta {\tilde{P}}_{dci}$ represents the change in AC-side loss in ith AC area due to the change in the power flow of the network, which occurs because of providing frequency support, together with the share of DC-side loss supported by the ith AC area. Unlike the lossless case, $\Delta P_1$ subsumes the magnitude of disturbance in the AC system along with the change in losses.

Let $\Delta {\widehat{P}}_i$ be the estimated change in the total loss from the ith area. When the two conditions mentioned in the statement of Corollary 1 are satisfied, we can rewrite (18) as

$$\begin{array}{c}\hfill -2f_0{\mathbf{H}}_{\mathbf{Nr}}\Delta \dot{\mathbf{f}}={\mathbf{K}}_{\mathbf{Nr}}\Delta \mathbf{f}+{\left[\begin{array}{cccc}{\displaystyle \sum _{j=2}^{N_r}}\Delta {\widehat{P}}_j& -\Delta {\widehat{P}}_2& \cdots & -\Delta {\widehat{P}}_{Nr}\end{array}\right]}^T\\ \hfill +{\left[\begin{array}{cccc}\Delta P_1& \Delta {\tilde{P}}_2& \cdots & \Delta {\tilde{P}}_{Nr}\end{array}\right]}^T\end{array}$$

(19)

Since $\Delta {\tilde{P}}_i=\Delta {\widehat{P}}_i,i=2,\cdots N_r$, (19) can be written as

$$\begin{array}{c}\hfill 2f_0{\mathbf{H}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}\Delta {\dot{f}}_1& \cdots & \Delta {\dot{f}}_{Nr}\end{array}\right]}^T=\\ \hfill -{\mathbf{K}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}\Delta f_1& \cdots & \Delta f_{Nr}\end{array}\right]}^T-(\Delta P_1+\sum _{j=2}^{N_r}\Delta {\widehat{P}}_j){\widehat{\mathbf{e}}}_{\mathbf{1}}\end{array}$$

(20)

Now, let us consider the first-order system with desired frequency dynamics, maintaining the ratios

$$\begin{array}{c}\hfill 2f_0{\mathbf{H}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}{r}_1& \cdots & r_{Nr}\end{array}\right]}^T\Delta {\dot{f}}^{*}=\\ \hfill -{\mathbf{K}}_{\mathbf{Nr}}{\left[\begin{array}{ccc}{r}_1& \cdots & r_{Nr}\end{array}\right]}^T\Delta f^{*}-(\Delta P_1+\sum _{j=2}^{N_r}\Delta {\widehat{P}}_j){\widehat{\mathbf{e}}}_{\mathbf{1}}.\end{array}$$

(21)

Subtracting (21) from (20), we clearly see that Theorem 2 holds. □

As described above, Corollary 1 requires the accurate estimation of $\Delta {\tilde{P}}_i$, denoted by

$$\begin{array}{c}\hfill \Delta {\widehat{P}}_i=\Delta {\widehat{P}}_{aci}+\Delta {\widehat{P}}_{dci}\end{array}$$

(22)

Next, we propose a model-reference estimation approach [30] to calculate $\Delta {\widehat{P}}_i$.

4.2. Estimation of Power Loss

Model-reference estimation of change in AC power loss: As stated earlier, the frequency dynamics in the ith AC area is represented by the following equation:

$$\begin{array}{c}\hfill 2H_{gi}{f}_0{\dot{f}}_i=k_{gi}\Delta f_i+\Delta P_{Ci}+\Delta {\tilde{P}}_{aci}\end{array}$$

(23)

where $\Delta P_{Ci}$ denotes the change in total converter power from both the positive and negative poles. We estimate the term $\Delta P_{Ci}+\Delta {\tilde{P}}_{aci}$ from this model by carrying out the following steps:

• Measuring the actual frequency deviation $\Delta f_i$ from the ith area;
• Using a PI controller to produce the estimate $(\Delta {\widehat{P}}_{Ci}+\Delta {\widehat{P}}_{aci})$ as the output while driving the error in frequency deviation $(\Delta {\widehat{f}}_i-\Delta f_i)$ to zero;
• Designing gains of the PI controller via the root locus method described in Section 6.1;
• Using (23) to output the estimated frequency deviation $\Delta {\widehat{f}}_i$ while using $(\Delta {\widehat{P}}_{Ci}+\Delta {\widehat{P}}_{aci})$ as the input.

This is illustrated in Figure 4.

As shown in Figure 4, from the actual plant, which, in this case, is the full-order model, we can also measure the change in converter power $\Delta P_{Ci}$. We subtract $\Delta P_{Ci}$ from the PI controller output to obtain an estimate $\Delta {\widehat{P}}_{aci}$ for the power loss in AC Area #i.

Estimating the change in DC power loss: The change in DC power loss is given by $\sum \Delta P_{Ci}$. Due to the DC voltage droop, this is distributed among the converters in the ratio of the DC voltage-droop coefficients $k_{vi}$. This means converters in the ith AC area, as an aggregate, will share the loss, amounting to ${\overline{k}}_{vi}\sum \Delta P_{Ci}$, after the DC voltage transients have settled. Therefore, the estimate for DC loss shared by the ith AC area, $\Delta {\tilde{P}}_{dci}$, is ${\overline{k}}_{vi}\sum \Delta P_{Ci}{|}_{mes}$. Finally, the total loss estimate of the ith AC area $\Delta {\widehat{P}}_i$ is calculated following (22).

To satisfy the two conditions postulated in Corollary I, the estimated loss $\Delta {\widehat{P}}_i$ from each Area$\#i$ is communicated to the disturbed Area$\#1$, while converter stations in individual areas apply the corresponding loss estimates to nullify their effect. Next, we describe the communication architecture and the corrective power reference control to achieve these objectives.

4.3. Proposed Corrective Power Reference Control

Figure 5 shows a flowchart of the sequence of actions taking place under the proposed corrective power reference control following a disturbance in Area$\#i^{*}$. Figure 6a shows the communication architecture enabling the proposed corrective control. Finally, Figure 6b shows a block diagram for producing the reference signal $\Delta P_{fgi}$ for frequency support from the ith AC area.

We summarize the whole process of the proposed control below:

• Following a loss of generation in AC Area $\#i^{*}$, it sends distress signals to other areas through fiber-optic links in the subsea cables of the MTDC grid.
• Based on the ratios prescribed by the operators, the participating areas design droop coefficients from (17) and activate the droop controls.
• All participating areas except Area$\#i^{*}$ start estimating $\Delta {\widehat{P}}_j$ and continue to communicate the estimates to the disturbed area, upon which Area $\#i^{*}$ calculates ${\displaystyle \sum _{j\ne i^{*}}}\Delta {\widehat{P}}_j$.
• As shown in Figure 6b, the estimates $\Delta {\widehat{P}}_j$ and ${\displaystyle \sum _{j\ne i^{*}}}\Delta {\widehat{P}}_j$ are used to modify the converter power references $\Delta P_{fgi}$ to (a) nullify the impacts of AC and DC losses in all AC areas except Area$\#i^{*}$ and (b) reproduce these losses in Area#$i^{*}$.

4.4. Fast Corrective Control

Despite compensating for losses, there is still a lingering need for the PI controller in Figure 2 to act faster—especially during the inertial support phase. However, increasing the PI gain beyond a particular value will render the system unstable [29]. Instead of completely relying on the PI controller, the loss that needs to be compensated for is divided into fast and slow components using a low-pass filter (here, $T=1$ s), as shown in Figure 7. The fast component is directly fed, after the PI compensator as a modification, to the current reference, and the slow component is fed into the frequency support corrective controller to modify the power reference. The reason behind doing this is that the slowly varying reference can be tracked by the outer loop, whereas the inner current control loop with a much higher closed-loop bandwidth can track the faster component more effectively.

Remark on stability: If the loss estimation process is accurate enough, then we are left with the $N_r$th-order model. In that scenario, stability and performance can be guaranteed by designing droop coefficients based on Equation (17). However, the PI controllers in the model-reference estimator cannot perform instantaneous loss estimation. Therefore, this leads to a closed-loop system of order $3N_r$. Since it is difficult to provide analytical stability guarantees of such a system, we performed a numerical root-locus-based stability analysis of the system when the PI controller gains are changed, while keeping $h_{fi}s$ and $k_{fi}$s at their designed values from (17).

Remarks on complexity of practical implementation: The proposed corrective controller requires communication of the distress signal, a change in converter powers, and an estimated change in losses among the converter stations, which can be easily performed through existing fiber-optic links in the subsea cables of the MTDC grid. Therefore, no additional infrastructure is needed for this. Our simulation results in Section 6 assume a 100 ms latency in this communication and show that such a delay does not impact the accuracy of ratio tracking. The model-reference estimation requires a PI controller and needs to solve a single linear state Equation (Equation (23)) in each area, which hardly adds any computational burden to the existing converter station controls. The model-reference loss estimation in individual areas depends on local measurements of frequency and converter real power outputs that are already used in existing controllers. Finally, the fast corrective control also needs a minimal modification, as shown in Figure 7.

5. Robustness Analysis with Modeling Error

Until now, we assumed that the actual values of the aggregated inertia constants $H_{gi}$s and aggregated inverse governor droop coefficients $k_{gi}$s are known, i.e., the plant model in the model-reference estimator is accurate. In this section, we relax that assumption and analyze the robustness of the proposed corrective controller in the presence of modeling errors. To this end, we perform our analysis on the $N_r$th-order system (10), where modeling errors due to unknown $H_{gi}$s and $k_{gi}$s are taken into account. For the ith area, let ${\overline{H}}_{gi}$ and ${\overline{k}}_{gi}$ be the nominal values of the aggregated inertia constant and inverse governor droop, respectively. We assume that the droop control design for ratio-based control following (17) is performed using these nominal values. Without loss of generality, we assume that the disturbed area is Area$\#1$. First, we derive the conditions that ensure that the proposed corrective control can perform accurate ratio tracking in (19) in the presence of modeling errors.

5.1. Conditions for Perfect Ratio Tracking

Assuming the reference model uses the nominal values and the measurements (with subscript ‘mes’) reflect the actual values, we present a lemma describing the condition for accurate ratio tracking in the $N_r$th-order model.

Lemma 1.

Under the assumption of instantaneous reference tracking by the PI controllers, the proposed corrective power reference control performs accurate ratio tracking in the $N_r$th-order model in the presence of unknown $H_{gi}$s and $k_{gi}$s if ${\left({\mathbf{H}}_{\mathbf{Nr}}-{\tilde{\mathbf{H}}}_{\mathbf{Nr}}\right)}^{-1}\left({\tilde{\mathbf{K}}}_{\mathbf{Nr}}-{\mathbf{K}}_{\mathbf{Nr}}\right)$ is Hurwitz and $\left({\tilde{\mathbf{K}}}_{\mathbf{Nr}}-{\mathbf{K}}_{\mathbf{Nr}}\right)$ is full-rank, where

$${\tilde{\mathbf{X}}}_{\mathbf{Nr}}=\left[\begin{array}{cc}0& -\left[\begin{array}{ccc}\left(x_{g2}-{\overline{x}}_{g2}\right)& \cdots & \left(x_{gNr}-{\overline{x}}_{gNr}\right)\end{array}\right]\\ \mathbf{0}& diag\left(\left[\begin{array}{ccc}\left(x_{g2}-{\overline{x}}_{g2}\right)& \cdots & \left(x_{gNr}-{\overline{x}}_{gNr}\right)\end{array}\right]\right)\end{array}\right]$$

(24)

with $x_{gi}=H_{gi}$, ${\overline{x}}_{gi}={\overline{H}}_{gi}$ for ${\tilde{\mathbf{H}}}_{\mathbf{Nr}}$ and $x_{gi}=k_{gi}$, ${\overline{x}}_{gi}={\overline{k}}_{gi}$ for ${\tilde{\mathbf{K}}}_{\mathbf{Nr}}$, respectively, and $\mathbf{0}\in {\mathbb{R}}^{Nr-1\times 1}$.

Proof of Lemma 1. 

With the assumptions leading to Lemma 1, and considering instantaneous reference tracking by the PI controllers, the reference model for the ith AC area can be represented by

$$2f_0{\overline{H}}_{gi}\Delta {\dot{f}}_i+{\overline{k}}_{gi}\Delta f_i=-\Delta u_i$$

(25)

where $\Delta u_i$ is the output of the PI controller. On the other hand, the actual reduced-order model satisfies

$$2f_0{H}_{gi}\Delta {\dot{f}}_i+k_{gi}\Delta f_i=-\left(\Delta P_{ci}+\Delta {\tilde{P}}_{aci}\right)$$

(26)

This leads to

$$\begin{array}{c}\Delta {\widehat{P}}_i-\Delta {\tilde{P}}_i=2f_0\left(H_{gi}-{\overline{H}}_{gi}\right)\Delta {\dot{f}}_i+\left(k_{gi}-{\overline{k}}_{gi}\right)\Delta f_i,\forall i\ne 1\hfill \\ \\ \Rightarrow \Delta P_1+{\displaystyle \sum _{j=2}^{N_r}}\Delta {\widehat{P}}_j=\Delta P_1+{\displaystyle \sum _{i=2}^{N_r}}\left(\Delta {\tilde{P}}_i+2f_0\left(H_{gi}-{\overline{H}}_{gi}\right)\Delta {\dot{f}}_i\right)\hfill \\ +{\displaystyle \sum _{i=2}^{N_r}}\left(k_{gi}-{\overline{k}}_{gi}\right)\Delta f_i.\hfill \end{array}$$

(27)

Using the above expressions, we can see that $\left(\Delta {\widehat{P}}_i-\Delta {\tilde{P}}_i\right)\ne 0$ in (19), i.e., the estimated loss is not equal to the actual loss in the ith area due to modeling errors. Thus, we modify (21) to obtain the desired frequency dynamics, where accurate ratio tracking is achieved, as follows:

$$\begin{array}{c}2f_0{\mathbf{H}}_{\mathbf{Nr}}{\left[r_1\cdots r_{Nr}\right]}^T\Delta {\dot{f}}^{*}+{\mathbf{K}}_{\mathbf{Nr}}{\left[r_1\cdots r_{Nr}\right]}^T\Delta f^{*}=\hfill \\ \begin{array}{c}\left[\begin{array}{cc}-\left(\Delta P_1+{\displaystyle \sum _{i=2}^{N_r}}\Delta {\widehat{P}}_i\right)& \left(\Delta {\widehat{P}}_2-\Delta {\tilde{P}}_2\right)\end{array}\right.\hfill \\ {\left.\begin{array}{cc}\cdots & \left(\Delta {\widehat{P}}_{Nr}-\Delta {\tilde{P}}_{Nr}\right)\end{array}\right]}^T\hfill \end{array}\hfill \end{array}$$

(28)

where $\Delta {\widehat{P}}_i-\Delta {\tilde{P}}_i=2f_0\left(H_{gi}-{\overline{H}}_{gi}\right)r_i\Delta f^{*}+\left(k_{gi}-{\overline{k}}_{gi}\right)r_i\Delta f^{*}$.

Subtracting (28) from (19), we obtain

$$\begin{array}{c}2f_0{\mathbf{H}}_{\mathbf{Nr}}\Delta \dot{\mathbf{F}}+{\mathbf{K}}_{\mathbf{Nr}}\Delta \mathbf{F}=2f_0{\tilde{\mathbf{H}}}_{\mathbf{Nr}}\Delta \dot{\mathbf{F}}+{\tilde{\mathbf{K}}}_{\mathbf{Nr}}\Delta \mathbf{F}\hfill \\ \Rightarrow 2f_0\left({\mathbf{H}}_{\mathbf{Nr}}-{\tilde{\mathbf{H}}}_{\mathbf{Nr}}\right)\Delta \dot{\mathbf{F}}=\left({\tilde{\mathbf{K}}}_{\mathbf{Nr}}-{\mathbf{K}}_{\mathbf{Nr}}\right)\Delta \mathbf{F}\hfill \end{array}$$

(29)

As in the proof of Theorem-II, $\Delta \mathbf{F}$ will remain zero following the disturbance if ${\left({\mathbf{H}}_{\mathbf{Nr}}-{\tilde{\mathbf{H}}}_{\mathbf{Nr}}\right)}^{-1}\left({\tilde{\mathbf{K}}}_{\mathbf{Nr}}-{\mathbf{K}}_{\mathbf{Nr}}\right)$ is Hurwitz. Also, in a steady state $\left({\tilde{\mathbf{K}}}_{\mathbf{Nr}}-{\mathbf{K}}_{\mathbf{Nr}}\right)\Delta \mathbf{F}=\mathbf{0}$, with $\Delta \mathbf{F}=\mathbf{0}$, if $\left({\tilde{\mathbf{K}}}_{\mathbf{Nr}}-{\mathbf{K}}_{\mathbf{Nr}}\right)$ has a trivial nullspace. Hence, Lemma 1 is proven. □

As observed above, for accurate ratio tracking, we need the PI compensators to perform instantaneous reference tracking. We know that this is not possible. Nevertheless, we demonstrate that in the presence of uncertainties in the plant, the PI controller can track $\Delta {\widehat{f}}_i$ asymptotically and demonstrate ratio tracking in a steady state.

5.2. Robustness Analysis of PI Controller in ith AC Area: Sliding-Mode Control Analogy

In this section, we show that a sliding-mode controller that is robust to modeling uncertainty takes the form of the PI compensator in the proposed corrective control. As shown in Figure 8, we pose the proposed corrective controller as a tracking control that follows a time-varying reference $\Delta {\widehat{f}}_i$. We rewrite (19) in the following single-input–single-output state-space form for the ith AC area, where the output of the system $\Delta f_i$ is used as a feedback signal for tracking control:

$$\begin{array}{c}\Delta \dot{\mathbf{f}}={\mathbf{A}}_{\mathbf{0}}\Delta \mathbf{f}+{\mathbf{A}}_{\mathbf{ffl}}\Delta \mathbf{f}+{\mathbf{B}}_{\mathbf{i}}\left(\Delta v_i+\Delta w_i\right)\hfill \\ y_i={\mathbf{C}}_{\mathbf{i}}\Delta \mathbf{f},i\ne 1.\hfill \end{array}$$

(30)

Here, ${\mathbf{A}}_{\mathbf{0}}=-{\displaystyle \frac{1}{2f_0}}\left({\overline{\mathbf{H}}}_{\mathbf{Nr}}^{-\mathbf{1}}{\overline{\mathbf{K}}}_{\mathbf{Nr}}\right),{\mathbf{A}}_{’’}=-{\displaystyle \frac{1}{2f_0}}\left({\mathbf{H}}_{\mathbf{Nr}}^{-\mathbf{1}}{\mathbf{K}}_{\mathbf{Nr}}-{\overline{\mathbf{H}}}_{\mathbf{Nr}}^{-\mathbf{1}}{\overline{\mathbf{K}}}_{\mathbf{Nr}}\right)$, ${\mathbf{B}}_{\mathbf{i}}\in {\mathbb{R}}^{Nr\times 1}$ is the ith column of ${\displaystyle \frac{1}{2f_0}}\left({\mathbf{H}}_{\mathbf{Nr}}^{-\mathbf{1}}\right),{\mathbf{C}}_{\mathbf{i}}={\widehat{\mathbf{e}}}_{\mathbf{i}}^{\mathbf{T}},\Delta v_i=\Delta {\widehat{P}}_i$, and $\Delta w_i=-\Delta {\tilde{P}}_i$. Matrices ${\overline{\mathbf{H}}}_{\mathbf{Nr}}$ and ${\overline{\mathbf{K}}}_{\mathbf{Nr}}$ are calculated using ${\overline{H}}_{gi}$ and ${\overline{k}}_{gi}$, respectively, in ${\mathbf{X}}_{\mathbf{Nr}}$ from (11). We assume that $\Delta w_i$ is piecewise continuous in t and sufficiently smooth in $(\Delta \mathbf{f},\Delta v_i)$ for $\left(t,\Delta \mathbf{f},\Delta v_i\right)\in \left[0,\left.\infty \right)\right.\times {\mathbb{R}}^{Nr}\times \mathbb{R}$. Clearly, ${\mathbf{A}}_{\mathbf{0}}$ and ${\mathbf{C}}_{\mathbf{i}}$ are known and ${\mathbf{A}}_{\mathbf{ffl}},{\mathbf{B}}_{\mathbf{i}}$, and $\Delta w_i$ could be uncertain.

For all possible uncertainties in ${\mathbf{B}}_{\mathbf{i}}$, we assume that the system

$$\begin{array}{c}\Delta \dot{\mathbf{f}}={\mathbf{A}}_{\mathbf{0}}\Delta \mathbf{f}+{\mathbf{B}}_{\mathbf{i}}\Delta v_i\hfill \\ y_i={\mathbf{C}}_{\mathbf{i}}\Delta \mathbf{f},i\ne 1\hfill \end{array}$$

(31)

has a relative degree of $\rho$ in ${\mathbb{R}}^{Nr}$ and ${\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}>0$.

Lemma 2.

The system represented in (31) has a relative degree of $\rho =1$ in ${\mathbb{R}}^{Nr}$ and ${\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}>0$ if $p>\underset{i\ne 1}{max}\left({\displaystyle \frac{r_i}{{\overline{k}}_{vi}}}{\overline{H}}_{gi}\right)$.

Proof of Lemma 2. 

System (26) is linear time-invariant, which can be expressed in a transfer function form $H\left(s\right)={\mathbf{C}}_{\mathbf{i}}{\left(\mathbf{sI}-{\mathbf{A}}_{\mathbf{0}}\right)}^{-\mathbf{1}}{\mathbf{B}}_{\mathbf{i}}$. In [24], we show that ${\mathbf{H}}_{\mathbf{Nr}}^{-\mathbf{1}}$ can be expressed as a rank-1 perturbed diagonal matrix, whose jth element of the ith column can be expressed as

$${\mathbf{H}}_{\mathbf{Nr}}^{-\mathbf{1}}(j,i)=2f_0{\mathbf{B}}_{\mathbf{i}}\left(j\right)=\left\{\begin{array}{c}{\displaystyle \frac{{\overline{k}}_{vj}}{w(h_j+H_{gj})}}\left({\displaystyle \frac{h_i}{h_i+H_{gi}}}\right),j\ne i\\ {\displaystyle \frac{1}{(h_i+H_{gi})}}\left(1+{\displaystyle \frac{{\overline{k}}_{vi}{h}_i}{w(h_i+H_{gi})}}\right),j=i\end{array}\right.$$

(32)

where $w={\displaystyle \sum _{i=1}^{N_r}}{\displaystyle \frac{{\overline{k}}_{vi}{H}_{gi}}{h_i+H_{gi}}}.$

Clearly,

$${\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}={\displaystyle \frac{1}{2f_0}}\left({\displaystyle \frac{1}{h_i+H_{gi}}}\right)\left(1+{\displaystyle \frac{{\overline{k}}_{vi}{h}_i}{w(h_i+H_{gi})}}\right)$$

(33)

The condition for relative degree $\rho =1$ of $H\left(s\right)$ is ${\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\ne 0$ [31], which becomes redundant when ${\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}>0$. One of the sufficient conditions for this is $(h_i+H_{gi})>0,\forall i$ and $\left(1+{\displaystyle \frac{{\overline{k}}_{vi}{h}_i}{w(h_i+H_{gi})}}\right)>0,\forall i$. At this point, let us consider the first inequality, which can be met by different constraints on parameter p. These are as follows: (i) $p>0$ for $i=1$ and (ii) $p>\underset{i\ne 1}{max}{\displaystyle \frac{r_i}{{\overline{k}}_{vi}}}\left|{\overline{H}}_{gi}-H_{gi}\right|$ or $p>\underset{i\ne 1}{max}{\displaystyle \frac{r_i}{{\overline{k}}_{vi}}}{\overline{H}}_{gi}$. However, due to modeling uncertainty, $H_{gi}$ is unknown. Therefore, the only realistic option is to choose $p>\underset{i\ne 1}{max}{\displaystyle \frac{r_i}{{\overline{k}}_{vi}}}{\overline{H}}_{gi}$, which also makes $p>0$ for $i=1$ a redundant constraint. This, in turn, implies that $h_i>0,\forall i$, which leads to $(h_i+H_{gi})>0,\forall i$ and $\left(1+{\displaystyle \frac{{\overline{k}}_{vi}{h}_i}{w(h_i+H_{gi})}}\right)>0,\forall i$. Therefore, ${\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}>0$. Hence, Lemma 2 is proven. □

Under the assumption that ${\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}>0$, we now consider the problem of designing a state feedback control law for the plant (30) that asymptotically tracks the reference signal $\Delta {\widehat{f}}_i$, while $\Delta {\widehat{f}}_i{|}_{mes}$ or, equivalently, $\Delta f_i$ (based on our assumption) is fed back (see Figure 8). We assume that $\Delta {\widehat{f}}_i$ and $\Delta {\dot{\widehat{f}}}_i$ are bounded for all $t\ge 0$. Also, since $\Delta {\widehat{f}}_i$ is a state variable, $\Delta {\dot{\widehat{f}}}_i$ is a piecewise continuous function in t.

We apply the principle of input–output linearization [31] to express (30) in the normal form using a change in variables to ${\left[\begin{array}{cc}ȷ& \xi \end{array}\right]}^T$, whose input–output map between $\Delta v_i$ and $\xi =y_i$ is a chain of $\rho$ integrators (i.e., a single integrator in this case) and the remaining state variables $ȷ\in {\mathbb{R}}^{N_r-\rho }$ are unobservable from the output $y_i$:

$$\begin{array}{c}\dot{\mathbf{ȷ}}={\mathbf{A}}_{\mathbf{c}}\mathbf{ȷ}+{\mathbf{B}}_{\mathbf{c}}{y}_i\hfill \\ \dot{\xi }={\mathbf{C}}_{\mathbf{i}}\left({\mathbf{A}}_{\mathbf{0}}+{\mathbf{A}}_{’’}\right)\Delta \mathbf{f}+{\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\left(\Delta v_i+\Delta w_i\right)\hfill \\ y_i=\xi =\Delta f_i\hfill \end{array}$$

(34)

We assume that (30) is a minimum-phase system, which, in turn, implies that $A_c$ is Hurwitz. Thus, we focus our attention on the state equation involving $\xi$. To ensure asymptotic tracking of $\Delta {\widehat{f}}_i$ and zero steady-state (if there exists one) error, we introduce an integrator in the following form: ${\dot{e}}_0=\xi -\Delta {\widehat{f}}_i=e_1$. The augmented input–output equation involving a chain of $\rho +1$ integrators can now be written as

$$\begin{array}{c}{\dot{e}}_0=e_1\hfill \\ {\dot{e}}_1={\mathbf{C}}_{\mathbf{i}}\left({\mathbf{A}}_{\mathbf{0}}+{\mathbf{A}}_{’’}\right)\Delta \mathbf{f}+{\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\left(\Delta v_i+\Delta w_i\right)-\Delta {\dot{\widehat{f}}}_i\hfill \end{array}$$

(35)

Sliding-mode control design: Sliding-mode control is a robust control strategy for nonlinear systems, and can be the state feedback controller of choice for plant (30) with uncertainty. We will demonstrate that the sliding-mode controller takes the form of a PI controller in this case.

We design a sliding-mode controller with a sliding manifold $s=k_0{e}_0+e_1=0$, where $k_0>0$. This leads to

$$\begin{array}{c}\dot{s}=k_0\left(\Delta f_i-\Delta {\widehat{f}}_i\right)+{\mathbf{C}}_{\mathbf{i}}\left({\mathbf{A}}_{\mathbf{0}}+{\mathbf{A}}_{’’}\right)\Delta \mathbf{f}\hfill \\ +{\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\left(\Delta v_i+\Delta w_i\right)-\Delta {\dot{\widehat{f}}}_i\hfill \end{array}$$

(36)

From Figure 8, $\Delta v_i=\Delta u_i-\Delta P_{ci}+\Delta {\tilde{P}}_{dci}$, which leads to

$$\begin{array}{c}\dot{s}={\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\Delta u_i+\mathrm{\Omega }\hfill \end{array}$$

(37)

where $\mathrm{\Omega }=k_0\left(\Delta f_i-\Delta {\widehat{f}}_i\right)+{\mathbf{C}}_{\mathbf{i}}\left({\mathbf{A}}_{\mathbf{0}}+{\mathbf{A}}_{’’}\right)\Delta \mathbf{f}-{\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\left(\Delta P_{ci}+\Delta {\tilde{P}}_{aci}\right)-\Delta {\dot{\widehat{f}}}_i$ includes uncertain terms.

We assume that $\mathrm{\Omega }$ is bounded. Suppose that $\left|{\displaystyle \frac{\mathrm{\Omega }}{{\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}}}\right|\le \mathrm{\Gamma }\left(\Delta \mathbf{f}\right)+{\kappa }_0\left|\Delta u_i\right|,\mathrm{\Gamma }\ge 0,0\le {\kappa }_0\le 1$, where $\mathrm{\Gamma }$ and ${\kappa }_0$ are known. We design a control input $\Delta u_i$ to force the trajectories towards the sliding manifold $s=0$. We use a Lyapunov function candidate $V={\displaystyle \frac{1}{2}}{s}^2$. Therefore,

$$\dot{V}=s\dot{s}={\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}s\Delta u_i+s\mathrm{\Omega }\le {\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\left\{s\Delta u_i+\left|s\right|\left(\mathrm{\Gamma }+{\kappa }_0\left|\Delta u_i\right|\right)\right\}$$

(38)

Let, $\beta \ge {\displaystyle \frac{\mathrm{\Gamma }\left(\Delta \mathbf{f}\right)}{1-{\kappa }_0}}+{\beta }_0,{\beta }_0>0$. We choose $\Delta u_i=-\beta \mathrm{sgn}\left(s\right)$, which leads to a discrete sliding-mode controller. It can be easily shown that

$$\dot{V}\le -{\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\left(1-{\kappa }_0\right){\beta }_0\left|s\right|$$

(39)

This inequality ensures that all trajectories starting from the sliding manifold will reach it in finite time, and the ones that are on it cannot leave it.

To avoid the chattering issues in such a controller, we can use the continuous approximation of the control law

$$\Delta u_i=-\beta sat\left({\displaystyle \frac{s}{\gamma }}\right)$$

(40)

where $\gamma$ is a small positive constant and $sat\left(·\right)$ is the saturation function defined by $sat\left(y\right)=\left\{\begin{array}{c}y,\left|y\right|\le 1\\ \mathrm{sgn}\left(y\right),otherwise\end{array}\right.$. It can be shown that $\dot{V}\le -{\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}\left(1-{\kappa }_0\right){\beta }_0\left|s\right|\forall \left|s\right|\ge \gamma$.

Control law (40) can be re-written as

$$\Delta u_i=sat\left({\displaystyle \frac{k_0\beta }{\gamma }}\int \left(\Delta {\widehat{f}}_i-\Delta f_i\right)dt+{\displaystyle \frac{\beta }{\gamma }}\left(\Delta {\widehat{f}}_i-\Delta f_i\right)\right)$$

(41)

It is clear from (41) that the control law leads to a PI controller, whose proportional and integral gains are ${\displaystyle \frac{\beta }{\gamma }}$ and ${\displaystyle \frac{k_0\beta }{\gamma }}$, respectively, and the output is limited between $\pm \beta$.

Remark 1.

(1) We have assumed the uncertainty in the plant to be bounded. For a large enough β and small positive γ, the sliding-mode control law (41) in the form of a PI controller asymptotically tracks $\Delta {\widehat{f}}_i$ in the presence of uncertainties.

(2) The PI controller guarantees zero error only under a steady state, i.e., when $\Delta {\widehat{f}}_i$ is a constant. Therefore, perfect ratio tracking is possible in a steady state if the conditions of Lemma 1 are satisfied.

(3) For a large enough β, the saturation limits are not hit, which implies that the output saturation is inactive in that case.

6. Simulation Results and Discussion

In this simulation study, the performance of the corrective controller is validated using the full-order nonlinear model of the test system in Figure 1. An AC disturbance with a 10% step reduction in generation from generator G7 of AC Area #4 (see Figure 1) is initiated at t = 1 s. For the sake of comparison with the tracking performance of the existing droop control [21,22] shown in Figure 3, the same prescribed ratio of 1:1.5:2 among AC areas #1, #2, and #4, with AC Area #3 as a non-participating area, is considered. A delay of 100 ms is assumed in communicating the $\Delta {\widehat{P}}_i$ of participating areas #1 and #2 to AC Area #4. Next, the PI controllers of the model-reference estimators and the droop coefficients are designed using the full-order model with accurate values of inertia constants ($H_{gi}$s) and inverse governor droop coefficients ($k_{gi}$s).

6.1. Design of PI Controller in Model-Reference Estimator

At the outset, we perform small-signal stability analysis of the full-order system to design the PI controller gains of the model-reference estimator shown in Figure 4. To that end, we vary the gains of the PI controllers in the same proportions while keeping the rest of the controller parameters constant. In particular, the values of $k_{fi}$ and $h_i$ are calculated using (17) for maintaining the prescribed ratios, and p is set to 1000. Figure 9 shows the variation in the eigenvalues, where the movement of the poles starts from the blue spectrum and ends at the red spectrum.

It can be observed that for the lower values of the PI gains, the system is unstable. The locus of the corresponding pair of eigenvalues with low PI gains of the model-reference estimator is shown in Figure 10. As the gains increase, the poles move towards the left half. We have chosen the PI gains corresponding to the pole-pair marked using triangles. It is worth noting that the eigenvalue pair under discussion has a frequency of $\approx 20$ Hz, which is much higher than the electromechanical frequency range.

6.2. Effect of $h_i$s and $k_{fi}$s on Stability of Full-Order System

We fix the PI controller gains of the model-reference estimators at their designed values obtained from the previous section. We vary the parameter p in (17) from $-1000$ to 1000 while maintaining the prescribed ratios 1:1.5:NP:2. It can be seen from the root locus plot shown in Figure 11 that a low-frequency (≈0.6 Hz) mode becomes unstable with $p=-1000$. As p increases, it moves towards the left half of s-plane.

An important thing to note is that the system is expected to be stable at $p=100$ per Theorem 1 for the Nth-order system. However, this is marginally unstable, since the actual system has PI controllers in the model-reference estimator for loss estimation. Nevertheless, the system becomes stable at $p=200$. Therefore, one should have a margin of safety during the droop coefficient design. To that end, p = 1000 was chosen.

6.3. Tracking Performance of Proposed Corrective Controller

First, the power references of all participating AC areas are modified utilizing the corrective control, as shown in Figure 6b. The frequency dynamics following the disturbance is shown in Figure 12a. As illustrated in Figure 12b, the ratios between the frequency deviations of generators G7 and G1, and G5 and G1, are exactly equal to the prescribed value after t = 5 s. Finally, there is minimal impact on the DC-side voltage, as shown in Figure 12c.

6.4. Improved Tracking Performance with Fast Corrective Controller

The correction term is separated into fast and slow components, and fed both before and after the PI controller, as shown in Figure 7. The same droop coefficients as in the previous study are used for comparison. Figure 13 shows the response of the system following the same disturbance. Both the frequency dynamics (Figure 13a) and the ratio tracking (Figure 13b) show improvement in the inertial support phase over and above the normal corrective control (Figure 12)—all this while not impacting the DC-side voltage shown in Figure 13c.

6.5. Performance with Incorrect System Parameters

The fast corrective control and normal corrective control methods are both robust to the parametric uncertainty in the model. Please refer to Section 5 for a formal analysis of the robustness of the corrective control in the presence of modeling uncertainty in the reduced-order model.

To demonstrate this, the plant model in model-reference estimation, along with the droop coefficients, are designed using nominal system parameters ${\overline{H}}_{gi}$s and ${\overline{k}}_{gi}$s, one of which is inaccurate. As before, p = 1000 and the normalized voltage-droop coefficients ${\overline{k}}_{v1}=0.5$, ${\overline{k}}_{v2}=0.25$, and ${\overline{k}}_{v4}=0.25$ are used in the droop controller design. The values of the nominal parameters are ${\overline{H}}_{g1}=228.15,{\overline{H}}_{g2}=185.25,{\overline{H}}_{g3}={\overline{H}}_{g4}=74.10,{\overline{k}}_{g1}=169.77,{\overline{k}}_{g2}={\overline{k}}_{g3}={\overline{k}}_{g4}=53.05$, wherein the units of the inertia constants and inverse governor droop coefficients are s and MW/Hz, respectively. Note that, except $k_{g1}$, the remaining nominal values are equal to the actual values in previous studies as well as this study. The actual value of $k_{g1}$ in previous studies as well as this study is $k_{g1}=212.21$, implying a 20% error in its nominal value.

Using the convention that the disturbed Area #4 is indexed as $i=1$, followed by areas 1 and 2, and noting that Area #3 is non-participating, we can write

$${\tilde{\mathbf{H}}}_{\mathbf{Nr}}={\mathbf{0}}_{\mathbf{3}\times \mathbf{3}};{\tilde{\mathbf{K}}}_{\mathbf{Nr}}=\left[\begin{array}{ccc}0& -42.44& 0\\ 0& 42.44& 0\\ 0& 0& 0\end{array}\right]$$

The eigenvalues of ${\left({\mathbf{H}}_{\mathbf{Nr}}-{\tilde{\mathbf{H}}}_{\mathbf{Nr}}\right)}^{-1}\left({\tilde{\mathbf{K}}}_{\mathbf{Nr}}-{\mathbf{K}}_{\mathbf{Nr}}\right)$ can be calculated as $-0.6605,-0.6082,$ and $-1.0000$. In addition, the matrix $\left({\tilde{\mathbf{K}}}_{\mathbf{Nr}}-{\mathbf{K}}_{\mathbf{Nr}}\right)$ is full-rank. Therefore, the conditions for Lemma 1 are satisfied. Similarly, one can verify that p = 1000 leads to $(h_1+H_{g1})=500$, $(h_2+H_{g2})=167$, $(h_3+H_{g3})=452$, $\left(1+{\displaystyle \frac{{\overline{k}}_{v1}{h}_1}{w(h_1+H_{g1})}}\right)=1.5$, $\left(1+{\displaystyle \frac{{\overline{k}}_{v2}{h}_2}{w(h_2+H_{g2})}}\right)=0.95$, and $\left(1+{\displaystyle \frac{{\overline{k}}_{v3}{h}_3}{w(h_3+H_{g3})}}\right)=1.38$. This ensures ${\mathbf{C}}_{\mathbf{i}}{\mathbf{B}}_{\mathbf{i}}>0$ and a relative degree of 1 for the system in (31), in spite of not meeting the sufficient condition of Lemma 2, which requires $p>1111$. As illustrated in Figure 14, the tracking performance of the fast corrective controller following the disturbance is nearly identical to the case without modeling errors. Also, a zero steady-state error in ratio tracking is achieved.

7. Conclusions and Future Work

The limitations of the droop controller, which was designed via a reduced-order model, were investigated, and disregarding losses was identified as a major source of its unsatisfactory performance. A new corrective control strategy was proposed to account for the losses through model-reference estimation, which were, in turn, compensated for by modifying the power references of converters in a specific manner. An improvement upon the corrective controller was also proposed that divided the correction term into fast and slow components. This enabled better ratio tracking in the inertial support phase. Sufficient conditions for perfect ratio tracking through corrective control in the presence of modeling uncertainty in the reduced-order model were established. In this context, it was also shown that a sliding-mode controller can be designed for the model-reference estimator, which is robust to bounded uncertainty. For a sufficiently large p value in the droop coefficient design, the sliding-mode controller assumes the form of a PI controller. Eigenvalue analysis of the full-order system revealed that such a sufficiently large value also introduces a safety margin in small-signal stability of the system in presence of the model-reference estimator. Nonlinear time-domain simulations were performed on a five-terminal MTDC grid connected to four asynchronous AC areas to demonstrate the effectiveness of the proposed corrective control. The results revealed that the corrective control strategy subsumes parametric uncertainty into the loss compensation mechanism and was able to perform ratio tracking as intended. It was also observed that the proposed method does not have any negative impact on the DC voltage dynamics of the MTDC grid.

This work assumes that each area participating in frequency support will procure inertial and primary frequency reserves as conventional synchronous generators are retired progressively. This assumption is valid, as it is already happening today. Such frequency reserves come from spinning reserve, flywheel, battery energy storage, and other technologies. Although not explicitly mentioned, we assume that the equivalent inertia constant $H_{gi}$ and governor droop $k_{gi}$ in the ith area includes the contribution of the frequency reserve resources in the corresponding area. In our simulations, a non-reheat turbine model is used as an equivalent. This assumes that (a) the fossil-fuel-fired reheat turbines are going to be decommissioned in a future grid and (b) the frequency reserve resources in each area will allow such an equivalent response from that area. The idea is that the areas will participate in a frequency reserve sharing market. In the future, research needs to be performed to quantify such resources from each area, given the prescribed ratio of frequency deviation. Sensitivity analysis should be performed to obtain a deeper understanding of the importance of AC- vs. DC-side losses in accurate ratio tracking. More advanced controllers like robust and adaptive controllers should be also be explored for frequency support across a range of operating conditions.