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Processes 2019, 7(8), 490; https://doi.org/10.3390/pr7080490
Article
Optimal Nonlinear Adaptive Control for Voltage Source Converters via Memetic Salp Swarm Algorithm: Design and Hardware Implementation
^{1}
NARI Technology Co. Ltd., Nanjing 211106, China
^{2}
Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming 650500, China
^{3}
College of Electric Power, South China University of Technology, Guangzhou 510640, China
^{*}
Author to whom correspondence should be addressed.
Received: 9 July 2019 / Accepted: 25 July 2019 / Published: 1 August 2019
Abstract
:Voltage source converter (VSC) has been extensively applied in renewable energy systems which can rapidly regulate the active and reactive power. This paper aims at developing a novel optimal nonlinear adaptive control (ONAC) scheme to control VSC in both rectifier mode and inverter mode. Firstly, the nonlinearities, parameter uncertainties, timevarying external disturbances, and unmodelled dynamics can be aggregated into a perturbation, which is then estimated by an extended state observer (ESO) called highgain perturbation observer (HGPO) online. Moreover, the estimated perturbation will be fully compensated through state feedback. Besides, the observer gains and controller gains are optimally tuned by a recent emerging biologybased memetic salp swarm algorithm (MSSA), the utilization of such method can ensure a desirably satisfactory control performance. The advantage of ONAC is that even though the operation conditions are constantly changing, the control performance can still be maintained to be globally consistent. In addition, it is noteworthy that in rectifier mode only the reactive power and DC voltage are required to be measured, while in inverter mode merely the reactive power and active power have to be measured. At last, in order to verify the feasibility of ONAC in practical application, a hardware experiment is implemented.
Keywords:
optimal nonlinear adaptive control; memetic salp swarm algorithm; voltage source converter; perturbation observer; hardware experiment1. Introduction
The rapidly growing energy demand has driven the fast deployment and development of renewable energy which involves advanced electronic devices to generate power. Generally speaking, such electrical power usually needs to be converted, e.g., DC (direct current)DC(direct current) boost or buck, AC (alternating current)DC (rectifier mode), DCAC (inverter mode), etc., to provide a proper power supply to satisfy various power demanders. In the past decade, thanks to the advancement of modern electronics technology, studies on voltage source converters (VSC) utilizing insulated gate bipolar transistors (IGBT) have gained considerable attentions and interests around the globe [1]. VSC has many advantages over the conventional current source converter (CSC), e.g., decoupled control of the active power and reactive power and the communication process can successfully be achieved without external voltage source supply. Several typical VSC applications can be traced to VSCbased high voltage direct current (VSCHVDC) systems [2], doubly fed induction generation (DFIG) [3], permanent magnetic synchronous generator (PMSG) [4], and photovoltaic (PV) inverter [5]. So far, VSC plays a major role in distributed generation (DG) systems [6,7,8] to ensure a reliable power supply.
In practice, it often requires VSC to operate rapidly to ensure the reactive power and DC voltage can be regulated in time when it operates at rectifier mode, while achieving the regulation of the reactive power and active power when it operates at inverter mode. Hence, the proper design of control systems plays an important role in VSC operation. Conventional proportional–integral–derivative (PID) control has acquired wide applications, since the onepoint linearization is always used when the control parameters are tuned, once the operation points change the overall control performance of PID controller may decline significantly [9]. Such issue become particularly severe for DG systems as they often operate under stochastic atmospheric conditions, such as random wind speed variation, solar irradiation, and temperature change.
To resolve this challenging issue, many advanced control strategies have been developed, such as feedback linearization control (FLC) [10] applied in VSCHVDC systems, which fully removes the nonlinearities of the system to guarantee a global control consistency under different operation conditions. Nevertheless, FLC requires a system model which can precisely describe the characteristics of the system and fully state measurements, thus it lacks robustness for the parameters of ideal model change and is difficult to be implemented in practice. To enhance the robustness, slidingmode control (SMC) [11] was applied to VSCHVDC system. However, chattering may emerge from SMC due to the use of discontinuous functions in the controller loop, together with an increased scale of controller structure complexity. Meanwhile, in reference [12], a passivitybased slidingmode control (PBSMC) scheme was employed for rotorside VSC of PMSG to efficiently undertake maximum power point tracking (MPPT) under different wind profiles. However, it requires many state measurements and an accurate system model which is hard to achieve in reality, thus making its implementation questionable. Moreover, a robust slidingmode control (RSMC) strategy was developed on the rotorside VSC of DFIG to enhance the robustness under modelling uncertainties [13], which largely increased the complexity and resulted in an overconservativeness due to the upper bound of perturbation used, which does not often occur in practice. In addition, model predictive control (MPC) has been applied on PV inverters to achieve a rapid power regulation in the presence of modelling uncertainties [14], which, however, needs to employ extra control loops, thus, the system complexity becomes significant. Besides, literature [15] adopted an effective control strategy, in which a disturbance estimatorbased predictive current controller is utilized for PV inverters to estimate the unknown PV inverter parameters and modelling uncertainties. However, it introduces a large number of estimator gains for which tuning is difficult.
Basically, the aforementioned nonlinear approaches have a relatively complex structure, thus their implementation is usually difficult. Based on extended state observer (ESO) [16,17,18], a nonlinear adaptive control (NAC) strategy for VSC operating in both rectifier mode and inverter mode is proposed in this paper. The combined influence of system nonlinearities, uncertain parameters, unmodelled dynamics, and timevarying external disturbances can be synthesized into a perturbation, which can be estimated through a highgain perturbation observer (HGPO) [19,20,21]. NAC only requires the measurement of DC voltage and reactive power for rectifier mode, and active power and reactive power for inverter mode. Thus, it provides the merit of inherently easy implementation in real systems.
However, NAC involves many parameters that need to be properly tuned, e.g., (a) observer gains which determine the perturbation estimation performance and (b) controller gains which influence the control performance. These gains are interactively coupled, thus, the optimal tuning of them is worth studying. To handle this issue, this paper aimed to design an optimal NAC (ONAC) strategy to achieve optimal parameter tuning of NAC for VSC, such that an optimal and robust control can be achieved under different operation scenarios. In particular, a newly emerging biology based metaheuristic algorithm was utilized, called memetic salp swarm algorithm (MSSA) [22], to tune the parameters of NAC more optimally and efficiently, such that the control performance of NAC can be optimized. More specifically, the salp chains adopted by MSSA are multiple independent ones, such a strategy can better balance exploration and exploitation, thus, a highquality optimum can be efficiently found. Besides this, a novel regroup operation using a virtual population is employed to globally coordinate different salp chains, such that the convergence stability can be considerably improved.
Lastly, the proposed ONAC is compiled and downloaded into the dSPACE processor as the realtime controller for VSC, in which the I/O interface of dSPACE simulator enables the realtime sampling of inputs from measurements and output control signals with IGBT converters. Several case studies were done to verify the hardware practicality of ONAC based inverter controller and rectifier controller.
The originality and novelty of this work can be summarized as follows:
 (a)
 The large parameter tuning burden of conventional NAC can be efficiently resolved via MSSA, which can efficiently and effectively seek the optimal controller and observer parameters with a high stability;
 (b)
 Various modelling uncertainties can be effectively estimated by HGPO in the realtime, which is then fully compensated by the controller online, such that great robustness can be realized;
 (c)
 A dSpace based hardware experiment was undertaken which validates the implementation feasibility of ONAC.
2. VSC Modelling
VSC can be operated in either rectifier mode (which can achieve the regulation of the reactive power and DC voltage) or inverter mode (which can achieve the regulation of the reactive power and active power). Figure 1 presents a typical structure of VSC. Here, the balanced condition has been taken into consideration, i.e., the parameters of three phases on the AC side are totally the same and the amplitude of voltages and currents are identical, while each phase shifts 120° between themselves. When the angular frequency is ω, the rectifier dynamics in dq rotational frame can be expressed as [9]:
where the rectifier is linked to the AC grid by the equivalent resistance R_{1} and inductance L_{1}, respectively, C_{1} represents the capacitance of DC bus, ${u}_{\mathrm{d}1}=\frac{{u}_{\mathrm{sd}1}{u}_{\mathrm{rd}}}{{L}_{1}}$ and ${u}_{\mathrm{q}1}=\frac{{u}_{\mathrm{sq}1}{u}_{\mathrm{rq}}}{{L}_{1}}$. ${i}_{\mathrm{L}}$ denotes the DC current that flows out of the rectifier.
$$\{\begin{array}{c}\frac{\mathrm{d}{i}_{\mathrm{d}1}}{\mathrm{d}t}=\frac{{R}_{1}}{{L}_{1}}{i}_{\mathrm{d}1}+\omega {i}_{\mathrm{q}1}+{u}_{\mathrm{d}1}\\ \frac{\mathrm{d}{i}_{\mathrm{q}1}}{\mathrm{d}t}=\frac{{R}_{1}}{{L}_{1}}{i}_{\mathrm{q}1}\omega {i}_{\mathrm{d}1}+{u}_{\mathrm{q}1}\\ \frac{\mathrm{d}{V}_{\mathrm{dc}1}}{\mathrm{d}t}=\frac{3{u}_{\mathrm{sq}1}{i}_{\mathrm{q}1}}{2{C}_{1}{V}_{\mathrm{dc}1}}\frac{{i}_{\mathrm{L}}}{{C}_{1}}\end{array}$$
The inverter dynamics in dq rotational frame are written as:
where the inverter is linked to the AC grid by the equivalent resistance R_{2} and inductance L_{2}, respectively, C_{2} represents the capacitance of DC bus, ${u}_{\mathrm{d}2}=\frac{{u}_{\mathrm{sd}2}{u}_{\mathrm{rd}}}{{L}_{2}}$ and ${u}_{\mathrm{q}2}=\frac{{u}_{\mathrm{sq}2}{u}_{\mathrm{rq}}}{{L}_{2}}$.${i}_{\mathrm{L}}$ denotes the DC current that flows into the inverter.
$$\{\begin{array}{c}\frac{\mathrm{d}{i}_{\mathrm{d}2}}{\mathrm{d}t}=\frac{{R}_{2}}{{L}_{2}}{i}_{\mathrm{d}2}+\omega {i}_{\mathrm{q}2}+{u}_{\mathrm{d}2}\\ \frac{\mathrm{d}{i}_{\mathrm{q}2}}{\mathrm{d}t}=\frac{{R}_{2}}{{L}_{2}}{i}_{\mathrm{q}2}\omega {i}_{\mathrm{d}2}+{u}_{\mathrm{q}2}\\ \frac{\mathrm{d}{V}_{\mathrm{dc}2}}{\mathrm{d}t}=\frac{3{u}_{\mathrm{sq}2}{i}_{\mathrm{q}2}}{2{C}_{2}{V}_{\mathrm{dc}2}}+\frac{{i}_{\mathrm{L}}}{{C}_{2}}\end{array}$$
On the rectifier side, the qaxis is defined to share the same phase with the AC grid voltage u_{s1}. Similarly, the qaxis and the AC grid voltage u_{s2} are defined to share the same phase on the inverter side. Under this circumstance, the values of u_{sd1} and u_{sd2} can be regarded as 0, while the values of u_{sq1} and u_{sq2} are deemed as identical with the magnitude of u_{s1} and u_{s2}. Hence, the reactive and active power can be described by
$$\{\begin{array}{l}{P}_{1}=\frac{3}{2}\left({u}_{\mathrm{sq}1}{i}_{\mathrm{q}1}+{u}_{\mathrm{sd}1}{i}_{\mathrm{d}1}\right)=\frac{3}{2}{u}_{\mathrm{sq}1}{i}_{\mathrm{q}1}\\ {Q}_{1}=\frac{3}{2}\left({u}_{\mathrm{sq}1}{i}_{\mathrm{d}1}+{u}_{\mathrm{sd}1}{i}_{\mathrm{q}1}\right)=\frac{3}{2}{u}_{\mathrm{sq}1}{i}_{\mathrm{d}1}\\ {P}_{2}=\frac{3}{2}\left({u}_{\mathrm{sq}2}{i}_{\mathrm{q}2}+{u}_{\mathrm{sd}2}{i}_{\mathrm{d}2}\right)=\frac{3}{2}{u}_{\mathrm{sq}2}{i}_{\mathrm{q}2}\\ {Q}_{2}=\frac{3}{2}\left({u}_{\mathrm{sq}2}{i}_{\mathrm{d}2}+{u}_{\mathrm{sd}2}{i}_{\mathrm{q}2}\right)=\frac{3}{2}{u}_{\mathrm{sq}2}{i}_{\mathrm{d}2}\end{array}.$$
3. Optimal Nonlinear Adaptive Control
3.1. Nonlinear Adaptive Control
The standard form of an uncertain nonlinear system can be expressed as follows:
where $x={\left[{x}_{1},{x}_{2},\cdots ,{x}_{n}\right]}^{\mathrm{T}}\in {R}^{n}$ represents the state variable vector; $y\in R$ and $u\in R$ denote the system output and control input, respectively; a(x): ${R}^{n}\mapsto R$ and b(x): ${R}^{n}\mapsto R$ represent the unknown smooth functions which describe the aggregation influence of nonlinearities, parameter uncertainties, and unmodelled dynamics; and d(t): ${R}^{+}\mapsto R$ means the timevarying external disturbances.
$$\{\begin{array}{c}\dot{x}=Ax+B\left(a\left(x\right)+b\left(x\right)u+d\left(t\right)\right)\\ y={x}_{1}\end{array}$$
The perturbation of system (4) can be obtained by [19,20,21]
where b_{0} represents a constant control gain defined by users, by which the uncertainties of the control gain b(x) can be synthesized into the perturbation.
$$\psi \left(x,u,t\right)=a\left(x\right)+\left(b\left(x\right){b}_{0}\right)u+d\left(t\right)$$
From system (4), the final state, e.g., x_{n}, can be further expressed as
$${\dot{x}}_{n}=a\left(x\right)+\left(b\left(x\right){b}_{0}\right)u+d\left(t\right)+{b}_{0}u=\psi \left(x,u,t\right)+{b}_{0}u.$$
Besides, the perturbation term can be expressed by defining an extended state, e.g., ${x}_{n+1}=\psi \left(x,u,t\right)$. Further, system (4) can be described by
$$\{\begin{array}{c}y={x}_{1}\\ {\dot{x}}_{1}={x}_{2}\\ \vdots \\ {\dot{x}}_{n}={x}_{n+1}+{b}_{0}u\\ {\dot{x}}_{n+1}=\dot{\psi}(\xb7)\end{array}.$$
Under the two assumptions given in [19,20,21], the extended state vector can be represented by ${x}_{\mathrm{e}}={\left[{x}_{1},{x}_{2},\cdots ,{x}_{n},{x}_{n+1}\right]}^{\mathrm{T}}$.
 A.1
 The selection of b_{0} needs to satisfy certain constraint: $\leftb\left(x\right)/{b}_{0}1\right\le \theta <1$, where θ is a positive constant.
 A.2
 The function $\psi \left(x,u,t\right):{R}^{n}\times R\times {R}^{+}\mapsto R$ and $\dot{\psi}\left(x,u,t\right):{R}^{n}\times R\times {R}^{+}\mapsto R$ are locally Lipschitz in their arguments with a bounded domain of interest, with $\psi $ (0,0,0) = 0 and $\dot{\psi}$ (0,0,0) = 0.
In this paper, $\tilde{x}=x\widehat{x}$ denotes the estimation error of x and $\widehat{x}$ means the estimation of x, while x* represents the reference of variable x. We designed an (n + 1)thorder HGSPO (highgain state and perturbation observer) for the expanded system (8) to evaluate the states and perturbation at the same time, as follows [19,20,21]
where $H={\left[{\alpha}_{1}/\epsilon ,{\alpha}_{2}/{\epsilon}^{2},\dots ,{\alpha}_{n}/{\epsilon}^{n},{\alpha}_{n+1}/{\epsilon}^{n+1}\right]}^{\mathrm{T}}$, thickness layer boundary of observer $0\le \epsilon \ll 1$, thus a highgain can be successfully obtained in this way; and α_{i}, i = 1, 2, $\cdots $, n + 1 represents the gains of Luenberger observer—proper choice of its value can help to ensure the poles of polynomial s^{n}^{+1} + α_{1}s^{n} + α_{2}s^{n}^{−1} + $\cdots $ + α_{n}_{+1} = (s + ${\lambda}_{\alpha}$)^{n}^{+1} = 0 are located at the open lefthalf complex plane at −λ_{α}, as follows
where λ_{α} represents the observer root utilized for ensuring the convergence of observer.
$${\dot{\widehat{x}}}_{\mathrm{e}}={A}_{0}{\widehat{x}}_{\mathrm{e}}+{B}_{1}u+H\left({x}_{1}{\widehat{x}}_{1}\right)$$
$${\alpha}_{i}={C}_{n+1}^{i}{\lambda}_{\alpha}^{i},i=1,2,\cdots ,n+1$$
The NAC for system (4) under the estimation of states and perturbation can be written as
where $K=\left[{k}_{1},{k}_{2},\cdots ,{k}_{n}\right]$ represents the feedback control gain which makes the matrix A_{1} = ABK Hurwitzian.
$$u=\frac{1}{{b}_{0}}\left[{x}_{1}^{*\left(n+1\right)}\widehat{\psi}(\xb7)K\left(\widehat{x}{x}^{*}\right)\right]$$
3.2. Memetic Salp Swarm Algorithm
MSSA is an emerging and promising metaheuristic algorithm which mimics the food hunting behavior of salp chains in deep oceans. Note that only the main idea of MSSA is presented in this section while more details are introduced in literature [21] for interested readers.
3.2.1. Optimization Framework
The memetic computing framework is utilized in MSSA for the purpose of improving the searching efficiency of SSA, which consists of two steps shown in Figure 2, as follows [22]:
 Local search in each chain: Several various salp chains are connected in parallel to constitute MSSA, in which a single salp chain consists of a host of salps. At each iteration, on the basis of foraging behavior of SSA, all the salp chains will undertake a local search while this process is independently carried out in each salp chain.
 Global coordination in virtual population: The whole species group of MSSA can merely be considered as a large number of memes, in which each meme is deemed as a unit of cultural evolution [23]. In particular, positive strategy is adopted in the selection of memes to enhance the communication ability among the salps in MSSA (memetic salp swarm algorithm). Besides, each individual can keep its own physical characteristics stable during the process of global coordination. Therefore, the salps gather together can be considered as a virtual population, where all individuals need to rearrange into many new salp chains, thus various salp chains can achieve a global coordination.
3.2.2. Local Search in Each Chain
The salp chain is composed of two groups, i.e., leader and follower. As illustrated by Figure 2, the leader always stays at the first position of the chain to decide the moving direction of the swarm, while all the other salps act as the followers to execute the command. In SSA (salp swarm algorithm), the position of each salp can be mathematically represented. Basically, the leader’s position in the mth salp chain during the foraging process is expressed as [24]:
where the superscript j denotes the jth dimension of the searching space; ${x}_{m1}^{j}$ represents the leader’s position in the mth salp chain; ${F}_{m}^{j}$ means the position of the food source; $l{b}^{j}$ and $u{b}^{j}$ represent the lower and the upper bounds of the jth dimension, respectively; ${c}_{1}$, ${c}_{2}$, and ${c}_{3}$ represent the random numbers, while ${c}_{2}$ and ${c}_{3}$ are uniformly distributed in the range of 0–1, respectively.
$${x}_{m1}^{j}=\{\begin{array}{c}{F}_{m}^{j}+{c}_{1}\left({c}_{2}\left(u{b}^{j}l{b}^{j}\right)+l{b}^{j}\right),\text{\hspace{1em}}\mathrm{if}\text{}{c}_{3}\ge 0\\ {F}_{m}^{j}{c}_{1}\left({c}_{2}\left(u{b}^{j}l{b}^{j}\right)+l{b}^{j}\right),\text{\hspace{1em}}\mathrm{if}\text{}{c}_{3}0\end{array}$$
Since ${c}_{1}$ acts as the most vital parameter in Equation (11) as it directly influences the tradeoff between global exploration and local exploitation, for the purpose of realizing a more appropriate balance, it has been designed based on the iteration number, which can be written as [24]:
where k represents the number of current iteration and k_{max} denotes the maximum iteration number.
$${c}_{1}=2{e}^{{\left(\frac{4k}{{k}_{\mathrm{max}}}\right)}^{2}}.$$
Moreover, the position of the followers is updated as follows [24]:
where ${x}_{mi}^{j}$ represents the position of the ith salp in the mth salp chain; n denotes the population size of each salp chain; and M means the salp chains number, respectively.
$${x}_{mi}^{j}=\frac{1}{2}\text{}\left({x}_{mi}^{j}+{x}_{m,i1}^{j}\right),\text{\hspace{1em}}i=2,3,\dots ,n;\text{}m=1,2,\dots ,M$$
3.2.3. Global Coordination in Virtual Population
For the purpose of enhancing the stability of convergence, the virtual population need to be rearranged and generate various new salp chains based on the fitness values of the salps. In particular, the regroup operation of the original population in MSSA is achieved according to the classical memetic algorithm called shuffled frog leaping algorithm (SFLA) [25], which is a widely utilized population partition principle. Therefore, through arranging the fitness value in descending order the maximum optimization of regroup operation can be realized. Here, the best solution is allocated to the salp chain #1, then the salp chain #2 will capture the secondbest solution, and so on. Hence, the mth salp chain can update the swarm according to
where ${x}_{mi}$ represents the position vector of the ith salp in the mth chain; ${f}_{mi}$ denotes the fitness value of the ith salp in the mth chain; F means the fitness value set of all the salps in descending order; and X represents the relevant position vector set of all the salps, i.e., a position matrix, by
where d represents the number of dimensions; and each row vector denotes the position vector of each salp.
$$\begin{array}{c}{Y}^{m}=\{\left({x}_{mi},{f}_{mi}\right){x}_{mi}=X\left(m+M\left(i1\right),:\right),\text{\hspace{1em}}{f}_{mi}=F\left(m+M\left(i1\right)\right),\\ i=1,2,\cdots ,n\},\text{\hspace{1em}}m=1,2,\cdots ,M\end{array}$$
$$X=\left[\begin{array}{ccc}{x}_{1}^{1}& {x}_{1}^{2}& \begin{array}{cc}\cdots & {x}_{1}^{d}\end{array}\\ {x}_{2}^{1}& {x}_{2}^{2}& \begin{array}{cc}\cdots & {x}_{2}^{d}\end{array}\\ \begin{array}{c}\vdots \\ {x}_{n\times M}^{1}\end{array}& \begin{array}{c}\vdots \\ {x}_{n\times M}^{2}\end{array}& \begin{array}{cc}\begin{array}{c}\ddots \\ \dots \end{array}& \begin{array}{c}\vdots \\ {x}_{n\times M}^{d}\end{array}\end{array}\end{array}\right]$$
3.2.4. Optimization Procedure of MSSA
In conclusion, Table 1 demonstrates the overall process of MSSA. When solving a practical optimization issue, the design of fitness function takes on the most important part in the whole procedure, and it needs to take the related objective function and constraints into consideration.
4. ONAC Design for VSC Based Rectifier and Inverter
4.1. Rectifier Controller Design
Select system output y_{r} = [y_{r1}, y_{r2} ]^{T} = [Q_{1}, V_{dc1}]^{T}, ${\mathrm{Q}}_{1}^{*}$ and ${\mathrm{V}}_{\mathrm{dc}1}^{*}$ act as the given references of the DC voltage and reactive power, respectively. Throughout this paper, denote ${\widehat{x}}_{i}$ and ${\tilde{x}}_{i}={x}_{i}{\widehat{x}}_{i}$ as the estimate and estimate error of system state x_{i}, respectively. Define the tracking error as e_{r} = [e_{r1,} e_{r2}]^{T} = [Q_{1} ${\mathrm{Q}}_{1}^{*}$, V_{dc1} ${\mathrm{V}}_{\mathrm{dc}1}^{*}$]^{T}, differentiate e_{r} for rectifier (1) and repeat such operation until the control input can appear clearly, as follows
where
and
$$\left[\begin{array}{l}{\dot{e}}_{\mathrm{r}1}\\ {\ddot{e}}_{\mathrm{r}2}\end{array}\right]=\left[\begin{array}{l}{f}_{\mathrm{r}1}{\dot{Q}}_{1}^{*}\\ {f}_{\mathrm{r}2}{\ddot{V}}_{\mathrm{dc}1}^{{}^{*}}\end{array}\right]+B\mathrm{r}\left[\begin{array}{l}{u}_{\mathrm{d}1}\\ {u}_{\mathrm{q}1}\end{array}\right]$$
$$\{\begin{array}{l}{f}_{\mathrm{r}1}=\frac{3{u}_{\mathrm{sq}1}}{2}\left(\frac{{R}_{1}}{{L}_{1}}{i}_{\mathrm{d}1}+\omega {i}_{\mathrm{q}1}\right)\\ {f}_{\mathrm{r}2}=\frac{3{u}_{\mathrm{sq}1}}{2{C}_{1}{V}_{\mathrm{dc}1}}\left[\omega {i}_{\mathrm{d}1}\frac{{R}_{1}}{{L}_{1}}{i}_{\mathrm{q}1}\frac{{i}_{\mathrm{q}1}}{{V}_{\mathrm{dc}1}}\left(\frac{3{u}_{\mathrm{sq}1}{i}_{\mathrm{q}1}}{2{C}_{1}{V}_{\mathrm{dc}1}}\frac{{i}_{\mathrm{L}}}{{C}_{1}}\right)\right]\\ \frac{1}{2{R}_{0}{C}_{1}}\left(\frac{3{u}_{\mathrm{sq}1}{i}_{\mathrm{q}1}}{2{C}_{1}{V}_{\mathrm{dc}1}}\frac{{i}_{\mathrm{L}}}{{C}_{1}}\frac{3{u}_{\mathrm{sq}2}{i}_{\mathrm{q}2}}{2{C}_{2}{V}_{\mathrm{dc}2}}\frac{{i}_{\mathrm{L}}}{{C}_{2}}\right)\end{array}$$
$$B\mathrm{r}=\left[\begin{array}{cc}\frac{3{u}_{\mathrm{sq}1}}{2{L}_{1}}& 0\\ 0& \frac{3{u}_{\mathrm{sq}1}}{2{C}_{1}{L}_{1}{V}_{\mathrm{dc}1}}\end{array}\right].$$
The determinant of matrix B_{r} is obtained as $\left{B}_{\mathrm{r}}\right=9{u}_{\mathrm{sq}1}^{2}/\left(4{C}_{1}{L}_{1}^{2}{V}_{\mathrm{dc}1}\right)$, which is nonzero in the entire operation range of the rectifier. Note that the only condition for zero value of $\left{B}_{\mathrm{r}}\right$ is that u_{sq1} = 0, such condition corresponds to the case that the connected AC power grid voltage becomes zero. In practice, when the power gird voltage is below 0.2 p.u., the protection devices of VSC are activated and disconnect/stop the VSC working [8,9]. It is the same to $\left{B}_{\mathrm{i}}\right$ under inverter mode. As a result, VSC will not encounter this extreme case in practice. Hence, the system (16) can be regarded as linearizable, let v_{r} = [ v_{r1,} v_{r2}]^{T} be the control input of the linear system
where k_{r1}, ${k}_{r1}^{\prime}$, and ${\mathrm{k}}_{\mathrm{r}2}^{\prime}$ represent the positive feedback gains.
$$\left[\begin{array}{c}{v}_{\mathrm{r}1}\\ {v}_{\mathrm{r}2}\end{array}\right]=\left[\begin{array}{c}{k}_{\mathrm{r}1}{e}_{\mathrm{r}1}\\ {k}_{\mathrm{r}1}^{\prime}{e}_{\mathrm{r}2}+{k}_{\mathrm{r}2}^{\prime}{\dot{e}}_{\mathrm{r}2}\end{array}\right]$$
Under the assumption that the nonlinearities are all unknown, the perturbations Ψ_{r1}(·) and Ψ_{r2}(·) can be written as
where the constant control gain B_{r0} can be expressed as
$$\left[\begin{array}{l}\Psi \mathrm{r}1(\cdot )\\ \Psi \mathrm{r}2(\cdot )\end{array}\right]=\left[\begin{array}{l}f\mathrm{r}1\\ f\mathrm{r}2\end{array}\right]+(B\mathrm{r}B\mathrm{r}0)\left[\begin{array}{c}{u}_{\mathrm{d}1}\\ {u}_{\mathrm{q}1}\end{array}\right]$$
$$B\mathrm{r}0=\left[\begin{array}{cc}b\mathrm{r}10& 0\\ 0& b\mathrm{r}20\end{array}\right].$$
Then, system (16) can be represented by
$$\left[\begin{array}{l}{\dot{e}}_{\mathrm{r}1}\\ {\ddot{e}}_{\mathrm{r}2}\end{array}\right]=\left[\begin{array}{l}\Psi r1(\cdot )\\ \Psi r2(\cdot )\end{array}\right]+Br0\left[\begin{array}{l}{u}_{d1}\\ {u}_{q1}\end{array}\right]\left[\begin{array}{l}{\dot{Q}}_{1}^{*}\\ {\ddot{V}}_{dc1}^{*}\end{array}\right].$$
Define z_{11} = Q_{1}, a secondorder HGPO is given by
$$\{\begin{array}{l}{\dot{\widehat{z}}}_{11}=\widehat{\Psi}\mathrm{r}1(\cdot )+\frac{\alpha \mathrm{r}1}{\epsilon}({z}_{11}{\widehat{z}}_{11})+b\mathrm{r}10{u}_{\mathrm{d}1}\\ {\dot{\widehat{\Psi}}}_{\mathrm{r}1}(\cdot )=\frac{\alpha \mathrm{r}2}{{\epsilon}^{2}}({z}_{11}{\widehat{z}}_{11})\end{array}.$$
Define ${z}_{11}^{\prime}={V}_{\mathrm{dc}1}$ and ${z}_{12}^{\prime}={\dot{z}}_{11}^{\prime}$, a thirdorder highgain state and perturbation observer (HGSPO) is obtained by
where $\alpha \mathrm{r}1$, $\alpha \mathrm{r}2$, ${{\alpha}^{\prime}}_{\mathrm{r}1}$, ${{\alpha}^{\prime}}_{\mathrm{r}2}$, and ${{\alpha}^{\prime}}_{\mathrm{r}3}$ represent positive constants with 1 $\gg \u03f5>0$.
$$\{\begin{array}{l}{{\dot{\widehat{z}}}^{\prime}}_{11}={{\widehat{z}}^{\prime}}_{12}+\frac{{{\alpha}^{\prime}}_{\mathrm{r}1}}{\epsilon}({{z}^{\prime}}_{11}{{\widehat{z}}^{\prime}}_{11})\\ {{\dot{\widehat{z}}}^{\prime}}_{12}={\widehat{\Psi}}_{\mathrm{r}2}(\cdot )+\frac{{{\alpha}^{\prime}}_{\mathrm{r}2}}{{\epsilon}^{2}}({{z}^{\prime}}_{11}{{\widehat{z}}^{\prime}}_{11})+\mathrm{br}20u\mathrm{q}1\\ {\dot{\widehat{\Psi}}}_{\mathrm{r}2}(\cdot )=\frac{{{\alpha}^{\prime}}_{\mathrm{r}3}}{{\epsilon}^{3}}({{z}^{\prime}}_{11}{{\widehat{z}}^{\prime}}_{11})\end{array}$$
The NACbased rectifier controller based on the estimations of perturbation can be given by
$$\left[\begin{array}{l}{u}_{\mathrm{d}1}\\ {u}_{\mathrm{q}1}\end{array}\right]={B}_{\mathrm{r}0}^{1}\left(\left[\begin{array}{l}\widehat{\Psi}\mathrm{r}1(\cdot )\\ \widehat{\Psi}\mathrm{r}2(\cdot )\end{array}\right]+\left[\begin{array}{l}{\dot{Q}}_{1}^{*}\\ {\ddot{V}}_{\mathrm{dc}1}^{{}^{*}}\end{array}\right]+{v}_{\mathrm{r}}\right).$$
4.2. Inverter Controller Design
Select system output y_{i} = [y_{i1}, y_{i2}]^{T} = [Q_{2}, P_{2}]^{T}, ${\mathrm{Q}}_{2}^{*}$ and ${P}_{2}^{*}$ act as the provided references of the active power and reactive power, respectively. The tracking error can be defined as e_{i} = [e_{i1}, e_{i2}]^{T} = [Q_{2}− ${\mathrm{Q}}_{2}^{*}$, P_{2}− ${P}_{2}^{*}$]^{T}, differentiate e_{i} for inverter (2) and repeat such operation until the control input appears, as follows
where
and
$$\left[\begin{array}{l}{\dot{e}}_{\mathrm{i}1}\\ {\dot{e}}_{\mathrm{i}2}\end{array}\right]=\left[\begin{array}{l}f\mathrm{i}1{\dot{Q}}_{2}^{*}\\ f\mathrm{i}2{\dot{P}}_{2}^{{}^{*}}\end{array}\right]+B\mathrm{i}\left[\begin{array}{l}{u}_{\mathrm{d}2}\\ {u}_{\mathrm{q}2}\end{array}\right]$$
$$\{\begin{array}{l}f\mathrm{i}1=\frac{3{u}_{\mathrm{sq}2}}{2}\left(\frac{R2}{L2}{i}_{\mathrm{d}2}+\omega {i}_{\mathrm{q}2}\right)\\ f\mathrm{i}2=\frac{3{u}_{\mathrm{sq}2}}{2}\left(\frac{R2}{L2}{i}_{\mathrm{q}2}+\omega {i}_{\mathrm{d}2}\right)\end{array}$$
$$B\mathrm{i}=\left[\begin{array}{cc}\frac{3{u}_{\mathrm{sq}2}}{2L2}& 0\\ 0& \frac{3{u}_{\mathrm{sq}2}}{2L2}\end{array}\right].$$
The determinant of matrix B_{i} is obtained as $\left{B}_{\mathrm{i}}\right{=9u}_{\mathrm{sq}2}^{2}{/(4L}_{2}^{2})$, which is nonzero in the whole operation range of the inverter. Such that system (26) can be regarded as linearizable. Let v_{i} = [v_{i1}, v_{i2}]^{T} be the control input of the linear system
where k_{i1} and ${k}_{\mathrm{i}1}^{\prime}$ represent positive feedback gains.
$$\left[\begin{array}{l}{v}_{\mathrm{i}1}\\ {v}_{\mathrm{i}2}\end{array}\right]=\left[\begin{array}{l}{k}_{\mathrm{i}1}{e}_{\mathrm{i}1}\\ {k}_{\prime}^{\mathrm{i}1}{e}_{\mathrm{i}2}\end{array}\right]$$
Under the assumption that the nonlinearities are all unknown, the perturbations Ψ_{r1}(·) and Ψ_{r2}(·) can be written as
where B_{i0} represents the constant control gain, as follows
$$\left[\begin{array}{l}\Psi \mathrm{i}1(\cdot )\\ \Psi \mathrm{i}2(\cdot )\end{array}\right]=\left[\begin{array}{l}f\mathrm{i}1\\ f\mathrm{i}2\end{array}\right]+(B\mathrm{i}B\mathrm{i}0)\left[\begin{array}{l}{u}_{\mathrm{d}2}\\ {u}_{\mathrm{q}2}\end{array}\right]$$
$$B\mathrm{i}0=\left[\begin{array}{cc}b\mathrm{i}10& 0\\ 0& b\mathrm{i}20\end{array}\right].$$
Then system (26) can be rewritten as
$$\left[\begin{array}{l}{\dot{e}}_{\mathrm{i}1}\\ {\dot{e}}_{\mathrm{i}2}\end{array}\right]=\left[\begin{array}{l}\Psi \mathrm{i}1(\cdot )\\ \Psi \mathrm{i}2(\cdot )\end{array}\right]+B\mathrm{i}0\left[\begin{array}{l}{u}_{\mathrm{d}2}\\ {u}_{\mathrm{q}2}\end{array}\right]\left[\begin{array}{l}{\dot{Q}}_{2}^{*}\\ {\dot{P}}_{2}^{*}\end{array}\right].$$
Similarly, define z_{21} = Q_{2} and ${z}_{21}^{\prime}={P}_{2}$, two secondorder HGPOs are designed as
where α_{i1}, α_{i2}, ${{\alpha}^{\prime}}_{\mathrm{i}1}$, and ${{\alpha}^{\prime}}_{\mathrm{i}2}$ denote positive constants.
$$\{\begin{array}{l}{\dot{\widehat{z}}}_{21}=\widehat{\Psi}\mathrm{i}1(\cdot )+\frac{{\alpha}_{\mathrm{i}1}}{\epsilon}({z}_{21}{\widehat{z}}_{21})+{b}_{\mathrm{i}10}{u}_{\mathrm{d}2}\\ {\dot{\widehat{\Psi}}}_{\mathrm{i}1}(\cdot )=\frac{{\alpha}_{\mathrm{i}2}}{{\epsilon}^{2}}({z}_{21}{\widehat{z}}_{21})\end{array}$$
$$\{\begin{array}{l}{{\dot{\widehat{z}}}^{\prime}}_{21}=\widehat{\Psi}\mathrm{i}2(\cdot )+\frac{{{\alpha}^{\prime}}_{\mathrm{i}1}}{\epsilon}({{z}^{\prime}}_{21}{{\widehat{z}}^{\prime}}_{21})+{b}_{\mathrm{i}20}{u}_{\mathrm{q}2}\\ \dot{\widehat{\Psi}}\mathrm{i}2(\cdot )=\frac{{{\alpha}^{\prime}}_{\mathrm{i}2}}{{\epsilon}^{2}}({{z}^{\prime}}_{21}{{\widehat{z}}^{\prime}}_{21})\end{array}$$
The NACbased inverter controller on the basis of the estimations of perturbation can be given by
$$\left[\begin{array}{l}{u}_{\mathrm{d}2}\\ {u}_{\mathrm{q}2}\end{array}\right]={B}_{\mathrm{i}0}^{1}\left(\left[\begin{array}{l}\widehat{\Psi}\mathrm{i}1(\cdot )\\ \widehat{\Psi}\mathrm{i}2(\cdot )\end{array}\right]+\left[\begin{array}{l}{\dot{Q}}_{2}^{*}\\ {\dot{P}}_{2}^{{}^{*}}\end{array}\right]+{v}_{\mathrm{i}}\right).$$
4.3. Optimal Control Parameter Tuning
The controller gains in Equations (19), (25), (29), and (35) and observer gains in Equations (23)–(24) and (33)–(34) are optimally tuned through MSSA under reactive power and active power regulation.
The ultimate purpose of the optimization is to maximumly reduce the tracking error of DC link voltage, reactive power, active power, perturbation estimation error, as well as the total control costs, the model for which can be described by
where ${\omega}_{i1}$ and ${\omega}_{i2}$ are two weights used to measure the magnitude of perturbation estimation error which are set to be 0.01; while ${\omega}_{i3}$ and ${\omega}_{i4}$ are the weights utilized to measure the magnitude of overall control costs which usually are set to be 0.25; note that the selection of weighting factor value is depended on the magnitude of controller output, perturbation estimation error, and control error. Normally, one should examine their magnitudes and make them have the same order of magnitude. In this paper, through trial and error, it can be found that perturbation estimation is 120 times higher (peak value occurs due to peaking phenomenon of HGPO) than that of control error while controller output is around six times higher than that of control error. Thus, their values are chosen as aforementioned to scale them to have the same order of magnitude. Lastly, T = 5 s represents the experiment time, while the control costs are restricted by the limits.
$$\begin{array}{c}\mathrm{Minimize}\text{}J(\mathrm{x})={{\displaystyle \int}}_{0}^{T}[\left{P}_{i}{P}_{i}^{*}\right+\left{Q}_{i}{Q}_{i}^{*}\right+\left{V}_{\mathrm{dc}}{V}_{\mathrm{dc}}^{*}\right+{\omega}_{i1}\left{\tilde{\Psi}}_{\mathrm{r}i}\right+{\omega}_{i2}\left{\tilde{\Psi}}_{\mathrm{i}i}\right+{\omega}_{i3}\left{u}_{\mathrm{d}i}\right+\\ {\omega}_{i4}\left{u}_{\mathrm{q}i}\right]\mathrm{d}t.\end{array}\phantom{\rule{0ex}{0ex}}\mathrm{subject}\text{}\mathrm{to}\{\begin{array}{c}{k}_{i1}^{min}\le {k}_{i1}\le {k}_{i1}^{max}\\ {k}_{i1}^{\prime min}\le {k}_{i1}^{\prime}\le {k}_{i1}^{\prime max}\\ {k}_{r1}^{min}\le {k}_{r1}\le {k}_{r1}^{max}\\ {k}_{ri}^{\prime min}\le {k}_{ri}^{\prime}\le {k}_{ri}^{\prime max}\\ {\alpha}_{ii}^{min}\le {\alpha}_{ii}\le {\alpha}_{ii}^{max}\\ {\alpha}_{ri}^{min}\le {\alpha}_{ri}\le {\alpha}_{ri}^{max}\\ {\alpha}_{ii}^{\prime min}\le {\alpha}_{ii}^{\prime}\le {\alpha}_{ii}^{\prime max}\\ {\alpha}_{rj}^{\prime min}\le {\alpha}_{rj}^{\prime}\le {\alpha}_{rj}^{\prime max}\\ {u}_{di}^{min}\le {u}_{di}\le {u}_{di}^{max}\\ {u}_{qi}^{min}\le {u}_{qi}\le {u}_{qi}^{max}\end{array},\text{}i=1,\text{}2\text{}\mathrm{and}\text{}j=1,\text{}2,\text{}3.$$
5. Experiment Results
5.1. Experimental Platform
For the purpose of verifying the practicability of ONAC, a hardware experiment was carried out while the experimental platform is illustrated by Figure 5. A strong AC/DC power supply was used to provide stable three phase AC and DC voltages. The power supply was connected with the converter by an IGBT converter [27] with an inductive filter. IGBT converters can be driven by pulse width modulation (PWM) signals. Compared with the conventional PWM or sinusoidal pulse width modulation (SPWM) techniques, the space vector modulation (SVM) can increase 15% more of the maximum output voltage and reduce the switching times. The voltage/current isolators measured the voltages and currents, which were then sent to the dSPACE simulator through analogue/digital (A/D) blocks [28].
The dSPACE simulator had quadcore AMD (advanced micro devices) processors (DS1104) and operated with DS5202 ACMC board, which was capable of generating highfrequency PWM [29] signals and providing highspeed A/D interfaces. The IGBT converter, inductive filter, voltage/current isolators, and AC/DC power supply were all from the LabVolt company, which provides various equipment with the userfriendly interfaces, accurate system parameters, and reliable hardware protections. The DC bus of the IGBT converter was protected by adopting a 100 Ω damping resistor with 1 kW rated power connected to the damping circuit built in to the IGBT converter to avoid potential damages caused by overvoltage or overcurrent in the DC bus. Finally, Table 2 demonstrates the guaranteed measurement error of DS1104 board.
The controller was embedded in the dSPACE platform (dSPACE Inc., Paderborn, Germany) shown in Figure 5, which measured the DC voltage, reactive power, and active power as inputs. Then, the PWM signals were generated with various duty cycles as controller outputs to the IGBT converter. Furthermore, the sampling frequency f_{s} and the PWM frequency f_{PWM} with SVPWM are given in Table 3. Here, sampling frequency was determined by the performance of industrial microprocessors of VSC, which usually ranges from 1 kHz to 3 kHz. Note that a lower sampling frequency might degrade the control precision while a higher sampling frequency might increase the computation burden. To make a tradeoff between the control precision and computation burden, this paper adopted the median sampling frequency (2 kHz) to validate the controller implementation feasibility. MSSA was run 30 times and the optimal outcomes (the controller gains and observer gains leading to the lowest fitness function) were undertaken in ONAC. The parameters of the controller are provided in Table 4, while the control inputs were limited as u_{di} ≤ 50 V and u_{qi} ≤ 50 V, i = 1,2, respectively. Based on SVPWM and ignoring the resistance in the steady state, the minimum DC voltage must satisfy the following inequality to ensure the converter is controllable and can work properly as [16]
$${V}_{\mathrm{dc}i}\ge \sqrt{3}\sqrt{{({u}_{\mathrm{sd}i}+\omega Liiqi)}^{2}+{({u}_{\mathrm{sq}i}\omega Li{i}_{\mathrm{d}i})}^{2}}$$
At last, the IGBT converter was used as the DC/AC converter, the antiparallel diodes were combined with the IGBT converters such that it could be operated as either a rectifier or an inverter.
5.2. Rectifier Controller Experiment
The configuration of the rectifier controller experiment is illustrated by Figure 6, in which the DC bus was connected to a resistive load to achieve a specific evaluation of control performance when the active power flowed from the AC grid to the DC cable.
The effectiveness of the ONACbased rectifier controller was tested at first, in which the initial DC voltage V_{dc1} and reactive power Q_{1} were adjusted to be 100 V and 0 Var, respectively. The DC voltage remained at 100 V for the whole period of the experiment. The reference of reactive power was set to be 15 Var, then dropped to 5 Var and was restored to 15 Var. After reactive power was stabilized at 15 Var, a resistive load R_{L} was connected to the DC bus given in Figure 6 to represent the DC cable. Then a 50% voltage drop of the AC grid occured to evaluate the system transient performance.
The experiment results are provided in Figure 7. It demonstrates that both the reactive power and DC voltage were rapidly regulated, while the DC voltage was able to be restored after the AC grid voltage drop occurred, which validates the effectiveness of the ONACbased rectifier controller.
5.3. Inverter Controller Experiment
The configuration of the inverter controller experiment is shown in Figure 8, where a DC power supply was utilized to output a stable DC voltage (representing a constant DC voltage regulated by the rectifier controller) such that the IGBT converter could work properly.
The effectiveness of the ONACbased inverter controller was then tested, the initial value of reactive power and active power were zero. The DC voltage V_{dc2} was maintained at 60 V for the whole period of the experiment. The controller was activated to increase the reactive power Q_{2} to be 20 Var at first; after reactive power Q_{2} was stabilized the active power P_{2} was increased to be 10 W, then the reactive power Q_{2} was decreased to be 10 Var. At last a 33.3% voltage drop of the AC grid occurred to evaluate the system transient performance.
The experiment outcomes are given in Figure 9. Obviously, the reactive power and active power were able to be regulated independently, and the system could also be restored after the voltage drop, which verified the effectiveness of the ONACbased inverter controller.
5.4. Set point Tracking
In order to evaluate the set point tracking performance of ONAC, both the simulation (set point curve) and experiment are compared in Figure 10. One can clearly observe that the experiment can closely track the set point, while their tiny difference resulted from the transmission delay of control signals or measurement noises during the experiment.
5.5. Disturbance Rejection
Disturbances are very common in many industrial processes which often have a malignant impact on the predesigned control performance of the studied system. As a result, disturbance rejection performance is a very crucial property for advanced controller design in industries [31,32,33]. This test aimed to evaluate the disturbance rejection performance of ONAC, while the perturbation estimation performance of HGPO under 20% voltage drop lasting for 15 ms (t = 0.1 s–0.115 s) of VSC under rectifier mode was recorded and illustrated in Figure 11. As shown in Figure 11, both perturbations ${\Psi}_{\mathrm{r}1}$ and ${\Psi}_{\mathrm{r}2}$ could be rapidly estimated by HGPO s under the voltage drop of VSC as the perturbation estimation error could be efficiently converged to 0. As a result, the disturbance was able to be effectively rejected as its realtime estimate could be rapidly obtained and then fully compensated by ONAC. Here, only the rectifier results are provided as it involved both the secondorder HGPO (Figure 11a) and thirdorder HGPO (Figure 11b), while similar results could be found in inverter mode. Note that the transient relatively large variation of perturbation when the disturbance occurred was due to the high gains used in HGPO.
5.6. Robustness to Parameter Uncertainties
Lastly, the robustness to parameter uncertainties was investigated in this section. Here, a series of plant model mismatches of equivalent resistance R_{1} and inductance L_{1} within ±20% variation around their nominal value were undertaken. Then, a 50% voltage drop lasting 100 ms at VSC was applied, in which the peak value of output power P was recorded. Figure 12 shows that the variation of peak value of active power P obtained by FLC [10] and ONAC were 73.4% and 26.5%, which clearly demonstrated that ONAC provided the highest robustness against VSC parameter uncertainties thanks to the realtime compensation of perturbation, while FLC was vulnerable to parameter uncertainties as it required an accurate VSC system model.
6. Conclusions
This paper designed an ONACbased rectifier controller and ONAC based inverter controller. It can provide significant robustness as various modelling uncertainties were aggregated into a perturbation, which was rapidly evaluated by HGPO and then compensated for by the proposed controller. Besides, ONAC merely needs the active power and reactive power, as well as DC voltage need to be measured, while a precise VSC system model is not required, thus the proposed controller can be readily implemented in practice. Moreover, the controller gains and observer gains are optimally tuned by an emerging and promising biology based metaheuristic algorithm called MSSA, which can effectively achieve an appropriate tradeoff between global exploration and local exploitation, thus a satisfactory control performance and perturbation estimation performance could be guaranteed. Finally, for the sake of verifying the practicability of the proposed controller, a hardware experiment utilizing dSPACE platform was undertaken. The experiment results showed that ONAC could achieve satisfactory control performance.
Future studies will apply ONACbased VSC on renewable energy systems, e.g., PV inverter, wind energy conversion systems, as well as electric energy storage systems.
Author Contributions
Preparation of the manuscript has been performed by Y.J., X.J., H.W., Y.F., W.G., B.Y., and T.Y.
Funding
The authors gratefully acknowledge the support of National Natural Science Foundation of China (51777078), and Yunnan Provincial Basic Research ProjectYouth Researcher Program (2018FD036).
Conflicts of Interest
The authors declare no conflict of interest.
Nomenclature
Variables  Abbreviations  
R_{1}, R_{2}  equivalent resistance, Ω  VSC  voltage source converter 
L_{1}, L_{2}  equivalent inductance, mH  CSC  current source converter 
${i}_{\mathrm{d}1}$, ${i}_{\mathrm{d}2}$  daxes components of the line currents, A  VSCHVDC  VSC based high voltage direct current 
${i}_{\begin{array}{c}q1\\ \end{array}}$, ${i}_{\begin{array}{c}q2\\ \\ \end{array}}$  qaxes components of the line currents, A  DFIG  doubly fed induction generation 
${u}_{\mathrm{d}1}$, ${u}_{\mathrm{d}2}$  daxes components of the line voltages, V  PMSG  permanent magnetic synchronous generator 
${u}_{\begin{array}{c}q1\\ \end{array}}$, ${u}_{\begin{array}{c}q2\\ \end{array}}$  qaxes components of the line voltages, V  PV  photovoltaic 
${V}_{\mathrm{dc}1}$, ${V}_{\mathrm{dc}2}$  DC voltages, V  DG  distributed generation 
${u}_{\mathrm{sd}1}$, ${u}_{\mathrm{sd}2}$  daxes components of the respective AC grid voltages, V  PID  proportionalintegralderivative 
${u}_{\begin{array}{c}sq1\\ \end{array}}$, ${u}_{\begin{array}{c}sq2\\ \end{array}}$  qaxes components of the respective AC grid voltages, V  FLC  feedback linearization control 
C_{1}, C_{2}  capacitance of DC bus, µF  SMC  slidingmode control 
${u}_{\mathrm{rd}}$, ${u}_{\mathrm{id}}$  daxes components of the converter input voltages, V  PBSMC  passivitybased slidingmode control 
${u}_{\begin{array}{c}rq\\ \end{array}}$, ${u}_{\begin{array}{c}iq\\ \end{array}}$  qaxes components of the converter input voltages, V  MPPT  maximum power point tracking 
U_{a}, U_{b}, U_{c}  abc threephase voltages at AC side, V  RSMC  robust slidingmode control 
L_{a}, L_{b}, L_{c}  abc threephase inductances at AC side, mH  MPC  model predictive control 
V_{1}~V_{6}  #1#6 valves of IGBT  ESO  extended state observer 
U_{d}  DClink voltage, V  NAC  nonlinear adaptive control 
I_{d}  DClink current, A  HGPO  highgain perturbation observer 
P_{1}, P_{2}  active powers transmitted from the AC grid to the VSC, W  MSSA  memetic salp swarm algorithm 
Q_{1}, Q_{2}  reactive powers transmitted from the AC grid to the VSC, W  PWM  pulse width modulation 
System parameters  SPWM  sinusoidal pulse width modulation  
P_{0}  rated active power, W  SVM  space vector modulation 
V_{0}  rated rms voltage, V  SSA  salp swarm algorithm 
I_{0}  rated rms current, A  DC  direct current 
f_{0}  rated frequency, Hz  AC  alternating current 
${L}_{1}^{\prime}$, ${L}_{2}^{\prime}$  filter inductance, mH  MSSA parameters  
f_{pwm}  PWM frequency, kHz  c_{1}, c_{2}, c_{3}  random numbers 
f_{s}  Sampling frequency, kHz  n  population size of each salp chain 
${C}_{1}^{\prime}$, ${C}_{2}^{\prime}$  DC bus capacitance, µF  M  number of salp chains 
R_{L}  load resistance, Ω  k_{max}  maximum iteration number 
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Figure 3.
The overall optimal nonlinear adaptive control (ONAC)based control framework of rectifier.
Figure 7.
System responses obtained in the rectifier controller experiment: (a) Average active power ${P}_{1}$; (b) average active power ${Q}_{1}$; (c) DC voltage ${V}_{\mathrm{dc}1}$; (d) control input ${V}_{\mathrm{dq}}$.
Figure 9.
System responses obtained in the inverter controller experiment. (a) Average active power ${P}_{2}$; (b) average active power ${Q}_{2}$; (c) DC voltage ${V}_{\mathrm{dc}2}$; (d) control input ${V}_{\mathrm{dq}}$.
Figure 10.
System responses obtained under set point tracking. (a) DC voltage ${V}_{\mathrm{dc}1}$ of rectifier; (b) average reactive power ${Q}_{1}$ of rectifier; (c) DC voltage ${V}_{\mathrm{dc}2}$ of inverter; (d) average reactive power ${Q}_{2}$ of inverter.
Figure 11.
Disturbance rejection performance obtained under 20% voltage drop of rectifier. (a) Perturbation ${\Psi}_{\mathrm{r}1}$ and (b) perturbation ${\Psi}_{\mathrm{r}2}$.
Figure 12.
Peak value of active power P obtained under a 50% voltage drop lasting 100 ms at VSC with 20% variation of the equivalent resistance R_{1} and inductance L_{1} of FLC and ONAC. (a) FLC and (b) ONAC.

Table 2.
The error specification of quadcore AMD (advanced micro devices) processors (DS 1104) board [30].
Component Parameters  A/D Converter  D/A Converter 

Offset error  ±5 mV  ±1 mV 
Gain error  Multiplexed channels: ±0.25%  ±0.1% 
Parallel channels: ±0.5%  
Offset drift  40 μV/K  130 μV/K 
Gain drift  25 ppm/K  25 ppm/K 
Signaltonoise ratio  Multiplexed channels: >80 dB  >80 dB 
Parallel channels: >65 dB 
Rated Active Power  P_{0}  300 W 

Rated rms voltage  V_{0}  30 V 
Rated rms current  I_{0}  10 A 
Rated frequency  f_{0}  50 Hz 
Filter inductance  L_{1}^{′}, L_{2}^{′}  60 mH 
PWM frequency  f_{PWM}  2 kHz 
Sampling frequency  f_{s}  10 kHz 
DC bus capacitance  C_{1}^{′}, C_{2}^{′}  1320 µF 
Load resistance  R_{L}  1200 Ω 
Rectifier Controller  

Controller gains  ${k}_{\mathrm{r}1}=40$  ${k}_{\mathrm{r}1}^{\prime}=400$  ${k}_{\mathrm{r}2}^{\prime}=40$ 
${b}_{\mathrm{r}10}=105$  ${b}_{\mathrm{r}20}=3000$  
Observer gains  ${\alpha}_{\mathrm{r}1}=80$  ${\alpha}_{\mathrm{r}2}=1600$  ${\alpha}_{\mathrm{r}1}^{\prime}=120=120$ 
${\alpha}_{\mathrm{r}2}^{\prime}=4800$  ${\alpha}_{\mathrm{r}3}^{\prime}=6.4\times {10}^{4}$  ε = 0.1  
Inverter controller  
Controller gains  ${k}_{i1}=20$  ${k}_{i1}^{\prime}=20$  ${b}_{i10}=100$ 
${b}_{i20}=100$  
Observer gains  ${\alpha}_{i1}=60$  ${\alpha}_{i2}=80$  ${\alpha}_{i1}^{\prime}=60$ 
${\alpha}_{i2}^{\prime}=900$  ε = 0.1 
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