Development of an efficient and rapid computational solar photovoltaic emulator utilizing an explicit PV model

doi:10.1016/j.solener.2024.112426

Solar Energy 太阳能

Volume 271, 15 March 2024, 112426
第 271 卷，2024 年 3 月 15 日，112426

https://doi.org/10.1016/j.solener.2024.112426 Get rights and content 获取权利和内容

Highlights 亮点

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Numerical computation of intricate PV equation complicates the structure of Photovoltaic emulator.
复杂的光伏方程的数值计算使光伏仿真器的结构变得复杂。
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A new emulator based on a straightforward explicit PV model is introduced.
介绍了一种基于直接显式光伏模型的新型仿真器。
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Integration of the explicit model into the PV features high simplicity and accuracy.
将显式模型集成到光伏系统中，具有高度的简便性和准确性。
• -
Developed PVE requires just one iteration to compute the PV model.
开发的 PVE 只需一次迭代即可计算 PV 模型。
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Proposed emulator is suitable for field PV testing.
拟议的仿真器适用于现场光伏测试。

Abstract 摘要

This paper presents a novel simplified and flexible Photovoltaic emulator (PVE) that effectively emulates the static and dynamic characteristics of a solar panel. By meticulously generating a benchmark for the entire system, the proposed PVE utilizes an original explicit PV model (EPVM). As an approximation of the implicit equation of the PV, the EPVM employs a Taylor series resolution. This methodology eliminates the necessity for numerous iterative computations, in contrast to previous frameworks, and as a result possesses formidable computational capabilities. Furthermore, it is clearly illustrated that the EPVM can be effortlessly incorporated into the PVE architecture, resulting in a straightforward and simplified procedure that contributes to its overall flexibility and simplicity. Using MATLAB/Simulink and a 200 W solar system, the developed PVE is investigated numerically. Through the application of nonlinear processes such as (maximum power point tracking) MPPT, the PVE's viability is confirmed. This study repeatedly emphasizes the superior qualities of the proposed PVE in simulating a solar panel in a straightforward, precise, and efficient manner, by conducting a comparative analysis with contemporary PVE designs. It is consistently determined through quantitative evaluations that the developed PVE achieves an efficiency exceeding 99 % and a maximum error rate of less than 0.7 %.
本文提出了一种新颖的简化、灵活的光伏模拟器（PVE），可有效模拟太阳能电池板的静态和动态特性。通过细致地生成整个系统的基准，所提出的 PVE 采用了原始的显式光伏模型（EPVM）。作为光伏隐式方程的近似值，EPVM 采用了泰勒级数解析法。与之前的框架相比，这种方法无需进行大量迭代计算，因此具有强大的计算能力。此外，EPVM 可以毫不费力地集成到 PVE 架构中，从而简化了程序，提高了整体灵活性和简便性。利用 MATLAB/Simulink 和 200 W 太阳能系统，对所开发的 PVE 进行了数值研究。通过应用（最大功率点跟踪）MPPT 等非线性过程，证实了 PVE 的可行性。本研究通过与当代 PVE 设计的比较分析，反复强调了所提出的 PVE 在以直接、精确和高效的方式模拟太阳能电池板方面的卓越品质。通过定量评估，一致认定所开发的 PVE 的效率超过 99%，最大误差率小于 0.7%。

Keywords 关键词

Photovoltaic (PV)

Explicit PV model (EPVM)

Photovoltaic emulator (PVE)

Solar system

光伏 (PV)显式光伏模型 (EPVM)光伏模拟器 (PVE)太阳能系统

Nomenclature 术语

Symbols 符号

E_g

Bandgap energy 带隙能

E_g-ref

Bandgap energy at STC STC 时的带隙能

Erro r_{pve}

PVE Error PVE 错误

Irradiance (W/m²)
辐照度（瓦/米 ² )

R_{p}

Shunt resistance 分流电阻

R_{s}

Series resistance 串联电阻

I_s 我 _s

Reverse saturation current
反向饱和电流

I_{s . r e f}

Reverse saturation current at STC
STC 时的反向饱和电流

n

Ideality factor 理想因素

V_{pv}

PV Voltage 光伏电压

I_{pv}

PV Current 光伏电流

f_{pve}

PVE buck converter switching frequency
PVE 降压转换器开关频率

f_{boost}

MPPT boost converter switching frequency
MPPT 升压转换器开关频率

V_{m}

Maximum power voltage 最大功率电压

I_{m}

Maximum power Current 最大功率电流

I_{pve}

PVE current PVE 当前

Temperature 温度

V_{pve}

PVE voltage PVE 电压

I_ph 我 _ph

Photocurrent 光电流

I_{ref}

Reference PVE Current 参考 PVE 电流

K_B

Boltzmann constant (1.38×10^-23 J/K)
波尔兹曼常数（1.38×10 ^-23 J/K）

K_i

Temperature coefficient of short circuit current
短路电流温度系数

Charge constant (1.602×10^-19C)
电荷常数（1.602×10 ^-19 C）

L_pve, r_L

PVE Inductance, and parasitic Inductance
PVE 电感和寄生电感

C_pve,r_c

PVE Capacitance, and parasitic Capacitance
PVE 电容和寄生电容

R_pve

PVE output resistance PVE 输出电阻

K_{p}, K_{i}

Proportional and, integral gain of PI controller
PI 控制器的比例增益和积分增益

L_{bost}

C_{bost}

Boost converter inductance and capacitance
升压转换器电感和电容

C_{in}

PVE-MPPT coupling capacitance
PVE-MPPT 耦合电容

G_{b}

Transfer function of boost
升压的传递函数

PVE DC Supply Voltage PVE 直流电源电压

d, D

PVE duty and MPPT control duty cycle
PVE 占空比和 MPPT 控制占空比

D_{\min}, D_{\max}

Minimum and maximum PVE duty cycle
最低和最高 PVE 工作周期

d_{\max}, d_{\max}

Minimum and maximum MPPT duty cycle
最小和最大 MPPT 占空比

V_{oc}

Open Circuit Voltage 开路电压

I_{sc}

Short Circuit Current 短路电流

Subscripts 下标

ref 档号

Reference 参考资料

STC

Standard test conditions 标准测试条件

Abbreviations 缩略语

Photovoltaic 光电

PVE

Photovoltaic Emulator 光伏模拟器

WOA

Whale Optimization algorithm
鲸鱼优化算法

SDM

Single Diode Model 单二极管型号

EPVM

Explicit PV model 显式光伏模型

Proportional Integral controller
比例积分控制器

MPPT

Maximum power point tracking
最大功率点跟踪

PWM

Pulse width modulation 脉宽调制

ANN

Artificial Neural Network
人工神经网络

LUT

Look up table 查询表

I-V

Current-voltage 电流-电压

V-R

Voltage resistor model 电压电阻器型号

RCM

Resistance comparison method
电阻比较法

DRM

Direct reference method 直接参照法

FLC

Fuzzy Logic controller 模糊逻辑控制器

FSL

Fifteen segment linearization
15 段线性化

2DR

2-diode circuit model 2 二极管电路模型

Proportional Integral 比例积分

MPPT

Maximum power point tracking
最大功率点跟踪

PVE

Photovoltaic emulator 光伏模拟器

BSC

Binary search computation
二进制搜索计算

I-R

Current-resistor model 电流电阻模型

Shift controller 换档控制器

BSC-SC

Binary search Computation-Shift controller PVE
二进制搜索计算-移位控制器 PVE

RCM

Resistance Comparison Method
电阻比较法

BSC-PI

Binary search Computation-PI controller PVE
二进制搜索计算-PI 控制器 PVE

IAE

Integral of Absolute Error
绝对误差积分

MAPE

Mean Absolute Percentage Error
平均绝对百分比误差

CVR

Constant Voltage Region 恒定电压区域

CCR

Constant Current Region 恒定电流区域

1. Introduction 1.导言

Renewable energy is widely recognized by esteemed institutions such as the International Energy Agency and the Renewable Energy Agency as a viable strategy for addressing the pressing issues of global warming and climate change. In recent years, particular attention has been directed towards solar energy, manifested through the utilization of solar Photovoltaic (PV) panels. The attributes of solar PV panels—including their adaptability, minimal reliance on moving components, and high dependability—render them highly applicable across a diverse range of scenarios. Nonetheless, despite these advantages, the successful integration of PV systems still presents substantial challenges.
国际能源机构和可再生能源机构等权威机构普遍认为，可再生能源是解决全球变暖和气候变化等紧迫问题的可行战略。近年来，人们特别关注太阳能，太阳能光伏（PV）板的利用就是其中之一。太阳能光伏板的特性，包括适应性强、对移动部件的依赖性最小以及可靠性高，使其在各种情况下都非常适用。然而，尽管有这些优势，光伏系统的成功集成仍然面临着巨大的挑战。

Prior to the deployment of PV systems, controlled testing and validation are essential prerequisites. However, a well-known hurdle arises from the fact that PV panels are intrinsically influenced by uncontrollable environmental factors [1], [2]. Consequently, the primary difficulty in testing these systems lies in the inability to impose a desired testing profile. In response to this challenge, the concept of photovoltaic emulation has emerged as a robust alternative for PV system testing. Photovoltaic emulators represent specialized power electronics systems designed to replicate both the static and dynamic attributes of solar panels [3].
在部署光伏系统之前，受控测试和验证是必不可少的先决条件。然而，众所周知的一个障碍是，光伏电池板本身会受到不可控环境因素的影响 [1]，[2]。因此，测试这些系统的主要困难在于无法施加理想的测试条件。为应对这一挑战，光伏仿真的概念应运而生，成为光伏系统测试的有力替代方案。光伏仿真器是专门的电力电子系统，旨在复制太阳能电池板的静态和动态属性[3]。

To effectively replicate the behavior of a solar panel, a Photovoltaic Emulator (PVE) necessitates both a reference PV model to establish the desired static response and a power stage controller to capture the intended dynamic characteristics of the PV panel. As a result, numerous contemporary PVE designs have been structured around these two pivotal components. It's important to note that for the sake of creating a straightforward and uncomplicated emulation test setup, these elements must inherently maintain simplicity.
为了有效复制太阳能电池板的行为，光伏模拟器（PVE）需要一个参考光伏模型来建立所需的静态响应，还需要一个功率级控制器来捕捉光伏电池板的预期动态特性。因此，许多当代 PVE 设计都围绕这两个关键组件展开。值得注意的是，为了创建一个简单明了的仿真测试装置，这些元件必须在本质上保持简洁。

As documented in various scientific literature, multiple approaches for referencing a PVE and ensuring reliable control have been devised. For instance, the PVE development in [4] employs a linearization strategy to generate a reference current for the PVE. This process involves decomposing the current–voltage (I-V) curve of the PV model into four segments, followed by the implementation of a Proportional-Integral (PI) controller to regulate the PVE's output based on the desired reference. However, it's worth noting that while this method is characterized by simplicity, it compromises the accuracy and dynamic response of the PVE due to the limited four-segment division of the I-V curve.
正如各种科学文献所记载的那样，人们设计了多种方法来参考 PVE 并确保可靠的控制。例如，[4] 中的 PVE 开发采用线性化策略为 PVE 生成参考电流。这一过程包括将光伏模型的电流-电压（I-V）曲线分解为四个部分，然后实施比例-积分（PI）控制器，根据所需的参考值调节 PVE 的输出。不过，值得注意的是，虽然这种方法的特点是简单，但由于 I-V 曲线的四段划分有限，因此会影响 PVE 的精度和动态响应。

To enhance the accuracy of the PVE, a higher number of I-V curve segments need to be integrated, as demonstrated in [5], [6]. Yet, achieving this increased accuracy introduces a higher level of complexity. Notably, [5], [6] relies heavily on an offline-trained artificial neural network (ANN), which poses challenges in real-time implementation. Integrating such models into the PVE introduces complexity to its structure, potentially overloading the computing processor and impeding the PVE's responsiveness.
如文献[5]和[6]所示，为了提高 PVE 的精度，需要集成更多的 I-V 曲线段。然而，要达到更高的精度，就需要引入更高的复杂度。值得注意的是，[5]、[6] 在很大程度上依赖于离线训练的人工神经网络 (ANN)，这给实时实施带来了挑战。将此类模型集成到 PVE 中会增加其结构的复杂性，可能会使计算处理器超载，并妨碍 PVE 的响应速度。

Similarly, an alternative method of providing reference voltage/current to the PVE is through the utilization of look-up-tables (LUTs) [7], [8], [9], [10], [11], [12], a practical approach widely adopted in PVE implementation. However, while feasible, this method heavily depends on pre-stored data and consequently places significant memory demands on the computing processor. This restricts the operating range of LUT-based PVEs to avoid computational burdens. Moreover, the LUT approach is specific to the PV panel, necessitating reconstruction whenever a new PV panel is introduced. Additionally, LUT-based PVEs suffer from the drawback that the equation calculating the I-V characteristics relies on linear interpolation between successive points in the table, compromising the overall accuracy of the PVE [13].
同样，为 PVE 提供参考电压/电流的另一种方法是利用查找表（LUT）[7]、[8]、[9]、[10]、[11]、[12]。然而，这种方法虽然可行，却严重依赖于预先存储的数据，因此对计算处理器的内存需求很大。这限制了基于 LUT 的 PVE 的操作范围，以避免计算负担。此外，LUT 方法只适用于光伏面板，因此每当引入一个新的光伏面板时，都必须进行重建。此外，基于 LUT 的 PVE 还存在一个缺点，即计算 I-V 特性的方程依赖于表中连续点之间的线性插值，从而影响了 PVE 的整体精度[13]。

An alternative avenue involves directly engaging with the I-V equations of the PV panel, presenting a distinct approach compared to the linearization and LUT methods. This approach promises superior reference accuracy compared to its predecessors. Consequently, several existing endeavors have embraced the direct incorporation of the PV model into the PVE framework [15], [16], [17]. Computation of the I-V equations of the PV panel to generate a reference voltage/current for the PVE can be executed through numerical techniques. Among these methods, the Newton-Raphson method has garnered considerable attention for solving PV equations. As demonstrated in [18], the voltage-resistor (V-R) equation of the PV model was calculated using the Newton-Raphson computation algorithm.
另一种方法是直接使用光伏板的 I-V 方程，与线性化和 LUT 方法相比，这是一种与众不同的方法。与前者相比，这种方法具有更高的参考精度。因此，现有的一些尝试已将 PV 模型直接纳入 PVE 框架 [15]、[16]、[17]。可以通过数值技术计算光伏板的 I-V 方程，为 PVE 生成参考电压/电流。在这些方法中，牛顿-拉夫逊法在求解 PV 方程方面受到了广泛关注。如文献 [18] 所示，光伏模型的电压-电阻 (V-R) 方程是通过牛顿-拉夫逊计算算法计算得出的。

While numerical computations might not impose significant strain on the processor, they necessitate multiple iterations to precisely emulate the solar panel. Unfortunately, these iterations introduce an adverse impact on the PVE's performance, leading to sluggish dynamic responses. In numerous testing scenarios, the PVE is required to operate within swiftly changing dynamic conditions. Consequently, the computations within the PVE structure must exhibit the agility to faithfully replicate both the dynamic and static characteristics of the solar panel. As a result, numerical computations might not consistently deliver the desired dynamic response expected from the PVE.
虽然数值计算可能不会对处理器造成很大的压力，但却需要多次迭代才能精确模拟太阳能电池板。遗憾的是，这些迭代会对 PVE 的性能产生不利影响，导致动态响应迟缓。在许多测试场景中，PVE 需要在快速变化的动态条件下运行。因此，PVE 结构中的计算必须表现出灵活性，以忠实地复制太阳能电池板的动态和静态特性。因此，数值计算可能无法始终如一地提供 PVE 所期望的动态响应。

Empirical evidence, as consistently indicated in [14], has revealed that using the current-resistor model of the PV, the Newton-Raphson algorithm can demand up to 200 iterations to converge toward a stable PVE output (see Fig. 1). This observation underscores the substantial challenge in liberating the PVE from iterative computations. Nevertheless, accomplishing such a feat would undoubtedly enhance the overarching PVE structure and its emulation performance significantly.
正如文献[14]中一致指出的那样，经验证据表明，使用光伏的电流电阻模型，牛顿-拉夫逊算法需要多达 200 次迭代才能收敛到稳定的 PVE 输出（见图 1）。这一观察结果凸显了将 PVE 从迭代计算中解放出来所面临的巨大挑战。尽管如此，完成这一壮举无疑将大大增强 PVE 的总体结构及其仿真性能。

Within the domain of numerical computations, a fresh approach was introduced by [19], presenting a novel strategy that combines the resistance comparison method (RCM) with the binary search computation (BSC) algorithm. This innovative approach achieves two notable outcomes. Firstly, the application of a resistance-based technique imbues the PVE with heightened output stability. Secondly, the utilization of the binary search computation algorithm significantly reduces the iteration requirements of the PVE. The study consistently exhibited that by adopting this approach, the computation iterations could be curtailed to 40, which represents a meaningful advancement. Nevertheless, the attainment of 40 iterations per computation only partially mitigates the iteration predicament faced by the PVE.
在数值计算领域，[19] 提出了一种新方法，将电阻比较法 (RCM) 与二进制搜索计算 (BSC) 算法相结合。这种创新方法取得了两个显著成果。首先，基于电阻技术的应用提高了 PVE 的输出稳定性。其次，二进制搜索计算算法的使用大大降低了 PVE 的迭代要求。研究一致表明，通过采用这种方法，计算迭代次数可减少到 40 次，这是一个有意义的进步。不过，每次计算迭代 40 次只能部分缓解 PVE 面临的迭代困境。

A refinement of this approach was put forth in [14], wherein the BSC algorithm was harnessed to efficiently calculate the current-resistor (I-R) model of the PVE. This improved methodology effectively lessened the iterative computation burden, attaining a reduced iteration count of less than 20. As corroborated by the literature and illustrated in Fig. 1, in terms of simplicity the BSC algorithm stands out as the most successful computation technique in the context of PVE implementation. However, it is evident that even with the BSC algorithm, a certain number of iterations remain necessary to attain accurate PVE outputs. Thus, the challenge of iterations remains a pertinent research gap that demands attention in order to augment the comprehensive performance of a PVE.
文献[14]对这种方法进行了改进，利用 BSC 算法有效地计算了 PVE 的电流-电阻（I-R）模型。这种改进的方法有效地减轻了迭代计算的负担，使迭代次数减少到 20 次以下。正如文献所证实和图 1 所示，就简便性而言，BSC 算法是 PVE 实现中最成功的计算技术。不过，很明显，即使采用 BSC 算法，要获得准确的 PVE 输出，仍然需要一定数量的迭代。因此，为了提高 PVE 的综合性能，迭代挑战仍然是一个需要关注的相关研究空白。

Regarding the control aspect, which stands as a pivotal component of the PVE, the Proportional-Integral (PI) controller occupies a prominent position as the most widely employed control strategy for PVEs. As indicated in Table 1, numerous existing PVE architectures are founded on the PI approach. The appeal of this control solution lies in its simplicity and adaptability, making it a preferred choice. Table 2.
在作为 PVE 重要组成部分的控制方面，比例-积分（PI）控制器作为 PVE 最广泛采用的控制策略占据了重要地位。如表 1 所示，许多现有的 PVE 架构都建立在 PI 方法的基础上。这种控制方案的吸引力在于其简单性和适应性，因此成为首选。表 2.

Table 1. A summary of some PVEs in the literature illustrating contemporary research gaps.
表 1.说明当代研究差距的一些 PVE 文献摘要。

Ref 参考文献	PVE Computation method PVE 计算方法	Power stage Control method 功率级控制方法	Key observation and remarks 主要意见和评论
[4]	Four segment Linearization 四段线性化	PI Controller and Buck converter PI 控制器和降压转换器	- Low accuracy introduced by linearization 线性化带来的低精度 - Low dynamic response 低动态响应
[5]	Four segment Linearization 四段线性化	PI Controller and Synchronous Buck converter PI 控制器和同步降压转换器	- Moderate accuracy due to linearization 由于线性化，精度适中 - Low dynamic response 低动态响应 - Complex due to integration of ANN model 因集成了 ANN 模型而变得复杂
[6]	Twelve segment Linearization 12 段线性化	Backstepping controller and Buck converter 反步进控制器和降压转换器	- High dynamic response due to nonlinear controller 非线性控制器带来高动态响应 - High complexity due to reliance on ANN 由于依赖 ANN，复杂性较高 - Control complication due to intricate control law 错综复杂的控制规律导致控制复杂化
[9]	LUT method LUT 方法	Adaptive state feedback controller and Buck Converter 自适应状态反馈控制器和降压转换器	- Lower control complexity compared to Backstepping 与反向步法相比，控制复杂度更低 - Narrow operating range due to limitations of LUT 由于 LUT 的限制，工作范围较窄 - High memory requirements imposed by LUT LUT 对内存要求较高
[15]	Direct PV model integration 直接光伏模型集成	PI Controller PI 控制器	- Moderate dynamic response due to slow computation 由于计算速度缓慢，动态响应适中 - High computational burden 高计算负担
[18]	V-R and Newton-Raphson’s method V-R 法和牛顿-拉斐逊法	Backstepping controller and Buck 反步进控制器和降压	- High dynamic response due to nonlinear controller 非线性控制器带来高动态响应 - Complicated control law 复杂的管制法 - Complex numerical computation involved by Newton Raphson 牛顿-拉斐尔森所涉及的复杂数值计算
[19]	RCM and BSC method RCM 和 BSC 方法	PI Controller PI 控制器	- Lower number of iterations required 所需迭代次数减少 - High dynamic response 高动态响应 - Iterations affect the performance of PI controller 迭代影响 PI 控制器的性能
[14]	I-R model and BSC method I-R 模型和 BSC 方法	PI controller PI 控制器	- Lowest number of iterations required 所需的最少迭代次数 - High dynamic response 高动态响应 - Iterations limit the performance of PI controller 迭代限制了 PI 控制器的性能
[20]	I-R model and BSC method I-R 模型和 BSC 方法	Shift Controller 轮班控制器	- Lowest number of iterations required 所需的最少迭代次数 - Very High dynamic response 极高的动态响应 - Fast computation PVE despite iterations 迭代后仍能快速计算 PVE

Table 2. Characteristics of the KC200GT solar panel at STC [24].
表 2.STC 的 KC200GT 太阳能电池板的特性 [24]。

Parameter 参数	Value 价值
Maximum, power 最大功率	200 W 200 W
Open circuit voltage, $V_{o c}$ 开路电压， $V_{o c}$	32.9 V 32.9 V
Short circuit current, $I_{s c}$ 短路电流， $I_{s c}$	8.21A
Maximum power voltage, $V_{m p}$ 最大电源电压， $V_{m p}$	26.3 V
Maximum power current, $I_{m p}$ 最大功率电流， $I_{m p}$	7.61A
Temperature coefficient of $V_{o c}, K_{v}$ $V_{o c}, K_{v}$ 的温度系数	−0.123 V/^oC -0.123 V/ ^o C
Temperature coefficient of $I_{s c}, K_{i}$ $I_{s c}, K_{i}$ 的温度系数	0.00318 A/^oC
Number of series cells 串联电池数量	54

The literature also highlights the emergence of advanced control methods, such as adaptive state feedback and nonlinear backstepping controllers, as potential alternative options for PVE control. However, the complexity associated with these controllers remains a significant hindrance to their practical implementation. Notably, the nonlinear backstepping controller demands full access to the system's states, a requirement that might not always be met for accurate measurements. Hence, the simplicity of the control stage holds vital importance for an enhanced PVE structure.
文献还强调了先进控制方法的出现，如自适应状态反馈和非线性反步进控制器，作为 PVE 控制的潜在替代选择。然而，与这些控制器相关的复杂性仍然是其实际应用的一大障碍。值得注意的是，非线性反步进控制器要求完全访问系统状态，而这一要求并不总能满足精确测量的要求。因此，控制阶段的简单性对于增强 PVE 结构至关重要。

Indeed, the studies conducted by [14], [19], [21] consistently affirm that the PI controller is highly suitable for PVE applications, especially when the reference computation is refined from an iterative perspective. Additionally, the Shift controller, introduced in [20], has been identified as a potent control strategy for PVEs. This controller is notable for its simplicity and rapid control solutions. The findings presented in [20] demonstrate that the Shift controller yields quicker computations compared to the PI alternative based on the Binary Search Computation (BSC) algorithm.
事实上，[14]、[19]、[21] 所做的研究一致肯定了 PI 控制器非常适合 PVE 应用，尤其是从迭代角度完善参考计算时。此外，[20] 提出的 Shift 控制器也被认为是 PVE 的有效控制策略。该控制器以其简单性和快速控制解决方案而著称。文献 [20] 中的研究结果表明，与基于二进制搜索计算 (BSC) 算法的 PI 替代方案相比，Shift 控制器的计算速度更快。

As uncovered through an in-depth exploration of existing literature, the incorporation of the PV model into the PVE utilizing numerical methods is susceptible to computational iterations. This tendency places limitations on both the static and dynamic performance of the PVE. Consequently, the persistent issue of iterations within the PVE persists as a noteworthy challenge and stands out as a substantial gap in current research. To effectively address such gaps, this paper introduces an innovative and efficient PVE that seamlessly integrates a straightforward referencing scheme.
通过对现有文献的深入研究发现，利用数值方法将 PV 模型纳入 PVE 时，容易出现计算迭代。这种趋势对 PVE 的静态和动态性能都造成了限制。因此，PVE 中持续存在的迭代问题一直是一个值得注意的挑战，也是当前研究中的一个重大空白。为了有效解决这些差距，本文介绍了一种创新而高效的 PVE，它无缝集成了一种直接的引用方案。

Given that the equations governing the PV system are nonlinear and intricate, employing numerical computations would unavoidably introduce iterations into the PVE system, which is far from ideal. To circumvent the burden of these multiple iterative processes, this study establishes and seamlessly integrates an explicit PV model (EPVM) into the PVE architecture. Rigorous demonstrations consistently indicate that the incorporation of the EPVM into the PVE framework yields a high degree of simplicity and flexibility. As a result, the proposed PVE distinguishes itself from existing counterparts by eliminating the need for multiple computational iterations, significantly reducing computation time.
鉴于管理光伏系统的方程是非线性且错综复杂的，采用数值计算将不可避免地在 PVE 系统中引入迭代，这是不理想的。为了避免多重迭代过程带来的负担，本研究建立了一个显式光伏模型（EPVM），并将其无缝集成到 PVE 架构中。严格的论证一致表明，将 EPVM 纳入 PVE 框架具有高度的简便性和灵活性。因此，所提出的 PVE 与现有的同类产品不同，无需进行多次计算迭代，大大减少了计算时间。

The comprehensive design of the proposed PVE is synergistically aligned with a Buck converter and PI controller, and its efficacy is verified through rigorous MATLAB/Simulink-based numerical investigation conducted on a 200 W solar system. Multiple iterations of testing, bolstered by comparisons conducted across various operational scenarios, firmly establish the competence of the proposed PVE. The primary contributions of this paper can be succinctly summarized as follows:
拟议 PVE 的综合设计与降压转换器和 PI 控制器协同配合，其功效通过在 200 W 太阳能系统上进行的基于 MATLAB/Simulink 的严格数值研究得到了验证。多次迭代测试以及在各种运行情况下进行的比较，牢固确立了拟议 PVE 的能力。本文的主要贡献可简明扼要地概括如下：

• -
Innovative incorporation of a clear and uncomplicated explicit PV model into the framework of the PVE.
在 PVE 框架中创新性地纳入一个清晰、简洁的明确 PV 模型。
• -
Distinguishing feature of the proposed PVE: absence of multiple computational iterations.
拟议 PVE 的显著特点：无需多次计算迭代。
• -
Noteworthy attributes of heightened flexibility and performance enhancement in the envisioned PVE architecture.
在设想的 PVE 架构中提高灵活性和性能的显著特征。
• -
Through comparison with PVEs based on the BSC algorithm-driven PI and shift controllers, the superiority of the proposed PVE is unmistakably established.
通过与基于 BSC 算法驱动的 PI 控制器和移位控制器的 PVE 进行比较，明确无误地确定了所提出的 PVE 的优越性。
• -
A prominent characteristic of the proposed PVE is its exceptional suitability for testing Maximum Power Point Tracking (MPPT) systems.
提议的 PVE 的一个突出特点是，它非常适合测试最大功率点跟踪 (MPPT) 系统。

The subsequent sections of this paper are structured as follows: In Section 2, an overview of the proposed PVE system is presented, followed by the mathematical depiction of the PV panel in Section 3. The explicit PV panel model, as well as its seamless incorporation into the PVE, is expounded upon in Section 4. The design specifics of the PI controller and the Buck converter are detailed in Section 5, which is succeeded by the presentation of the MPPT test arrangement in Section 6. Section 7 delves into the presentation and discourse of the outcomes, while the paper concludes in Section 8.
本文后续章节的结构如下：第 2 节概述了拟议的 PVE 系统，第 3 节介绍了光伏面板的数学描述。第 4 节阐述了显式光伏板模型及其与 PVE 的无缝结合。第 5 节详细介绍了 PI 控制器和降压转换器的设计细节，第 6 节介绍了 MPPT 测试安排。第 7 节是对结果的介绍和讨论，第 8 节是本文的结论。

2. Synoptic description of the proposed PVE structure
2.拟议 PVE 结构的综合说明

Illustrated in Fig. 2, the envisaged PVE comprises three essential components: a power stage encompassing a DC power supply and a Buck converter, an explicit PV model, and a power stage controller operating in the form of a PI controller. Operating as a closed-loop system, this setup generates the system's reference signal via the explicit PV model, while the PVE's output current and voltage serve as the feedback measurements within the closed-loop arrangement. The discrepancy between the reference and the feedback measurement is rectified by the synergistic interplay between the PI controller and the Buck converter, persisting until the sought-after static and dynamic responses are successfully attained. The development and integration of an explicit PV model into a PVE structure (see Fig. 2), represents the prime technical contribution brought in by this paper, which will be fully established in section 4.
如图 2 所示，设想中的 PVE 包括三个基本组件：由直流电源和降压转换器组成的功率级、明确的 PV 模型以及以 PI 控制器形式运行的功率级控制器。作为一个闭环系统，该装置通过显式光伏模型生成系统的参考信号，而 PVE 的输出电流和电压则作为闭环布置中的反馈测量值。通过 PI 控制器和降压转换器之间的协同作用，纠正参考测量和反馈测量之间的差异，直至成功实现所需的静态和动态响应。将显式光伏模型开发并集成到 PVE 结构中（见图 2），是本文的主要技术贡献，这将在第 4 节中全面阐述。

3. Modelling of solar panel
3.太阳能电池板建模

The model of the PV panel is an important element for a PVE. Thus prior to assessing the PVE, the model of the panel under emulation must be accurate. Several models of the solar panel have been established in the literature [22], [23]. In this study, we use the single diode model (SDM). This model is specially chosen for is ability to balance model accuracy and complexity. The electrical structure of this model is presented in Fig. 3. It is made of a photocurrent source

I_{ph}

, a diode

D_{1}

that models the nonlinearity of the PV cell, parasitic shunt and series resistance

R_{sh}

, and

R_{s}

respectively. Under a uniform operation regime, the current

I_{pv}

and voltage

V_{pv}

of the PV cell obeys the mathematical relationship in Eq. (1).(1)

I_{pv} = I_{ph} - I_{s} [\exp (\frac{(V_{pv} + I_{pv} R_{s})}{a}) - 1] - \frac{V_{pv} + I_{pv} R_{s}}{R_{p}}

Where

a = n N_{s} K_{B} T / q

. The terms

I_{ph}

I_{s}

are respectively the photocurrent and the saturation current of the diode. Additionally,

n

is the diode ideality factor,

N_{s}

is the number of cells,

k

is the Boltzmann’s constant having the value

1.381 \times 10^{- 23} J / K

q

is the charge constant

q = 1.602 \times 10^{- 19} C

, and

T

is the temperature of the cell in kelvin. The complete modelling of the solar panel requires five key parameters [24] namely

I_{ph}

I_{s}, n, R_{s}

, and

R_{p}

. The first two parameters can be obtained from the following equation:(2)

I_{ph} = \frac{G}{1000} (I_{ph, r e f} + K_{i} (T - T_{ref})

(3)

I_{s} = I_{s, r e f} {(\frac{T}{T_{ref}})}^{3} e x p [\frac{q}{k} (\frac{E_{g, r e f}}{T_{ref}} - \frac{E_{g}}{T})], E_{g} = E_{g, r e f} (1 - 0.000267 (T - T_{ref})

Where G is the incident solar irradiance in

W / m^{2}

I_{ph - r e f}

is the photocurrent at standard test conditions (STC) i.e. 100 W/m² and 25 °C, K_i is the temperature coefficient of short circuit current,

E_{g}

is the band-gap energy which has the reference value

E_{g, r e f} = 1.12 e V

for silicon material. Additionally, the variation of the shunt resistance with irradiance G can be modeled mathematically as:(4)

R_{p} = R_{p, r e f} (\frac{1000}{G})

Where

R_{p, r e f}

is the shunt resistance at STC, and

R_{p}

is any arbitrary value of the shunt at given irradiance
光伏电池板的模型是 PVE 的重要元素。因此，在评估 PVE 之前，被模拟的电池板模型必须准确无误。文献[22]、[23]中已经建立了多个太阳能电池板模型。在本研究中，我们使用单二极管模型（SDM）。之所以选择该模型，是因为它能够在模型精度和复杂性之间取得平衡。该模型的电气结构如图 3 所示。它由一个光电流源

I_{ph}

、一个模拟光伏电池非线性的二极管

D_{1}

、寄生并联电阻

R_{sh}

和串联电阻

R_{s}

组成。在均匀工作状态下，光伏电池的电流

I_{pv}

和电压

V_{pv}

遵循公式 (1) 中的数学关系。 (1)

I_{pv} = I_{ph} - I_{s} [\exp (\frac{(V_{pv} + I_{pv} R_{s})}{a}) - 1] - \frac{V_{pv} + I_{pv} R_{s}}{R_{p}}

式中

a = n N_{s} K_{B} T / q

。项

I_{ph}

、

I_{s}

分别为二极管的光电流和饱和电流。此外，

n

是二极管理想系数，

N_{s}

是电池数量，

k

是波尔兹曼常数，其值为

1.381 \times 10^{- 23} J / K

，

q

是电荷常数

q = 1.602 \times 10^{- 19} C

，

T

是电池温度（开尔文）。太阳能电池板的完整建模需要五个关键参数 [24]，即

I_{ph}

、

I_{s}, n, R_{s}

和

R_{p}

。前两个参数可通过下式求得： (2)

I_{ph} = \frac{G}{1000} (I_{ph, r e f} + K_{i} (T - T_{ref})

(3)

I_{s} = I_{s, r e f} {(\frac{T}{T_{ref}})}^{3} e x p [\frac{q}{k} (\frac{E_{g, r e f}}{T_{ref}} - \frac{E_{g}}{T})], E_{g} = E_{g, r e f} (1 - 0.000267 (T - T_{ref})

其中，G 是以

W / m^{2}

为单位的入射太阳辐照度；

I_{ph - r e f}

是标准测试条件 (STC) 即 100 W/m ² 和 25 °C 时的光电流；K 是短路电流的温度系数；

E_{g}

是带隙能，其参考值为硅材料的

E_{g, r e f} = 1.12 e V

。此外，并联电阻随辐照度 G 变化的数学模型为 (4)

R_{p} = R_{p, r e f} (\frac{1000}{G})

其中，

R_{p, r e f}

为 STC 时的分流电阻，

R_{p}

为给定辐照度下分流电阻的任意值。

Therefore, SDM require the identification of five parameters as previously highlighted. Several intelligent optimization methods have been covered in existing literatures on parameter identification of solar models [24], [25], [26], [27], [28]. In this work we employ the whale optimization algorithm (WOA) to optimally identify these parameters [29]. The WOA has been established as a successful optimization algorithm, especially faced with implicit equations of the PV nature [30], [31]. Throughout this paper, a 200 W solar panel of the type KC200GT is consistently adopted.
因此，如前所述，SDM 需要识别五个参数。关于太阳能模型的参数识别，现有文献中已有多种智能优化方法 [24]、[25]、[26]、[27]、[28]。在这项工作中，我们采用鲸鱼优化算法（WOA）来优化识别这些参数[29]。鲸鱼优化算法已被认为是一种成功的优化算法，尤其是在面对光伏性质的隐式方程时[30]、[31]。本文始终采用 KC200GT 型 200 W 太阳能电池板。

The five parameters of the KC200GT solar panel obtained through WOA are found to be

I_{ph, r e f} = 8.2288 A, I_{s . r e f} = 2.3246 \times 10^{- 10}, n = 0.97736 . R_{s} = 0.34483, R_{p} = 150.6921

. A plot of the complete model curves and data from the manufacturer of the solar panel as illustrated in Fig. 4, reveals a very strong agreement. This endorses the suitability of the available model. Thus this model will be consistently adopted throughout the remainder of this paper.
通过 WOA 获得的 KC200GT 太阳能电池板的五个参数为

I_{ph, r e f} = 8.2288 A, I_{s . r e f} = 2.3246 \times 10^{- 10}, n = 0.97736 . R_{s} = 0.34483, R_{p} = 150.6921

。如图 4 所示，完整的模型曲线与太阳能电池板制造商提供的数据非常吻合。这证明了现有模型的适用性。因此，本文其余部分将始终采用该模型。

4. Explicit PV model based integration into the PVE structure
4.将基于 PV 模型的显式集成纳入 PVE 结构

The equation presented in Eq. (1) serves as evidence that the equations governing the PV model are not solely nonlinear but also intricate in nature. Consequently, resolving this equation numerically invariably leads to multiple iterations during the computation process. The introduction of an explicit rendition of this equation is a concrete solution to circumvent the challenges posed by iterative computations inherent in numerical methods. To address this concern, this paper introduces an explicit PV model (EPVM). The ensuing steps outline the process of constructing the EPVM.
式 (1) 中的方程表明，光伏模型的控制方程不仅是非线性的，而且性质复杂。因此，在计算过程中，对该方程进行数值求解必然会导致多次迭代。引入该方程的显式演绎是规避数值方法固有的迭代计算带来的挑战的具体解决方案。为了解决这一问题，本文引入了显式光伏模型（EPVM）。接下来的步骤将概述构建 EPVM 的过程。

A close observation of Eq. (1) confirms that the inherent nonlinearity and intricacy of the PV equation comes from the exponential term

e x p (\frac{I_{pv} R_{s}}{a})

. The proposed EPVM in this paper is based on the fact that fact that a third order polynomial approximation of this term losses the intricacy of the equation. Thus, using Taylor’s theory, this approximation can be obtained as follow:(5)

e^{x} \approx 1 + x + \frac{1}{2!} x^{2} + \frac{1}{3!} x^{3}, x = \frac{I_{pv} R_{s}}{a}

Therefore, based on the above Equation, the nonlinear intricate relation of Eq. (1) is reduced to an implicit and straightforward polynomial form as follows:(6)

a_{1} x^{3} + b x^{2} + c x + d = 0

(7)

\{\begin{matrix} a_{1} = \frac{1}{6} I_{s} e x p (\frac{V_{pv}}{a}), \\ b = \frac{1}{2} I_{s} e x p (\frac{V_{pv}}{a}) \\ c = I_{s} e x p (\frac{V_{pv}}{a}) + a (\frac{1}{R_{s}} + \frac{1}{R_{p}}) \\ d = I_{s} e x p (\frac{V_{pv}}{a}) + \frac{V_{pv}}{R_{p}} - I_{ph} - I_{s} \end{matrix}

Where the polynomial coefficients

a_{1}, b, c,

and

d

of Eq. (6) are defined according to Eq.(7). In this realm, it is evident that Eq. (6) allows for a direct determination of x, that is subsequently inserted into the relation

x = \frac{I_{pv} R_{s}}{a}

to obtain

I_{pv}

. The third order polynomial in Eq. (6) will yield a real solution if the condition

B^{2} - 4 A C > 0

is satisfied [32], where

A = b^{2} - 3 a_{1} c, B = b c - 9 a_{1} d, C = c^{2} - 3 b d

. It is evident that this condition will always be fulfilled given that

x

always admits to a real definite value. Also, because its value will always be unique, the following solution of Eq. (6) must be obtained:(8)

x = \frac{- b - \sqrt[3]{M_{1}} - \sqrt[3]{M_{2}}}{3 a_{1}}

(9)

\{\begin{matrix} M_{1} = A b + \frac{3 a_{1} (- B + \sqrt{B_{2} - 4 A C}}{2} \\ M_{2} = A b + \frac{3 a_{1} (- B - \sqrt{B_{2} - 4 A C}}{2} \end{matrix}

Where the terms

M_{1},

and

M_{2}

in Eq. (8) are computed using Eq. (9). Using Eq. (9) and the expression

x = \frac{I_{pv} R_{s}}{a}

, a direct equation of the PV current is expressed as:(10)

I_{pv} = a \frac{- b - \sqrt[3]{M_{1}} - \sqrt[3]{M_{2}}}{3 a_{1} R_{s}}

In this context, Eq. (10) constitutes the explicit PV model (EPVM) for the solar panel. Integrating the EPVM is consequently accomplished by directly incorporating this expression into the PVE framework. It's noteworthy that the EPVM relies on the five parameters of the SDM. The simplicity of the EPVM is further accentuated as these parameters can be easily derived from the available accurate SDM and conveniently inputted into the EPVM equation within the PVE structure. Thus, within the PVE architecture, given the provision of inputs T and G, along with the availability of feedback voltage from the PVE, a reference current can be explicitly supplied to the controller employing the subsequent expression:(11)

I_{ref} = f (G, T, V_{pve}) = a \frac{- b - \sqrt[3]{M_{1}} - \sqrt[3]{M_{2}}}{3 a_{1} R_{s}}

Where

V_{pve}

is the output voltage of the PVE that is feedback into the system as input. To emulate conditions different from STC, the explicit model consistently uses the straightforward relations in Eqs. (2), (3), (4). Thus it is evident that referencing the PVE using the proposed EPVM follows a straightforward process that simplifies its integration into the PVE structure. More specifically, computation of the PV based on the EPVM requires just one iteration. Thus there is no need for multiple iterations
仔细观察公式 (1) 可以发现，光伏方程固有的非线性和复杂性来自指数项

e x p (\frac{I_{pv} R_{s}}{a})

。本文提出的 EPVM 基于这样一个事实，即对该项进行三阶多项式近似会减弱方程的复杂性。因此，利用泰勒理论，可以得到如下近似值： (5)

e^{x} \approx 1 + x + \frac{1}{2!} x^{2} + \frac{1}{3!} x^{3}, x = \frac{I_{pv} R_{s}}{a}

因此，根据上述等式，式 (1) 的非线性错综复杂关系可简化为如下隐式和直接多项式形式： (6)

a_{1} x^{3} + b x^{2} + c x + d = 0

(7)

\{\begin{matrix} a_{1} = \frac{1}{6} I_{s} e x p (\frac{V_{pv}}{a}), \\ b = \frac{1}{2} I_{s} e x p (\frac{V_{pv}}{a}) \\ c = I_{s} e x p (\frac{V_{pv}}{a}) + a (\frac{1}{R_{s}} + \frac{1}{R_{p}}) \\ d = I_{s} e x p (\frac{V_{pv}}{a}) + \frac{V_{pv}}{R_{p}} - I_{ph} - I_{s} \end{matrix}

其中，式 (6) 的多项式系数

a_{1}, b, c,

和

d

是根据式 (7) 定义的。在这种情况下，很明显式 (6) 可以直接确定 x，然后将其插入关系式

x = \frac{I_{pv} R_{s}}{a}

中，得到

I_{pv}

。如果满足条件

B^{2} - 4 A C > 0

，公式 (6) 中的三阶多项式将得到实解 [32]，其中

A = b^{2} - 3 a_{1} c, B = b c - 9 a_{1} d, C = c^{2} - 3 b d

。很明显，鉴于

x

总是实数定值，这个条件将始终得到满足。此外，由于其值总是唯一的，因此必须得到公式 (6) 的以下解： (8)

x = \frac{- b - \sqrt[3]{M_{1}} - \sqrt[3]{M_{2}}}{3 a_{1}}

(9)

\{\begin{matrix} M_{1} = A b + \frac{3 a_{1} (- B + \sqrt{B_{2} - 4 A C}}{2} \\ M_{2} = A b + \frac{3 a_{1} (- B - \sqrt{B_{2} - 4 A C}}{2} \end{matrix}

其中，式 (8) 中的项

M_{1},

和

M_{2}

是通过式 (9) 计算得出的。利用公式 (9) 和

x = \frac{I_{pv} R_{s}}{a}

表达式，光伏电流的直接方程表示为 (10)

I_{pv} = a \frac{- b - \sqrt[3]{M_{1}} - \sqrt[3]{M_{2}}}{3 a_{1} R_{s}}

在这种情况下，式 (10) 构成了太阳能电池板的显式光伏模型 (EPVM)。因此，通过将该表达式直接纳入 PVE 框架，就可以完成对 EPVM 的整合。值得注意的是，EPVM 依赖于 SDM 的五个参数。 EPVM 的简易性进一步凸显，因为这些参数可以轻松地从可用的精确 SDM 中导出，并方便地输入 PVE 结构中的 EPVM 方程。因此，在 PVE 结构中，考虑到输入 T 和 G 的提供，以及 PVE 反馈电压的可用性，可通过后续表达式向控制器明确提供参考电流： (11)

I_{ref} = f (G, T, V_{pve}) = a \frac{- b - \sqrt[3]{M_{1}} - \sqrt[3]{M_{2}}}{3 a_{1} R_{s}}

其中

V_{pve}

是作为输入反馈到系统中的 PVE 输出电压。为了模拟与 STC 不同的条件，显式模型始终使用公式 (2)、(3) 和 (4) 中的直接关系。由此可见，使用建议的 EPVM 参考 PVE 的过程简单明了，可简化其与 PVE 结构的整合。更具体地说，根据 EPVM 计算 PV 只需一次迭代。因此无需多次迭代

5. Buck converter and PI control design
5.降压转换器和 PI 控制设计

As illustrated in the synopsis of Fig. 1, the power stage and controller are two other key elements of the PVE. To ensure an accurate static and dynamic capturing of the actual solar panel, these subsystems must be properly designed. As was previously highlighted, this study uses a DC-DC buck converter in its power stage. This paper specially choses this converter topology for its simple structure, versatility and fewer number of component. The electrical structure of a practical Buck converter is presented in Fig. 5. It consist of two storage components

L_{pve},

and

C_{pve}

that is the PVE inductance and capacitance. The system is designed in continuous conduction mode, ensuring continuous flow of current. The converter is regulated using the switch sw at a switching frequency

f_{pve}

through a pulse width modulation technique. Furthermore, the converter is powered by a fixed DC-source E. The diode

D_{2}

completes the inner structure of the converter. In addition, the parasitic components in the inductance and capacitance that is

r_{L},

and

r_{c}

renders the converter practical. The converter is coupled with a load that could be resistive, in that case

R_{pve}

. The small signal transfer function (T.F) of the Buck converter can be obtained as [14]:(12)

G_{b} (s) = \frac{1 / L_{pve} C_{pve}}{s^{2} + (1 / C_{pve} R_{pve}) s + 1 / C_{pve} L_{pve}}

如图 1 的概要所示，功率级和控制器是 PVE 的另外两个关键要素。为确保准确捕捉实际太阳能电池板的静态和动态，必须对这些子系统进行适当设计。如前所述，本研究在功率级中使用了直流-直流降压转换器。本文特别选择了这种拓扑结构的转换器，因为它结构简单、用途广泛且元件数量较少。实用降压转换器的电气结构如图 5 所示。它由两个存储元件

L_{pve},

和

C_{pve}

组成，即 PVE 电感和电容。系统设计为连续导通模式，确保电流连续流动。转换器通过脉宽调制技术，利用开关 sw 以

f_{pve}

的开关频率进行调节。此外，转换器由固定的直流电源 E 供电，二极管

D_{2}

完善了转换器的内部结构。此外，电感和电容中的寄生元件

r_{L},

和

r_{c}

使转换器变得实用。在这种情况下，

R_{pve}

与转换器耦合的负载可能是电阻性的。降压转换器的小信号传递函数（T.F）可求得 [14]： (12)

G_{b} (s) = \frac{1 / L_{pve} C_{pve}}{s^{2} + (1 / C_{pve} R_{pve}) s + 1 / C_{pve} L_{pve}}

In order to regulate the output of the converter, a simple PI controller is employed. Because the controller is inherently linear, its gains

K_{p},

and

K_{i}

must be properly designed. The operation of the PVE as a close loop system can be visualized in Fig. 6. Therefore, to accurately design the controller, the close loop system can be subject to a step-response in reference input and the gains regulated using a tuning method.
为了调节转换器的输出，采用了一个简单的 PI 控制器。由于控制器本身是线性的，因此必须正确设计其增益

K_{p},

和

K_{i}

。图 6 显示了 PVE 作为闭环系统的运行情况。因此，为了准确设计控制器，可以对闭环系统的参考输入进行阶跃响应，并使用调整方法调节增益。

The structure of the proposed PVE is presented in Fig. 7. It is clearly seen that the PVE requires two external inputs and two other internal inputs. The external inputs are G and T, set by the PVE operator. The PVE feedback measurements

V_{pve},

and

I_{pve}

represent the internal inputs to the PVE. Therefore, from a practical point of view, the proposed PVE requires just two measurement sensors that is current and voltage.
拟议的 PVE 结构如图 7 所示。可以清楚地看到，PVE 需要两个外部输入和另外两个内部输入。外部输入为 G 和 T，由 PVE 运算符设置。PVE 的反馈测量

V_{pve},

和

I_{pve}

代表 PVE 的内部输入。因此，从实用角度来看，拟议的 PVE 只需要两个测量传感器，即电流和电压。

6. MPPT application of the proposed PVE
6.拟议 PVE 的 MPPT 应用

The straightforward functioning of the Power Voltage Emulator (PVE) is attributed to the linear nature of a resistive load. However, this simplicity can be challenged when the PVE is connected to a load exhibiting nonlinearity. When confronted with such a nonlinear load, the PVE must maintain both its dynamic and static characteristics to effectively replicate the behavior of a solar panel. In photovoltaic (PV) connected setups, various nonlinear processes come into play during operation. Among these processes, a critical component is the maximum power point tracking (MPPT) controller, responsible for optimizing power transfer from the PV to the load [1], [2].
功率电压仿真器 (PVE) 的直接功能归功于电阻负载的线性特性。然而，当 PVE 连接到非线性负载时，这种简单性就会受到挑战。在面对这种非线性负载时，PVE 必须保持其动态和静态特性，才能有效地复制太阳能电池板的行为。在光伏（PV）连接装置中，运行期间会出现各种非线性过程。在这些过程中，最大功率点跟踪 (MPPT) 控制器是一个关键组件，负责优化从光伏到负载的功率传输 [1]，[2]。

The configuration of the proposed system, which couples the PVE with the MPPT controller, is depicted in Fig. 8. Here, the MPPT controller interacts with a nonlinear load, represented by a DC-DC boost converter driven by an MPPT algorithm. Additionally, the specific Boost converter arrangement utilized in this study is shown in Fig. 9. As seen, it features two main elements required for the effective implementation of an MPPT i.e. the boost inductance

L_{boost},

and capaticnace

C_{boost}

. Additionally, the input PVE-MPPT coupling capacitance

C_{in}

is used to efficiently couple the PVE with the MPPT subsystem. In steady operation, the resistance seen by the PVE is modified by the boost converter according to the duty cycle as follows:(13)

R_{pve} = R_{o} {(1 - d)}^{2}

A plot of the PVE resistance

R_{pve}

against the MPPT’s duty cycle for different values of output resistance

R_{o},

is presented in Fig. 11. It can be seen that the resistance seen by the PVE in an MPPT configuration is nonlinear. Such nonlinearity presents more emulation challenges as opposed to a scenario when the PVE is coupled with a linear resistive load
图 8 描述了将 PVE 与 MPPT 控制器耦合在一起的拟议系统配置。在这里，MPPT 控制器与非线性负载相互作用，非线性负载由 MPPT 算法驱动的直流-直流升压转换器表示。此外，本研究中使用的升压转换器的具体布置如图 9 所示。从图中可以看出，它具有有效实施 MPPT 所需的两个主要元素，即升压电感

L_{boost},

和电容

C_{boost}

。此外，输入 PVE-MPPT 耦合电容

C_{in}

用于将 PVE 与 MPPT 子系统有效耦合。在稳定运行时，升压转换器根据占空比对 PVE 看到的电阻进行如下修改： (13)

R_{pve} = R_{o} {(1 - d)}^{2}

图 11 显示了不同输出电阻

R_{o},

值下 PVE 电阻

R_{pve}

与 MPPT 占空比的关系图。可以看出，在 MPPT 配置中，PVE 看到的电阻是非线性的。与 PVE 与线性电阻负载耦合的情况相比，这种非线性带来了更多的仿真挑战

Hence, the operation of the PVE-MPPT system hinges entirely on the equation designated as (12), where the MPPT's duty cycle is controlled through a well-suited MPPT algorithm. Among the various options, the perturb and observe (P&O) algorithm, renowned for its popularity, emerges as an appropriate choice for this context. The core principle of the P&O algorithm involves adjusting the converter's duty cycle in accordance with Eq. (13), iteratively pursuing the tracking of the emulator's maximum power point. To accomplish this with efficiency, the PVE's current and voltage need to be sampled at an optimal perturbation interval, allowing the MPPT controller to discern between consecutive perturbations. The process flow of the P&O algorithm, enhanced with a sample and hold function, is visually represented in Fig. 10.
因此，PVE-MPPT 系统的运行完全取决于 (12) 等式，其中 MPPT 的占空比是通过一种合适的 MPPT 算法来控制的。在各种选择中，以其受欢迎程度而闻名的扰动和观测（P&O）算法是最合适的选择。P&O 算法的核心原理是根据公式 (13) 调整转换器的占空比，反复跟踪仿真器的最大功率点。为了高效实现这一目标，需要以最佳扰动间隔对 PVE 的电流和电压进行采样，使 MPPT 控制器能够分辨连续扰动。图 10 直观地展示了采用采样和保持功能的 P&O 算法的流程。

7. Results and discussions
7.结果和讨论

A series of simulations and experiments conducted in MATLAB/Simulink environment are conducted to assess the efficacy of the proposed PVE. All the numerical investigations are conducted with the 200 W KC200GT solar panel whose model accuracy was confirmed in second section of this paper. The Buck converter of the PVE is consistently designed according to [20], [21], [33]. The converter parameters are designed for an allowable ripple output of 1 % [33]. The parameters of this converter are displayed in Table 3. The controller on the other hand is designed based on a step-response of the closed system of the PVE as shown in Fig. 6. The gains of the controller

K_{i}

and

K_{p}

are tuned in MATLAB/Simulink using the controller tuner application. The controller is designed to have a critically damped response, featuring no overshoot, with settling time of around 2 ms at the PVE load of

R_{pve} = 1 Ω

. The time (step response) and frequency response of the closed loop PVE system are illustrated in Fig. 12, Fig. 13, revealing a settling time of 2.2 ms, phase margin of 180 °C and infinite gain margin. Thus the designed system consistently obeys the desired criteria. It should be noted that the aforementioned PI controller are consistently used throughout all the investigation in this study.
本文在 MATLAB/Simulink 环境中进行了一系列模拟和实验，以评估所提议的 PVE 的功效。所有的数值研究都是使用 200 W KC200GT 太阳能电池板进行的，其模型的准确性已在本文第二部分得到确认。PVE 的降压转换器是根据 [20]、[21]、[33] 设计的。转换器参数的设计允许输出纹波为 1 % [33]。该转换器的参数如表 3 所示。控制器则根据 PVE 闭合系统的阶跃响应进行设计，如图 6 所示。控制器

K_{i}

和

K_{p}

的增益在 MATLAB/Simulink 中使用控制器调节器应用程序进行调节。控制器设计为具有临界阻尼响应，无过冲，在 PVE 负载

R_{pve} = 1 Ω

时，稳定时间约为 2 毫秒。闭环 PVE 系统的时间（阶跃响应）和频率响应如图 12 和图 13 所示，显示出 2.2 毫秒的稳定时间、180 °C 的相位裕度和无限的增益裕度。因此，所设计的系统始终符合预期标准。值得注意的是，上述 PI 控制器在本研究的所有调查中都得到了一致使用。

Table 3. Parameters of the PVE and MPPT controller.
表 3.PVE 和 MPPT 控制器的参数。

PVE parameters PVE 参数				MPPT controller parameters MPPT 控制器参数
Parameter 参数	Value 价值	Parameter 参数	Value 价值	Parameter 参数	Value 价值	Parameter 参数	Value 价值
$E, L_{p v e}$	48 V, 3.219mH 48 V，3.219mH	$K_{p}$	0.1163	$L_{boost}$	10mH	$Δ d$	0.01
$C_{p v e}$	80uF	$K_{i}$	32.15	$C_{boost}$	150uF, 150uF、	$T_{p}$	15 ms 15 毫秒
$f_{p v e}$	31 kHz 31 千赫	$r_{C}$	0.134 Ω 0.134 Ω	$C_{in}$	100uF	$R_{o}$	35 Ω 35 Ω
$r_{L}$	0.8631 Ω 0.8631 Ω	$[D_{\min} D_{\max}]$	[0.05 0.9]	$f_{boost}$	20 kHz 20 千赫	$[d_{\min} d_{\max}]$	[0.05 0.9]

Consequently, upon achieving a desirable design, the controller's resultant parameters, as outlined in Table 3, are incorporated into the configuration of the PVE, visually depicted in Fig. 7. Within this framework, two principal experiments are undertaken: firstly, an evaluation of steady-state accuracy, and secondly, an assessment of dynamic response. These experiments are elaborated upon in the subsequent sections.
因此，在获得理想的设计后，控制器的结果参数（如表 3 所示）将被纳入 PVE 的配置中，如图 7 所示。在此框架内，进行了两项主要实验：首先是稳态精度评估，其次是动态响应评估。这些实验将在随后的章节中详细阐述。

1.
Steady-state response assessment of the PVE
PVE 的稳态响应评估

In the steady-state assessment, the PVE is evaluated for emulation accuracy against the actual solar panel. In order to quantify this assessment, the PVE error will be calculated as follows:(14)

Erro r_{pve} = \frac{|I_{pve} - I_{pv}|}{I_{pv}} \times 100

Where

I_{pv}

is the current generated by the original PV model of the solar panel. In order to position the propose PVE with respect to the literature, it is consistently compared against the Binary search Computation (BSC)-PI controller based PVE proposed in [14] and the BSC-Shift controller (SC)-PVE developed in [20]. It should be noted that throughout this study, the designed value of 0.00007 is consistently considered as the shift controller parameter. For a steady-state evaluation, the different PVEs are coupled to different resistive loads running from the short-circuit to open-circuit.
在稳态评估中，将根据实际太阳能电池板对 PVE 的仿真精度进行评估。为了量化这一评估，PVE 误差的计算方法如下： (14)

Erro r_{pve} = \frac{|I_{pve} - I_{pv}|}{I_{pv}} \times 100

其中

I_{pv}

是太阳能电池板原始光伏模型产生的电流。为了根据文献对 PVE 进行定位，我们将其与 [14] 中提出的基于二进制搜索计算 (BSC)-PI 控制器的 PVE 和 [20] 中开发的 BSC 移位控制器 (SC)-PVE 进行了比较。值得注意的是，在整个研究过程中，0.00007 的设计值始终被视为移位控制器参数。在稳态评估中，不同的 PVE 与从短路到开路的不同电阻负载耦合。

It can be seen in Fig. 14 that all the PVEs exhibit a noticeable performance in reproducing the static characteristics of the PV panel. More specifically, they show a very strong agreement with the PV model.
从图 14 中可以看出，所有 PVE 在再现光伏板的静态特性方面都有显著的表现。更具体地说，它们与光伏模型的一致性非常高。

However, it is seen that the PVEs do not capture the entire static characteristics of the PV I-V curve. This is due to the

[D_{\min} D_{\max}]

duty cycle constraint introduced into the PVE. This constraint protects the switch of the buck converter from switching stresses at very low duty cycle. Consequently, it limits the static operation of the PVE. Thus if the load coupled with the PVE is out of the operating range of the PVE, the duty cycle constraint keeps the PVE output constant. This explains why the PVE does not accurately emulate the solar panel at extreme load conditions. The minimum resistance below which the PVE may not accurately reproduce the solar panel is found to be approximately 0.2 Ω at 1000 W/m² and 1 Ω at 200 W/m². Conversely the maximum resistance above which the emulator does not efficiently capture the panel’s static characteristics is found to be approximately 50 Ω at 1000 W/m² and 70 Ω at 200 W/m². It is noteworthy that above this maximum values, the PVE may reproduce the static characteristics of the solar panel at an extremely slow response.
然而，我们发现 PVE 并不能捕捉到光伏 I-V 曲线的全部静态特性。这是由于在 PVE 中引入了

[D_{\min} D_{\max}]

占空比约束。该约束可保护降压转换器的开关在占空比极低的情况下免受开关应力的影响。因此，它限制了 PVE 的静态运行。因此，如果与 PVE 耦合的负载超出了 PVE 的工作范围，占空比约束将使 PVE 输出保持恒定。这就解释了为什么在极端负载条件下，PVE 无法准确模拟太阳能电池板。在 1000 W/m ² 和 200 W/m ² 条件下，PVE 无法准确模拟太阳能电池板的最小电阻分别约为 0.2 Ω 和 1 Ω。相反，在 1000 W/m ² 和 200 W/m ² 时，仿真器不能有效捕捉面板静态特性的最大电阻分别约为 50 Ω 和 70 Ω。值得注意的是，超过这个最大值时，PVE 可能会以极慢的响应速度再现太阳能电池板的静态特性。

Moreover, the quantitative precision of the Photovoltaic Emulators (PVEs) is exemplified in Fig. 15, showcasing an impressive performance across various PVEs with an error margin of less than 2 %. Additionally, a visual analysis reveals that the proposed PVE exhibits a smaller error than the alternative PVEs at lower emulation output resistances. However, beyond an output resistance of 22 Ω for an irradiance of 1000 W/m2, the performance of the proposed PVE lags behind the alternative PVEs. This outcome is expected and can be attributed to the approximation introduced by the Explicit Photovoltaic Model (EPVM) in solving the PV equations explicitly.
此外，图 15 举例说明了光伏仿真器 (PVE) 的定量精度，展示了各种 PVE 令人印象深刻的性能，误差幅度小于 2%。此外，直观分析表明，在较低的仿真输出电阻下，建议的 PVE 比其他 PVE 误差更小。然而，在辐照度为 1000 W/m2 时，当输出电阻超过 22 Ω 时，拟议 PVE 的性能就会落后于其他 PVE。这一结果在意料之中，可归因于显式光伏模型（EPVM）在显式求解光伏方程时引入的近似值。

Nevertheless, upon closer examination of the distinct error curves, it becomes evident that the proposed PVE demonstrates comparable accuracy to the alternative PVEs based on numerical computations. From observation of Fig. 15, it is concluded that the accuracy of the PVE measured as the function of the relative error, is less than 0.7 %.
然而，在仔细观察不同的误差曲线后，可以明显看出，建议的 PVE 与基于数值计算的其他 PVE 具有相当的精度。通过观察图 15，可以得出结论：以相对误差函数衡量的 PVE 精度低于 0.7%。

The alternative PVEs rely on numerically evaluating the PV model, which consequently necessitates multiple iterations to converge towards a stable PVE output. This is consistent with the findings in [14], as shown in Fig. 14, where the alternative PVEs require as many as 18 iterations for convergence.
替代 PVE 依赖于对 PV 模型进行数值评估，因此需要多次迭代才能收敛到稳定的 PVE 输出。这与 [14] 的研究结果一致，如图 14 所示，替代 PVE 需要多达 18 次迭代才能收敛。

In contrast, the proposed PVE requires at most one iterations to generate a stable reference for the PVE. The implications of these iterations on the computational time of the PVE will be thoroughly assessed in the subsequent sections of this paper.
相比之下，拟议的 PVE 最多需要一次迭代才能为 PVE 生成稳定的参考。这些迭代对 PVE 计算时间的影响将在本文后续章节中进行全面评估。

2.
Dynamic response assessment of the PVE
PVE 的动态响应评估

To evaluate the dynamic responsiveness of the PVE configurations, they are subjected to distinct dynamic scenarios. Initially, the PVEs are connected to various constant load settings, maintained consistently across multiple experiments, in combination with varying environmental test conditions. As the current controller naturally ensures a satisfactory response within the constant current region (CCR) of the I-V curve [14], [19], the chosen PVE loads in this investigation are set to operate within the constant voltage region (CVR) of the I-V curve. This region presents greater challenges for achieving stable and efficient emulation. Consequently, the dynamic behavior of the different PVEs under diverse environmental and load conditions is presented in Fig. 16, Fig. 17, Fig. 18.
为了评估 PVE 配置的动态响应能力，我们对它们进行了不同的动态测试。起初，PVE 连接到各种恒定负载设置，并在多个实验中与不同的环境测试条件保持一致。由于电流控制器自然会确保在 I-V 曲线的恒定电流区 (CCR) 内做出令人满意的响应 [14]，[19]，因此本研究中选择的 PVE 负载设置为在 I-V 曲线的恒定电压区 (CVR) 内运行。该区域对实现稳定高效的仿真提出了更大的挑战。因此，不同 PVE 在不同环境和负载条件下的动态行为如图 16、图 17 和图 18 所示。

The visual analysis clearly demonstrates that all the PVEs successfully capture the dynamic characteristics of the PVE. However, the BSC-PI-PVE exhibits the slowest response time among them. The computation time of each PVE variant is documented under different conditions, as shown in Table 4. It is noted that since the PVE is a dynamic system, the computation time in this paper refers to the output settling time of the PVE, measured by a 2 % criteria. It is evident that the proposed PVE exhibits the swiftest computation speed, as evidenced by its shortest execution time. Specifically, the proposed PVE achieves a computation time of under 22 ms, in contrast to the longer computation time of the BSC algorithm, which records a time exceeding 100 ms. The inclusion of the shift controller (SC) contributes to the faster computational ability of the BSC-SC relative to the BSC-PI configuration, owing to the inherently rapid computation characteristics of the shift controller.
直观分析清楚地表明，所有 PVE 都成功地捕捉到了 PVE 的动态特性。不过，BSC-PI-PVE 的响应时间是其中最慢的。如表 4 所示，记录了每个 PVE 变体在不同条件下的计算时间。需要注意的是，由于 PVE 是一个动态系统，本文中的计算时间指的是 PVE 的输出稳定时间，以 2% 的标准来衡量。很明显，拟议的 PVE 具有最快的计算速度，其最短的执行时间就是最好的证明。具体来说，拟议的 PVE 计算时间不到 22 毫秒，而 BSC 算法的计算时间较长，超过 100 毫秒。与 BSC-PI 配置相比，由于移位控制器（SC）固有的快速计算特性，BSC-SC 的计算能力更快。

Table 4. Settling time of PVEs at different operation condition.
表 4.不同运行条件下 PVE 的沉降时间。

Operation Conditions 运行条件	Proposed PVE 拟议的 PVE	BSC-PI-PVE	BSC-SC-PVE
$G = 1000 W / m^{2}, R_{p v e} = 5 Ω$	7.95 ms 7.95 毫秒	25.65	24.85 ms 24.85 毫秒
$G = 1000 W / m^{2}, R_{p v e} = 25 Ω$	8.85 ms 8.85 毫秒	79 ms 79 毫秒	26.35 ms 26.35 毫秒
$G = 1000 W / m^{2}, R_{p v e} = 40 Ω$	10.55 ms 10.55 毫秒	100	25.45 ms 25.45 毫秒
$G = 1000 W / m^{2}, R_{p v e} = 50 Ω$	10.80 ms 10.80 毫秒	100 ms 100 毫秒	26.3 ms 26.3 毫秒
$G = 500 W / m^{2}, R_{p v e} = 10 Ω$	13 ms 13 毫秒	39.65	24 ms 24 毫秒
$G = 500 W / m^{2}, R_{p v e} = 25 Ω$	10.40 ms 10.40 毫秒	79.75 ms 79.75 毫秒	26 ms 26 毫秒
$G = 500 W / m^{2}, R_{p v e} = 40 Ω$	11.40 ms 11.40 毫秒	>100 ms >100 毫秒	25 ms 25 毫秒
$G = 500 W / m^{2}, R_{p v e} = 50 Ω$	11.45 ms 11.45 毫秒	>100 ms >100 毫秒	26.10 ms 26.10 毫秒
$G = 200 W / m^{2}, R_{p v e} = 30 Ω$	21.9 ms 21.9 毫秒	91.75 ms 91.75 毫秒	24.55 ms 24.55 毫秒
$G = 200 W / m^{2}, R_{p v e} = 40 Ω$	19.45 ms 19.45 毫秒	>100 ms >100 毫秒	24.20 ms 24.20 毫秒
$G = 200 W / m^{2}, R_{p v e} = 60 Ω$	17.95 ms 17.95 毫秒	>100 ms >100 毫秒	24.90 ms 24.90 毫秒
$G = 200 W / m^{2}, R_{p v e} = 70 Ω$	17.85 ms 17.85 毫秒	>100 ms >100 毫秒	24.85 ms 24.85 毫秒

The remarkable computational efficiency of the proposed PVE can be attributed to its direct and iteration-free calculation approach. This attribute further underscores the performance and computational efficacy of the integrated EPVM.
拟议的 PVE 计算效率高，这要归功于其直接、无迭代的计算方法。这一特性进一步凸显了集成式 EPVM 的性能和计算效率。

Hence, based on the observations depicted in Fig. 16, Fig. 17, Fig. 18, it becomes evident that iterations play a pivotal role in constraining the computational performance of the PVE. The employment of the Explicit Photovoltaic Model (EPVM) has the distinct advantage of rendering the referencing of the PVE entirely independent of computational iterations.
因此，根据图 16、图 17 和图 18 所示的观察结果，可以明显看出迭代在限制 PVE 计算性能方面起着关键作用。采用显式光伏模型（EPVM）的明显优势是，PVE 的参考完全不受计算迭代的影响。

In practical operational scenarios, real solar panels might encounter fluctuations in resistive load. From an emulation perspective, the PVE should possess the capability to replicate the dynamic characteristics of a solar panel when confronted with such circumstances. In essence, a change in the load induces a perturbation in the PVE's operating point. Therefore, the robustness of the controller and the overall computational prowess of the PVE should enable it to swiftly respond to such abrupt scenarios.
在实际操作场景中，真正的太阳能电池板可能会遇到电阻负载波动的情况。从仿真的角度来看，PVE 应具备在这种情况下复制太阳能电池板动态特性的能力。从本质上讲，负载的变化会引起 PVE 工作点的扰动。因此，控制器的鲁棒性和 PVE 的整体计算能力应使其能够迅速应对这种突发情况。

To concretely evaluate the performance of the PVEs in a varying load scenario, they are subjected to abrupt chang

Outline 概要

Cited by (8) 引用 (8)

Figures (34) 数字 (34)

Tables (5)

Solar Energy 太阳能

Highlights 亮点

Abstract 摘要

Keywords 关键词

Nomenclature 术语

Symbols 符号

Subscripts 下标

Abbreviations 缩略语

1. Introduction 1.导言

2. Synoptic description of the proposed PVE structure
2.拟议 PVE 结构的综合说明

3. Modelling of solar panel
3.太阳能电池板建模

4. Explicit PV model based integration into the PVE structure
4.将基于 PV 模型的显式集成纳入 PVE 结构

5. Buck converter and PI control design
5.降压转换器和 PI 控制设计

6. MPPT application of the proposed PVE
6.拟议 PVE 的 MPPT 应用

7. Results and discussions
7.结果和讨论

Solar Energy 太阳能

Development of an efficient and rapid computational solar photovoltaic emulator utilizing an explicit PV model利用显式光伏模型开发高效、快速的太阳能光伏仿真计算器

Highlights 亮点

Abstract 摘要

Keywords 关键词

Nomenclature 术语

Symbols 符号

Subscripts 下标

Abbreviations 缩略语

1. Introduction 1.导言

2. Synoptic description of the proposed PVE structure2.拟议 PVE 结构的综合说明

3. Modelling of solar panel3.太阳能电池板建模

4. Explicit PV model based integration into the PVE structure4.将基于 PV 模型的显式集成纳入 PVE 结构

5. Buck converter and PI control design5.降压转换器和 PI 控制设计

6. MPPT application of the proposed PVE6.拟议 PVE 的 MPPT 应用

7. Results and discussions7.结果和讨论

Development of an efficient and rapid computational solar photovoltaic emulator utilizing an explicit PV model
利用显式光伏模型开发高效、快速的太阳能光伏仿真计算器

2. Synoptic description of the proposed PVE structure
2.拟议 PVE 结构的综合说明

3. Modelling of solar panel
3.太阳能电池板建模

4. Explicit PV model based integration into the PVE structure
4.将基于 PV 模型的显式集成纳入 PVE 结构

5. Buck converter and PI control design
5.降压转换器和 PI 控制设计

6. MPPT application of the proposed PVE
6.拟议 PVE 的 MPPT 应用

7. Results and discussions
7.结果和讨论