FOPDT model and CHR method based control of flywheel energy storage integrated microgrid

Abstract

The main causes of frequency instability or oscillations in islanded microgrids are unstable load and varying power output from distributed generating units (DGUs). An important challenge for islanded microgrid systems powered by renewable energy is maintaining frequency stability. To address this issue, a proportional integral derivative (PID) controller is designed in this article. Firstly, islanded microgrid model is constructed by incorporating various DGUs and flywheel energy storage system (FESS). Further, considering first order transfer function of FESS and DGUs, a linearized transfer function is obtained. This transfer function is further approximated into first order plus time delay (FOPTD) form to design PID control strategy, which is efficient and easy to analyze. PID parameters are evaluated using the Chien-Hrones-Reswick (CHR) method for set point tracking and load disturbance rejection for 0% and 20% overshoot. The CHR method for load disturbance rejection for 20% overshoot emerges as the preferred choice over other discussed tuning methods. The effectiveness of the discussed method is demonstrated through frequency analysis and transient responses and also validated through real time simulations. Moreover, tabulated data presenting tuning parameters, time domain specifications and comparative frequency plots, support the validity of the proposed tuning method for PID control design of the presented islanded model.

Introduction

Energy use and its management are vital to economic growth, environmental sustainability, and our everyday existence. Fossil fuels, when burnt, produce heat and electricity, resulting in the production of a significant amount of greenhouse gases, solar radiation, and air pollution. Fossil fuels include gas, oil, and coal. These sources primarily contribute to global climate change, accounting for a significant percentage of carbon dioxide emissions and greenhouse gas emissions. It becomes very important to reduce these emissions and find more cleaner and sustainable alternatives. Renewable energy sources (RESs) are recently accepted as more cleaner alternatives over fossil fuels¹. These sources are basically infinite in nature and naturally replenish over time. Use of these is essential for guaranteeing a sustainable energy future, lowering reliance on fossil fuels, and reducing the effects of climate change.

In recent trends, renewable sources are significantly included in distributed power system (DPS)². The DPSs are located mainly near load centres or at load centres. These power systems work isolated or are included in micro-grids. The idea of microgrids is become a viable approach for implementing DPS because of many benefits like providing efficient and low-cost energy, improvement in stability and operation of regional grid, and reduction in peak loads. A microgrid is essentially a small-scale electricity distribution network that uses a combination of conventional and alternative energy sources to provide power to designated areas^3,4. Microgrids’ primary goal is to effectively manage a variety of distributed generation units (DGUs) and energy storage systems (ESSs) in order to meet the loads’ energy requirements.

A variety of microgrid models using RESs are available in literature^5–11. Kamal et al. performed a study on a standalone microgrid model utilizing sourses like biogass, solar photovoltaic, biomass, micro-hydropower, biogas, and wind energy in¹². In¹³, optimal planning of microgrid including wind turbines (WTs) and photovoltaics (PVs), is conducted. A microgrid model with solar PV, WT, and ESS incorporating batteries and a flywheel is presented in¹⁴. An islanded microgrid model is designed in^15–20 by combining ESS incorporating batteries and flywheel, with WT, solar PV, fuel cells (FCs), and a diesel engine generator (DEG). An electrification of hybrid microgrids is done using RES like solar PV and WT in²¹. In²², optimization and planning of microgrids, including PV, battery-ESS, and diesel generators, is carried out for a residential complex. Alzahrani et al., in²³, conducted energy optimization in real-time for smart homes integrated with RES like PV and wind energy. As microgrids consist of variety of components, higher order transfer function is generally obtained. As the number of components contributing in microgrid increases, the order of the transfer function also increases. Higher-order models may provide higher accuracy, but they are computationally expensive and difficult to interpret. Hence, such higher order models are approximated to lower orders. Approximate models are often preffered over higher-order ones due to their simplicity, computational efficiency, and ease of interpretation. For control design purposes, a most common approximation is the first order plus dead time (FOPDT) model^24,25.

As RESs used in microgrid are not consistent, proper regulation of frequency becomes essential. Appropriate control schemes for increasing or decreasing real power generation must be implemented in a microgrid in order to keep the frequency control within a permissible limit²⁶. Researchers frequently use PID controllers in microgrids; they^27–35. A PI controller for microgrid with wind pitch controllers, smart controllable water heaters, and diesel generators is shown in³⁶. A multi-staged PID controller implementation for a microgrid is described in³⁷. A PD filter is presented in the first stage, and a PI controller is developed for the second stage. PID controllers possess several advantages regarding robustness, easy implementation, and flexible tuning. These also offer tuning by simultaneously performing all three actions, i.e., derivative, integral, and proportional. These regulate frequency, power flow, and voltage within microgrids, by appropriately tuning the controller parameters, thereby improving stability, efficiency, and reliability. For tuning these controller parameters, various methods like Ziegler–Nichols (ZN)^38–40, Cohen-Coon^41–43, etc., are used in the literature. While utilizing these methods to tune controllers in microgrids possessing higher orders (HOs), the process becomes tedious and complex. To overcome this, approximations are performed on HO models to obtain reduced-order transfer functions.

This study introduces design of PID controller for an islanded microgrid integrated with RESs and flywheel energy storage system (FESS). The microgrid model used for analysis consists of uncontrollable DGUs like solar PV plant and WT generators and controllable DGUs like micro turbine generator (MTG), bio-diesel engine generator (BDEG), biogass turbine generator (BGTG) and diesel engine generator (DEG) units. As RESs are integrated with microgrid, FESS is utilized, along with controllable and uncontrollable DGUs, to store excess power at times of low demand and deliver it same in times of high demands. A first-order plus time delay (FOPDT) model is evaluated as an approximation of islanded microgrid. Further, for maintain stable power flow and frequency limits, PID controller is introduced in the microgrid. The Chien-Hrones-Reswick (CHR) method is used to tune the PID controller’s parameters for set point tracking (STP) and load disturbance rejection (LDR) at 0% and 20%, respectively. The practicality of CHR based PID controller design used to minimize frequency deviation, is investigated. Further, the performance of a CHR-tuned PID controller in frequency control of an islanded microgrid is evaluated by demonstrating frequency deviation plots and efforts required by controller to mitigate the frequency deviation. The major contribution of this study are listed as:

• An islanded microgrid model, consisting of controllable and uncontrollable DGUs, and integrated with RESs, is investigated for frequency control analysis. FESS is employed as an energy storage device in islanded microgrid for surplus energy storage during less demand and as an energy source during excess load demands.

• PID controller design is adopted and its parameters are obtained from FOPDT model derived by approximating the islanded microgrid model. The obtained parameters are tuned using CHR tuning rules for SPT and LDR for 0% and 20% overshoots, as a control strategy for mitigating frequency variation in islanded microgrid and its parameters

• Frequency deviations of model, with and without CHR-based PID controller, are plotted for comparative analysis of all four CHR-based controllers and for validation, real time simulations on OPAL-RT platform are also performed. Responses of control efforts exhibited by each CHR-based PID controller are also plotted to providing further clarification of their efficacy.

The flow of this paper proceeds as follows: Section “Islanded micro-grid architecture” describes the overall architecture of the islanded microgrid as well as the mathematical model. Section “First order plus delay time model of IMG” presents an FOPDT model for islanded microgrid and its block diagram. Section Tuning rules for Chien–Hrones–Reswick method” discusses the CHR tuning rules for PID controller design. Section “Results and discussion” presents the frequency deviation and time domain analysis of islanded microgrid model with CHR-based PID controller. Finally, our conclusions are summarized in Section “Conclusion”.

Islanded micro-grid architecture

Figure 1 provides the schema of islanded microgrid (IMG) considered in this study. IMG constitutes a complex network of components, including DGUs, ESSs, loads, controllers, and power converters⁴⁴. DGUs, which can be conventional or non-conventional, play a vital role in IMGs, with solar and wind DGUs being prominent examples. However, non-conventional DGUs are subject to various factors, such as weather conditions and geographical constraints, leading to unpredictable output power. It carries diverse DGUs, including PV, WTG, MTG, BDEG, BGTG, and DEG. Additionally, ESS namely, FESS, is also integrated into the model.

Figure 1.

Schema of considered IMG.

Details of components of IMG

Solar PV plant (SPVP)

SPVP converts solar energy into electrical energy. Surface temperature and radiation intensity are two factors that primarily affect the amount of electrical energy that SPV panels produce. To model dynamics of SPVP, its first-order transfer function (FOTF) i.e. $G_{\mathit{PV}}(s)$, is given by

$$\begin{array}{c}\hfill G_{\mathit{PV}}(s)=\frac{K_{\mathit{PV}}}{1+s{T}_{\mathit{PV}}}\end{array}$$ 1

In (1), $T_{\mathit{PV}}$ and $K_{\mathit{PV}}$ denote time constant and gain of SPVP, respectively.

Wind turbine generator (WTG)

Wind serves as input for generating electrical energy through a WTG system⁴⁵. The model of WTG system can be described by a FOTF, $G_{\mathit{WTG}}(s)$, as given by

$$\begin{array}{c}\hfill G_{\mathit{WTG}}(s)=\frac{K_{\mathit{WTG}}}{1+s{T}_{\mathit{WTG}}}\end{array}$$ 2

In (2), $T_{\mathit{WTG}}$ shows time constant of WTG system. And, $K_{\mathit{WTG}}$ denotes gain of the WTG system.

Micro-turbine generator (MTG)

Basically, MTG is a compact power generation source which obtains energy from liquid or gaseous fuels⁴⁶. The model of MTG can be characterized by FOTF, $G_{\mathit{MTG}}(s)$, given by

$$\begin{array}{c}\hfill G_{\mathit{MTG}}(s)=\frac{K_{\mathit{MTG}}}{1+s{T}_{\mathit{MTG}}}\end{array}$$ 3

In (3), $T_{\mathit{MTG}}$ is time constant of MTG. However, $K_{\mathit{MTG}}$ is gain of MTG unit.

Biodiesel engine generator (BDEG)

Biodiesel, a renewable fuel derived from plant sources through chemical processes, serves as an eco-friendly alternative in diesel engines for generating electrical power via combustion reactions⁴⁷. Given is the BDEG model.

$$\begin{array}{c}\hfill G_{\mathit{BDEG}}(s)\approx \frac{K_{\mathit{BDEG}}}{1+s{T}_{\mathit{BDEG}}}\end{array}$$ 4

In (4), $T_{\mathit{BDEG}}$ represents time constant, and $K_{\mathit{BDEG}}$ shows gain of BDEG system, respectively.

Biogas turbine generator (BGTG)

Anaerobic digestion, a process involving the breakdown of biodegradable organic materials and animal waste, yields biogas, a renewable energy source⁴⁷. Biogas is utilized as fuel in biogas power plants to generate both heat and electricity. The dynamic behavior of a biogas power plant can be mathematically represented by a FOTF, denoted as $G_{\mathit{BGTG}}(s)$, which describes the relationship between its input and output variables. Given is the transfer function, $G_{B}GTG(s)$.

$$\begin{array}{c}\hfill G_{\mathit{BGTG}}(s)\approx \frac{K_{\mathit{BGTG}}}{1+s{T}_{\mathit{BGTG}}}\end{array}$$ 5

The parameters $K_{\mathit{BGTG}}$ and $T_{\mathit{BGTG}}$ represent gain and time constants of BGTG system, respectively.

Diesel engine generator (DEG)

DEG serves as a conventional standby power source, typically employed for generating electrical power through the combustion of fuel⁴⁵. It plays a vital role in providing backup power supply in various applications, ensuring uninterrupted operation during grid outages or emergencies. To accurately represent the dynamic behavior of a DEG system, it can be modeled using a FOTF, $G_{\mathit{DEG}}(s)$, as outlined in (6).

$$\begin{array}{c}\hfill G_{\mathit{DEG}}(s)=\frac{K_{\mathit{DEG}}}{1+s{T}_{\mathit{DEG}}}\end{array}$$ 6

In (6), the parameters $K_{\mathit{DEG}}$ and $T_{\mathit{DEG}}$ represent gain and time constants of DEG system, respectively.

Flywheel energy storage system (FESS)

FESS serves as a quick-reaction (ESS) and a critical component in storing surplus energy during periods of low demand and releasing it in case of emergencies or peak demand situations¹⁹. This feature stabilizes grid frequency by mitigating oscillations brought on by RESs like solar and wind. By offering prompt assistance in the event of unexpected load fluctuations or disruptions, FESS additionally improves grid resilience by decreasing dependence on conventional fossil fuel-fueled generators for frequency regulation. Because of its quick reaction time typically measured in milliseconds it is well-suited to managing abrupt changes in the dynamics of the power supply and demand, guaranteeing the dependable and effective operation of microgrids. The FOTF of FESS is given by:

$$\begin{array}{c}\hfill G_{\mathit{FESS}}(s)=\frac{K_{\mathit{FESS}}}{1+s{T}_{\mathit{FESS}}}\end{array}$$ 7

In (7), $K_{\mathit{FESS}}$ is gain of FESS. On the other hand, $T_{\mathit{FESS}}$ is time constant of FESS.

Generator dynamics

The generator dynamics IMG model are given by

$$\begin{array}{c}\hfill G_{\mathit{IMG}}(s)=\frac{1}{D+sM}\end{array}$$ 8

where D is damping constant. While, M is inertial constant. A larger value of D means stronger damping, leading to faster dissipation of oscillatory motion and improved stability. While, a lower damping constant may lead to slower damping and increased susceptibility to oscillatory motion and instability. Inertia reflects the system’s ability to resist changes in its operating state and is crucial for maintaining stability. A higher inertia constant signifies greater kinetic energy storage capacity, resulting in more robust performance against disturbances and fluctuations in load or generation.

Total power generation and frequency deviation

The total generated power, $P_{\mathit{TOTAL}}$, is computed as the sum of power generated by uncontrollable DGUs ($P_{\mathit{unc}}$), controllable DGUs ($P_{\mathit{con}}$), and ESD ($P_{\mathit{ESDs}}$), as expressed in (9). In (9), the uncontrollable power ($P_{\mathit{unc}}$) comprises the power generated by SPVP panels and WTG set. The expression for $P_{\mathit{unc}}$ is given by (10). The controllable power, $P_{\mathit{con}}$, encompasses output powers of MT, BGDG, BGTG, and DEG units, respectively. This is given by (11). The power from energy storage device, $P_{\mathit{ESD}}$, comprises the exchangeable power of FESS. The $P_{\mathit{ESDs}}$ is provided in (12). $P_{\mathit{TOTAL}}$ is utilized to fulfill the load demand ($P_L$). The net power $(P_{\mathit{net}})$ represents power balance equation of the model which is depicted in (13).

$$\begin{array}{cc}\hfill P_{\mathit{TOTAL}}=& P_{\mathit{unc}}+P_{\mathit{con}}+P_{\mathit{ESDs}}\hfill \end{array}$$ 9

$$\begin{array}{cc}\hfill P_{\mathit{unc}}=& P_{\mathit{PV}}+P_{\mathit{WTG}}\hfill \end{array}$$ 10

$$\begin{array}{cc}\hfill P_{\mathit{con}}=& P_{\mathit{MTG}}+P_{\mathit{BDEG}}+P_{\mathit{BGTG}}+P_{\mathit{DEG}}\hfill \end{array}$$ 11

$$\begin{array}{cc}\hfill P_{\mathit{ESD}}=& P_{\mathit{FESS}}\hfill \end{array}$$ 12

$$\begin{array}{cc}\hfill P_{\mathit{net}}=& P_{\mathit{PV}}+P_{\mathit{WTG}}+P_{\mathit{MT}}+P_{\mathit{BDEG}}+P_{\mathit{BGTG}}+P_{\mathit{DEG}}\pm P_{\mathit{FESS}}-P_L\hfill \end{array}$$ 13

$P_{\mathit{net}}$, is calculated as the disparity between $P_L$ and $P_{\mathit{TOTAL}}$. Frequency deviations ($\mathrm{\Delta }f$) are generated in response to the difference between $P_{\mathit{net}}$ and $P_{\mathit{TOTAL}}$. The relationship between $\mathrm{\Delta }f$ and $P_{\mathit{net}}$ is defined as:

$$\begin{array}{c}\hfill \mathrm{\Delta }f=\frac{1}{Ms+D}{P}_{\mathit{net}}\end{array}$$ 14

$\mathrm{\Delta }f$ can be minimized by ensuring equilibrium between $P_{\mathit{TOTAL}}$ and $P_L$.

Equivalent block diagram of IMG

IMG system, as depicted in Fig. 1, is further presented through an equivalent FOTF model in Fig. 2. IMG constitutes of two control loops i.e. primary loop and secondary loop. FESS is incorporated in primary control loop, while controllable DGUs are integrated in secondary control. Additionally, uncontrollable DGUs, such as SPVP and WTG, interconnected with loads, are treated as dependent components or disturbances due to their fluctuating power generation capabilities.

Figure 2.

Transfer function based block diagram of islanded microgrid.

The primary control mechanism involves adjusting power through FESS, while the secondary control mechanism utilizes controllable DGUs to address varying load demands. The mathematical formulations of DGUs and FESS are presented in (1)–(8). FOTFs corresponding to DGUs and FESS, along with their numerical values, utilized within IMG model, are tabulated in Table 1.

Table 1.

Mathematical models^48–50.

Component	Transfer function	Parameters
Distributed generating units
DEG	$G_{\mathit{DEG}}(s)=\frac{K_{\mathit{DEG}}}{1+s{T}_{\mathit{DEG}}}$	$T_{\mathit{DEG}}=2$, $K_{\mathit{DEG}}=0.003$
BDEG	$G_{\mathit{BDEG}}(s)\approx \frac{K_{\mathit{BDEG}}}{1+s{T}_{\mathit{BDEG}}}$	$T_{\mathit{BDEG}}=0.55$, $K_{\mathit{BDEG}}=1$
BGTG	$G_{\mathit{BGTG}}(s)\approx \frac{K_{\mathit{BGTG}}}{1+s{T}_{\mathit{BGTG}}}$	$T_{\mathit{BGTG}}=1.48$, $K_{\mathit{BGTG}}=1$
MTG	$G_{\mathit{MTG}}(s)=\frac{K_{\mathit{MTG}}}{1+s{T}_{\mathit{MTG}}}$	$T_{\mathit{MTG}}=1.5$, $K_{\mathit{MTG}}=1$
WTG	$G_{\mathit{WTG}}(s)=\frac{K_{\mathit{WTG}}}{1+s{T}_{\mathit{WTG}}}$	$T_{\mathit{WTG}}=1.5$, $K_{\mathit{WTG}}=1$
SPVP	$T{F}_{\mathit{PV}}(s)=\frac{K_{\mathit{PV}}}{1+s{T}_{\mathit{PV}}}$	$T_{\mathit{PV}}=1.8$, $K_{\mathit{PV}}=1$
Energy storage device
FESS	$G_{\mathit{FESS}}(s)=\frac{K_{\mathit{FESS}}}{1+s{T}_{\mathit{FESS}}}$	$T_{\mathit{FESS}}=0.1$,
Generator dynamics
Generator dynamics	$G_{\mathit{IMG}}(s)=\frac{1}{D+sM}$	$D=0.03$,

It is imperative to note that SPVP and WTG units exhibit highly stochastic behaviors influenced by weather conditions, resulting in variable power output. Similarly, load variations significantly impact system dynamics. These fluctuations in the output power of WTG, SPVP, and load collectively contribute to frequency deviation, which are represented as disturbances in frequency control analysis.

The equivalent model depicted in Fig. 2 is represented in Fig. 3. This model employs FOTF representations to characterize the dynamics of both DGUs and FESS, as well as the overall system dynamics. The combined effect of the transfer functions of DGUs and FESS is encapsulated in a forward path transfer function, denoted as $G_{\mathit{MODEL}}(s)$. This transfer function serves as a mathematical representation of IMG model and can be formally expressed in transfer function form, as provided by

$$\begin{array}{c}\hfill G_{\mathit{MODEL}}(s)=\frac{\mathrm{\Delta }f(s)}{\mathrm{\Delta }{P}_{\mathit{net}}(s)}\end{array}$$ 15

where

$$\begin{array}{cc}\hfill \mathrm{\Delta }f(s)=& \sum _{k=0}^{n-1}{N}_k{s}^k\hfill \end{array}$$ 16

$$\begin{array}{cc}\hfill \mathrm{\Delta }{P}_{\mathit{net}}(s)=& \sum _{k=0}^n{D}_k{s}^k\hfill \end{array}$$ 17

$N_k$ for $k=0,1,2,3,\cdots ,(n-1)$ and $D_k$ for $k=0,1,2,3,\dots ,n$, in (16) and (17), represent numerator and denominator coefficients, respectively.

Figure 3.

Equivalent model of IMG with DGUs and FESS.

The values of numerator and denominator of $G_{\mathit{MODEL}}(s)$, depicted in (16) and (17), are provided in (18) and (19).

$$\begin{array}{cc}\hfill \mathrm{\Delta }f=& 0.3089s^5+3.831s^4+8.005s^3+5.927s^2+1.602s+0.09009\hfill \end{array}$$ 18

$$\begin{array}{cc}\hfill \mathrm{\Delta }{P}_{\mathit{net}}=& 0.03907s^7+0.05397s^6+1.675s^5+2.043s^4+1.139s^3+0.028s^2+0.02341s+0.0006\hfill \end{array}$$ 19

First order plus delay time model of IMG

Approximation is a technique used to approximate a higher-order model into a lower-order one. Generally, the lower-order model, especially the first-order plus dead time (FOPTD) model, is widely employed for identification and modeling of higher-order (HO) systems²⁴. The FOPTD model is characterized by three key parameters: the time constant, $T_m$, the time delay, $T_d$, and the gain, K. These parameters are integral to general mathematical representation of FOPDT model as depicted in (20). Accurately determining these parameters is essential for creating dependable approximate models capable of mimicking the dynamic behavior of system. These models are crucial for analyzing HO systems effectively.

Various methods exist for estimating the FOPDT parameters, including the Skogestad half rule⁵¹, Taylor series approximation, curve fitting techniques, relay feedback methods⁵², impulse response analysis⁵³, and step response analysis⁵⁴. The generalized mathematical expression for the FOPTD model is depicted as:

$$\begin{array}{c}\hfill {\varphi }_F(s)=\frac{K}{1+s{T}_m}{e}^{-T_d.s}\end{array}$$ 20

In (20), ${\varphi }_F(s)$ represents the approximate FOTF of IMG. Here, K denotes the gain, $T_m>0$ signifies the time constant, and $T_d>0$ indicates the time delay. The forward path transfer function of the IMG, as denoted in (15), exhibits HO, specifically seventh order. To simplify its representation, it is approximated into FOPDT form utilizing step response method. The step response of IMG is illustrated with tangent and inflection point in Fig. 4, while the parameters derived from this response are detailed in Table 2. Referring to these identified parameters, FOPDT model for IMG⁵⁵ is constructed, as demonstrated in (21).

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\varphi }_F(s)=\frac{149.7}{1+24.39s}{e}^{-0.748s}\end{array}\end{array}$$ 21

The step response of both IMG and FOPDT models is presented in Fig. 5. Notably, both responses exhibit similar characteristics, with peak values of 150.04 and 149.56, respectively. Further, Bode plot, shown in Fig. 6, provides insights into frequency domain responses of the models. The time domain specifications of both models are summarized in Tables 3. Furthermore, Nyquist plot shown in Fig. 7, also depicts similar behaviour for both IMG and FOPDT model. Based on analyses presented in Table 3 and Figs. 5, 6, 7, it can be inferred that FOPDT model effectively replicates the behavior of IMG.

Figure 4.

Step response of IMG with tangent.

Table 2.

Parameters of step response.

Parameters of step response
Time constant	Steady-state gain ($T_m$)	Time delay ($T_d$)
149.7	24.39	0.748

Figure 5.

Step response of IMG and FOPDT.

Figure 6.

Bode plot of IMG and FOPDT.

Table 3.

Time domain specifications.

Time domain specifications	Islanded microgrid	Approximated model
Rise time	44.201	53.5987
Peak	150.04	149.566
Settling time	79.665	96.175
Overshoot	0	0
Peak time	146.81	178.601
Undershoot	0	0

Figure 7.

Nyquist plot of IMG and FOPDT.

Tuning rules for Chien–Hrones–Reswick method

This article considered IMG, which contains various conventional and non-conventional sources to produce electrical power. Due to the integration of various sources and ESDs, ensuring frequency stability of IMG is major concern. Proper coordination among generation and storage units is desirable to limit frequency within its predefined limits. Imbalance between sources and storage leads to frequency deviation. To mitigate this deviation, proportional integral derivative (PID) controller is designed and implemented in this contribution.

PID controller

PID controller stands as a widely employed feedback controller within various industrial applications, offering commendable control performance across diverse system dynamics^56–59. Illustrated in Fig. 8, the block diagram of the PID controller outlines its operational mechanism. Operating across three modes i.e. proportional, integral, and derivative, PID controller incorporates three variables. Its mathematical representation in s domain is provided in (22).

$$\begin{array}{c}\hfill C(s)=[K_p+\frac{K_i}{s}+K_{d}s]E(s)\end{array}$$ 22

In (22), C(s) represents output of the controller, while E(s) denotes disparity between the desired and actual values. The parameters $K_i$ and $K_d$ signify the integral and derivative gains respectively, with $K_p$ indicating proportional gain. The effectiveness of PID controller mainly depends on its tuning parameters. This article employs Chien-Hrones-Reswick method to tune PID parameters.

Figure 8.

Block diagram of PID controller.

Chien–Hrones–Reswick method

Chien-Hrones-Reswick (CHR) method stands as a closed-loop tuning approach for control design, emerging from a modified version of Ziegler-Nichols (ZN) method. ZN’s drawbacks, including high overshoot and parameter sensitivity within closed-loop systems, are addressed by CHR through its focus on process identification via servo control and regulatory control. This method prioritizes enhanced damping and emphasizes load disturbance rejection (LDR) and set point tracking (SPT). To achieve this, SPT and LDR modes are subdivided into categories with 0% and 20% overshoot. CHR method employs the time constant ($T_m$) and delay time ($T_d$) of the step response of model to formulate tuning rules for SPT and LDR, incorporating a new variable, $a=\frac{K{T}_d}{T_m}$, representing the ratio of delay time to time constant with steady-state value (K) of step response. Tuning rules for fine-tuning of SPT (0% and 20% overshoot) and LDR (0% and 20% overshoot) are tabulated in Tables 4-5. These rules serve as foundation for evaluating PID controller parameters.

Table 4.

CHR method for SPT for 0% and 20%.

CHR method for SPT
Parameter	With $0\%$ overshoot	With $20\%$ overshoot
$K_p$	0.6/a	0.95/a
$K_i$	$0.6/(a{T}_m)$	$0.7/(a{T}_m)$
$K_d$	$0.3T_d/a$	$0.45T_d/a$

Table 5.

CHR method for LDR for 0% and 20%.

CHR method for LDR
Parameter	With $0\%$ overshoot	With $20\%$ overshoot
$K_p$	0.95/a	1.2/a
$K_i$	$0.4/(a{T}_d)$	$0.6/(a{T}_d)$
$K_d$	$0.4T_d/a$	$0.5T_d/a$

Fig. 9 illustrates a schematic diagram of PID controller integrated with an approximate model of IMG. Within this diagram, X(s) represents the desired value, E(s) denotes the error, $\varphi (s)$ signifies the control signal, and D(s) represents the disturbance input. The variables $P_{\mathit{net}}$ and $\mathrm{\Delta }f$ are respectively the input and output of FOPDT model applied to IMG.

Figure 9.

PID controller with IMG and FOPDT.

Results and discussion

An IMG model is simulated and analysed in this paper. For comparative analysis, the simulations are performed by considering IMG with PID controller and without PID controller. The controller is tuned using CHR method for SPT and LDR for nullifying the frequency deviations. The responses obtained for 0% overshoot for both, SPT and LDR, and and 20% overshoot for both, SPT and LDR, are compared and analysed on the basis of time domain specifications.

A response of frequency deviation of IMG without PID controller is shown in Fig. 10. In order to maintain the stability of system, it becomes essential to dampen the under-damped oscillations observed in Fig. 10. This is achieved through the utilization of CHR tuning method in design and implementation of PID controller. The CHR technique provides tuning rules for SPT and LDR with 0% and 20% overshoot, which are tabulated in Tables 4-5. By obeying these rules, tuning parameters for CHR based PID controllers are evaluated in Table 6. Table 6 includes tuning parameters for CHR based PID controller for 0% and 20% overshoots for SPT and LDR, respectively.

Figure 10.

Frequency deviations of IMG without controller.

Table 6.

CHR based PID controller parameters.

Tuning parameters for CHR based PID controller
Tuning methods	$K_p$	$K_i$	$K_d$
$CH{R}_{SP-0\%}$	0.1307	0.0054	0.0489
$CH{R}_{SP-20\%}$	0.2069	0.0063	0.0733
$CH{R}_{LDR-0\%}$	0.2069	0.1165	0.0652
$CH{R}_{LDR-20\%}$	0.2614	0.1747	0.0815

Further, by using the values from Tables 6, transfer functions for CHR based PID controller for 0% SPT, CHR based PID controller for 20% SPT, CHR based PID controller for 0% LDR, and CHR based PID controller for 20% LDR are presented as $G_{SP-0\%}$, $G_{SP-20\%}$, $G_{LDR-0\%}$, and $G_{LDR-20\%}$ in (23), (24), (25) and (26).

$$\begin{array}{cc}\hfill G_{SP-0\%}=& 0.1307+\frac{0.0054}{s}+0.0489s\hfill \end{array}$$ 23

$$\begin{array}{cc}\hfill G_{SP-20\%}=& 0.2069+\frac{0.0063}{s}+0.0733s\hfill \end{array}$$ 24

$$\begin{array}{cc}\hfill G_{LDR-0\%}=& 0.2069+\frac{0.1165}{s}+0.0652s\hfill \end{array}$$ 25

$$\begin{array}{cc}\hfill G_{LDR-20\%}=& 0.2614+\frac{0.1747}{s}+0.0815s\hfill \end{array}$$ 26

The frequency deviation response of CHR based PID controllers along with that of system is shown in Fig. 11. Time domain specifications obtained for all considered CHR based PID controllers are tabulated in Tables 7 and 8. Table 7 demonstrated time domain specifications for CHR based PID controller for SPT while Table 8 specified time domain specifications for CHR based PID controller for LDR. In case of SPT, it is seen that CHR based PID controller with 20% overshoot performed better by settling significantly faster than the response exhibited by CHR based PID controller with 0% overshoot. While in case of LDR, it is observed that CHR based PID controller with 20% overshoot settled faster than the response exhibited by CHR based PID controller with 0% overshoot, hence showcasing comparitively better response. Overall, comparing the best performing CHR based PID controller for SPT and LDR, it is seen that CHR based PID controller for LDR with 20% overshoot exhibits the finest tuning among all the considered controllers.

Figure 11.

Comparative analysis of frequency deviation of CHR-PID controller.

Table 7.

Specifications corresponding to CHR-PID for set point tracking for 0% and 20%.

Specifications
Method	$PID-CH{R}_{SP-0\%}$	$PID-CH{R}_{SP-20\%}$
Rise time	1.5910	1.4833
Peak	1.3610	1.0760
Settling time	14.8596	4.6564
Overshoot	36.1034	7.5965
Peak time	3.8966	3.0471
Undershoot	0	0

Table 8.

Specifications corresponding to CHR-PID for load disturbance rejection for 0% and 20%.

Specifications
Method	$PID-CH{R}_{LDR-0\%}$	$PID-CH{R}_{LDR-20\%}$
Rise time	1.4825	1.2352
Peak	1.0757	1.0938
Settling time	8.4973	4.0260
Overshoot	7.5670	9.3828
Peak time	2.9966	2.5592
Undershoot	0	0

To validate the responses presented in Fig. 11, obtained from simulations performed in MATLAB platform, real-time simulations are performed on OPAL-RT platform. Frequency deviation signals for IMG are captured on 5-channel DSOs, which are presented in Fig. 12. The simulations are run for 100 sec. In Fig. 12, channel 1 i.e. CH1 indicates frequency deviations in IMG without controller implementation, channel 2 i.e. CH2 shows frequency deviations in IMG with CHR based PID controller with 0% overshoot for SPT, channel 3 i.e. CH3 denotes frequency deviations in IMG with CHR based PID controller with 20% overshoot for SPT, channel 4 i.e. CH4 indicates frequency deviations in IMG with CHR based PID controller with 0% overshoot for LDR and channel 5 i.e. CH5 indicates frequency deviations in IMG with CHR based PID controller with 20% overshoot for LDR. Comparing responses from Figs. 11 and 12, it is observed that both figures showcase almost similar responses. Hence, the obtained outcomes from Fig. 11 are validated.

Figure 12.

Comparative analysis of frequency deviation of CHR-PID controller on OPAL-RT platform.

Additionally, amplitude of control signal needed to reduce frequency deviation of IMG is determined by plotting control-signals of CHR-based PID controllers. Control signals for CHR based PID controllers are displayed in Fig. 13. From Fig. 13, it is seen that PID controller tuned using CHR method having 20% overshoot with LDR performed well in mitigating the frequency deviations.

Figure 13.

Comparative analysis of control efforts to maintain frequency of IMG.

Conclusion

This study analysed frequency control of FOPDT model of IMG, integrated with RESs and FESS along with controllable and uncontrollable DGUs. This study is concluded as follows:

• PID controller is employed in IMG to diminish the frequency deviations and maintain a steady power flow.

• The controller is designed by exploiting CHR tuning rules for SPT and LDR for 0% and 20% overshoots.

• Simulations are performed on MATLAB simulink and OPAL-RT software to obtain frequency deviations responses for CHR based PID controller for SPT with 0%, SPT with 20%, LDR with 0% and LDR with 20%, along with that of IMG, and it is observed that on both the platforms the obtained responses are almost similar.

• Controller efforts required to mitigate frequency deviations are also presented in form of responses.

• The derived time domain specifications are also tabulated. It is concluded from these outcomes that CHR based PID controller for LDR with 20% overshoot overperformed all other controllers by settling faster than other controllers.

After all the analysis, there is still scope for improvement as CHR method is only intended to satisfy certain overshoot requirements, usually 20% or 0%. Also, this method relies solely on the FOPDT model of the system. Hence it becomes esential to obtain FOPDT model in order to apply the CHR method. This lack of adaptability may prove to be a hindrance when various performance attributes are needed. To overcome this, in future research, learning-based optimization methods can be used to tune PID controllers by merging intelligent control strategies. Fractional-order PID controllers can also be used as an extension of PID controllers to control frequencies of IMG.

Author contributions

All authors contributed to the study, conception, and design. All authors commented on the manuscript. All authors read and approved the final manuscript.

Consent for publication

Authors transfer to Springer the publication rights and warrant that our contribution is original.

Funding

The authors did not receive support from any organization for the submitted work.

Availability of data and materials

The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Ethical approval

Ethical approval This paper does not contain any studied with human participants or animals performed by any of the authors.” should be changed to “This paper does not contain any studied with human participants or animals performed by any of the authors.