Advancing Load Frequency Control in Multi-Resource Energy Systems Through Superconducting Magnetic Energy Storage

Abstract

Given the fundamental importance of the power grid in both supply and demand, frequency stability is critical to the reliable and stable function of energy systems. When energy is stored in the system, it mitigates problems caused by various disturbances that interrupt the energy system’s operation. The energy storage system (ESS) stores excess energy and returns it to the system by reducing power oscillations and improving stability and dependability. Superconducting magnetic energy storage (SMES) is one strategy for storing energy in the power system. As a rotational storage system, its quick dynamic response is a significant advantage. This device can quickly release a substantial amount of energy. A gas power plant in one area, along with a steam and a hydropower plant in another, constitute a multi-resource energy system. This paper’s primary objective is to study and model how SMES affects the dynamic behavior of this energy system. The state-space representation of the power system’s dynamic behavior is given by first-order differential equations. This power system has a complexity of fifteen orders. The outcomes of the simulation using MATLAB software are presented in the time domain, and its correctness is shown by analyzing the power system’s modes. The results show that placing an SMES unit not only eliminates oscillations and frequency deviation but also reduces the induction time in the time responses of power in the connection line and frequency deviation. Different modes are considered for the energy system, and the effect of the power storage unit is shown by presenting the simulation results.

1. Introduction

The global energy demand is constantly increasing which makes modern power systems more complex [1]. This complexity shows from technological point of view such as shifts in demand patterns because of new types of electrical loads like electric vehicle charging stations [2], heating systems [3], battery storage [4] etc. Furthermore, there is a growing dependence on decentralized generation facilities. Consequently, the electricity industry has expanded significantly. On the generation side, the need for energy worldwide, along with concerns about climate change and the use of the fossil fuels has accelerated the shift toward renewable power generation mainly from sources like wind turbines and solar panels [5,6].

Multiple renewable energy sources are located in remote areas that require extended transmission lines to supply power to where it is needed. At the same time, global population growth and the expansion of industrial and commercial centers have increased the electricity demand [7,8]. Recently, there has been a substantial effort to generate more electricity from renewable sources. Nonrenewable fossil fuels are not a sustainable option for the future. Their rapid depletion which worsens energy challenges and growing environmental concerns highlights the need to prioritize renewable energy [9,10]. Hence, reducing the consumption of nonrenewable resources is crucial, and renewable energy will play a key role in the future by supporting lower greenhouse gas emissions [11,12].

The goal of the energy system is to temporarily equalize the amount of electrical power that is generated and consumed. The lack of energy storage means that the operation is not economical and the production rate of the network will follow the demand curve. Uncontrolled industrial, commercial and residential loads may cause power system instability, shortages and outages [13,14]. The application of various robust and adaptive control techniques across different areas of electrical engineering, ranging from robotic systems [15,16] to power system stability analysis [17], demonstrates the effectiveness of these control schemes in managing complex and dynamic challenges. Frequency in power systems is very sensitive to load changes. Various disturbances in the power system bring about the oscillation of different network variables such as frequency, electric power in transmission lines and the power system moving out of the balance point [18,19]. Frequency and voltage must stay within specified ranges for the electrical power grid to operate effectively. Voltage and frequency are both impacted by reactive and active power; however, these two issues can be addressed independently [20,21].

Uncertainty and unpredictability of load demand affect diverse functioning levels throughout the energy system, which leads to changes in frequency and exchange of energy between areas. The load frequency control framework is used in power systems by supplying power from active production plants to respond to the continuous development of loads and to sustain the exchange of power among connected parts. In addition to helping to boost the stability of the energy, this system ensures that the error of the permanent state of frequency changes is zero and that the exchange power is in the programmed value [22,23].

The intrinsic variability of energy produced from renewable plants, such as solar and wind energy, necessitates a focus on energy storage technology to reduce pollution, sweeten dynamic performance and optimize the economic utilization of the power system. Energy storage systems (ESS) are paramount elements in contemporary energy systems that serve to enhance system reliability and stability. Among the different types of energy storage technologies, we can mention mechanical types such as compressed air energy storage (CAES), electrical types like capacitor systems, thermal types such as heat storage, and electrochemical types such as batteries. The response speed is one of the important parameters of the power storage facilities because the goal is to tune the rapid changes in power demand in the energy system [24,25]. Magnetic power storage facilities using superconductors are one of the methods of power storage. Superconductor magnetic energy storage (SMES) is used to store energy, enhance energy stability, and reduce fluctuations [26,27]. Among the reasons for employing superconducting magnetic power storage facilities compared to other methods such as hydraulic pumping or compressed air, we can point out minimal losses, outstanding reliability, and minimal time delays for charging and discharging [28,29]. In [30], for a power system with high integration of renewable energy sources, mainly high wind power penetration, a heuristic-based optimization technique along with the use of SMES units is suggested for robust load frequency control. The results of the study show that substituting several renewable energy sources for a small number of traditional generator units decreases system inertia, negatively impacting the frequency stability of the energy system and ultimately weakening its performance. In [31], the authors describe an adaptive neural control system for a power system’s SMES, enabling better management of load frequency. The proposed SMES regulation scheme addresses challenges in real-world implementation. This work involves a voltage source converter and a two-quadrant chopper as components of the SMES power conversion system. In each control region, control is executed using an online neural regulator and a neural estimator. The neural estimator identifies the dynamics of the control area around an operating point, while the neural controller uses an introduced variable dependent on the area control error and the change in stored energy in the SMES coil to generate the power command for the corresponding SMES unit.

Power supply networks, control structures, and emergency frameworks are the three categories that can be used to classify the applications of SMES facilities. Up to this point, several investigations have been suggested on the utilization of SMES, which is capable of rapidly exchanging both reactive and active power with the electrical grid. Some of the applications of SMES include the improvement of the operation of wind energy sources [32], the enhancement of the damping of oscillations among areas of the electrical infrastructure [33], the enhancement of the quality of electricity in charging stations for electric vehicles [34], the enhancement of the stability of the energy framework in the presence of wind farms [35], and the management of the power of a wind energy system employing the dual-feed induction generator [36]. Through the utilization of various energy storage systems, the frequency of a power system can be regulated. This includes the improvement of inertial gain through the utilization of SMES [37], the control of frequency in a two-zone power system through the utilization of SMES [38], the regulation of frequency through the usage of battery energy storage systems in an island microgrid [39] and the enhancement of frequency and power profile in an island hybrid energy system through the utilization of SMES [40].

To enhance the frequency stability of the system, the virtual inertial regulation system that is based on SMES is successfully applied in a microgrid that is powered by renewable energies [41]. In this example, the dynamic characteristics of the architecture of the virtual inertial regulatory system are displayed. The results of the simulation demonstrate how the control system described above can compensate for the loss of inertia that the system experiences as a result of the penetration of renewable energy sources.

An integrated regulation framework based on a fuzzy logic strategy and a superconducting magnetic energy storage unit is proposed in [42] to solve frequency regulation problems for an island microgrid. The simulation results show that frequency oscillations produced by rapid load changes and the discontinuous nature of renewable energy sources are reduced by SMES units.

The effect of SMES in sweetening the transient stability of multi-machine energy systems linked to combined renewable energy systems is investigated in [43]. The simulation results are shown by creating symmetric errors in different places. In the proposed system, the predictive controller model has been used for the voltage source converter and inverter for more satisfactory output voltage regulation and to enrich the stability of the energy system.

The combination of LFC and SMES in interconnected power systems is suggested in [44] to increase the frequency variation damping speed and connection line capacity. In the proposed control method, the pattern search optimization algorithm is used to determine the coefficients of the regulation framework and the energy storage in the optimization function. The simulation results demonstrate the effect of SMES in a connected energy system in mitigating oscillations and improving power flow responses in the connection line.

An adaptive neural fuzzy system controller to reduce diverse load frequency regulation problems in a two-zone hydrothermal energy system is investigated in [45], in which a combination of superconducting magnetic power source facilities and thyristor-controlled phase shifting (TCPS) is used to improve LFC performance. The use of these two types of devices will stop the initial frequency drop and power deviations of the connection line after a sudden load disturbance.

The imbalance of production load and consumption load can cause the demand load to be larger than the rated power of the generators. This load disturbance can cause problems in the load frequency regulation framework. The created load difference can be compensated by the output power of the SMES system. In this paper, the aim is to study the effect of SMES on the dynamic function of the two-area energy system. The first area consists of two power plants, including hydro and steam, and the second area consists of a gas power plant. The simulation outcomes indicate that the load frequency regulation framework based on SMES has a faster response compared to the conventional load control system. The structure of the article is as follows. In Section 2, the linearized model of the magnetic energy storage superconducting system is illustrated. In Section 3, the studied system is introduced, and its equations are stated. In Section 4, the simulation outcomes employing MATLAB simulation software for different modes are shown. The conclusion is given in Section 5.

2. Superconducting Magnetic Energy Storage System

Specifically, the SMES structure is a device that utilizes direct current and stores energy within the magnetic field [46]. This system makes use of a high-inductance circuit that is constructed out of superconductors to store energy. The high efficiency of the power storage process is due to the superconductivity of the coil. This framework has a fast dynamic response. Superconducting energy storage units are usually made with a high capacity to flatten the consumption curve and a low capacity to increase the damping of fluctuations and boost the stability of the energy system. Among the most important properties of superconductors, we can mention the small resistance to the passage of direct current and the ability to pass high current density, the ability to produce strong magnetic fields, and the tunneling property. The SMES system does not consume active power or reactive power in a steady state [47,48].

The magnetic coil is linked to the alternative current network through the inverter power converter facilities (power conversion unit) and under normal conditions it is charged to the required amount through the power system network. When the load suddenly increases, the superconducting coil delivers the stored power to the energy grid through the power converter before the governor or any other regulation unit is activated and vice versa [49]. The model in Figure 1 is used for SMES in frequency response analysis, where frequency changes are considered as input signals and active power generation changes (ΔP_SMES) are considered as output signals [50,51]. Coil voltage changes (ΔE_d) are proportionally controlled using frequency changes. For the coil to respond instantly to the subsequent demand, the current must be swiftly returned to its nominal value following the disturbance. Consequently, the system implements the negative feedback signal (K_d). The time constant of the converter T_dc and the inductance of the winding L are defined. The rated current of the I_do coil is considered [52,53]. The linear representation of superconducting magnetic power storage as a post-phase-pre-phase compensator for automatic generation control is shown in Figure 2 [54,55]. The time constant and gain of SMES are shown as T_SMES and K_SMES, respectively, and T₁, T₂, T₃, and T₄ are time constants [56].

3. Load Frequency Control

Frequency deviation is one of the problems caused by the non-linearity and complicatedness of the energy system. Frequency deviation is a permanent problem, which is caused by the non-constant and continuous change of demand, so changing the generated power is necessary to keep the frequency constant at the nominal value. Among the advantages of the interconnected energy grid, we can point out the balancing of the mismatch in demand and supply, the integration of intermittent renewable power resources, and the usage of remote power facilities [57].Hierarchical control, shown in Figure 3, is one method for lowering frequency oscillations; it typically consists of 3 levels: first-level, second-level, and third-level. An emergency management loop can be necessary to bring the power system’s frequency back to its nominal value in critical conditions where the frequency deviates significantly from it. For power system operation in normal mode, minor frequency fluctuations are reduced by first level control [58,59].

In the first loop, frequency transients are prevented by using the governor’s drop property, which can cause a steady state error. Regarding the storage power in the energy system, the second level regulation loop is the frequency load control, which is used to restore the frequency in abnormal operation conditions. Load frequency regulation is one of the most critical control manners in the energy system, which maintains the system frequency and power deviations in their nominal levels. In the conditions of creating a severe imbalance between demand and supply, restoring the frequency to the nominal value is done in the third level control [60,61].

4. Configuration and Modeling of the Studied System

A power system consists of various areas of production facilities, which, in order to increase fault tolerance in the entire energy system, are linked to each other through lines [62,63]. Therefore, when a sudden change in load occurs in one area, the energy required by this area is supplied from other areas through connecting lines. Keeping the frequency stable against random changes of active loads, or, in other words, unknown external disturbances, is one of the most important tasks of the frequency load control system. Adjusting the power exchange error between areas is one of the other tasks of this control system.

4.1. Linearized Multi-Resource Energy System Model

The studied power system, according to Figure 4, consists of connecting two regions to each other by a transmission line, each region having a separate SMES. Load changes are applied as input signals in both areas. Area 1 includes two steam and hydro power plants, and area 2 consists of only one gas power plant. Each area has a superconducting magnetic power storage system, which is considered with the first-order conversion function, where K_SMES and T_SMES are the coefficient and time constant, respectively.

Governor and turbine are displayed in the first area for each power plant with three state variables and in the second area with four state variables. In each block area, load and mass diagrams are displayed with the first-order transfer function, where M_I and C_D are the combined inertia constant and damping constant, respectively. Therefore, the order of the system, considering seven state variables for the first area, five state variables for the second area, one state variable for transmission power in the line, and one state variable for the SMES system in each area, will be equal to 15. Each area has two inputs, which include load changes and speed set point changes. The state variables are specified in Figure 5, where the combined inertia constant and damping constant equivalent to rotating mass and load are represented by M_Ii and C_Di, respectively, for I = 1, 2, and T_TL is the synchronizing coefficient. Therefore, for the two-area energy system without a controller, two main inputs are considered, which are the changes in the load consumption. The parameters of the different blocks that make up the power system for three gas turbines, hydro turbines, and steam turbine power plants are listed in

Table 1. Parameters of gas turbine power plant blocks.

Subsystem	Parameter	Symbol
Feeding system	Time constant of gas turbine fuel	TF
	Gas turbine ignition reaction time delay	TCR
Compressor discharge system	Time constant of compressor discharge volume	TCD
Speed governor	Time constant controlling the gas turbine speed	TL
	Gas turbine speed governor delay time constant	TG
	Speed governor adjustment	SgG
Valve position	Valve positioner gain	CV
	Valve positioner time constant	TV

,

Table 2. Parameters of the blocks that make up the steam turbine power plant.

Subsystem	Parameter	Symbol
Turbine	Steam turbine time constant	TT
Speed governor	Speed governor adjustment	SgS
	Governor time constant	TG
Reheater	Compressive power section	FH
	Reheater time constant	TH

, and

Table 3. Parameters of the blocks that make up the hydro turbine power plant.

Subsystem	Parameter	Symbol
Turbine	Water start time	TW
Speed governor with drop transient compensation	Speed governor adjustment	SgH
	Governor time constant	TG
	Reset time	TR
	Slope ratio	TP

, respectively.

To equalize SMES in each area, the first-order transfer function with K_SMES gain and T_SMES time constant is used. State space modeling approaches are widely used in power system stability studies. The energy systems under study equations in the state space are as follows:

$${\displaystyle \frac{d}{dt}}{y}_1=-{\displaystyle \frac{1}{M_I}}(C_D{y}_1+y_2-y_{13}-y_{14}-w_1)$$

(1)

$${\displaystyle \frac{d}{dt}}{y}_2=-{\displaystyle \frac{1}{T_H}}{y}_2-({\displaystyle \frac{F_H-T_T}{T_T{T}_H}})y_3+{\displaystyle \frac{F_H}{T_T}}{y}_4$$

(2)

$${\displaystyle \frac{d}{dt}}{y}_3={\displaystyle \frac{-1}{T_T}}{y}_3+{\displaystyle \frac{1}{T_T}}{y}_4$$

(3)

$${\displaystyle \frac{d}{dt}}{y}_4=-{\displaystyle \frac{C_G}{T_G{R}_S}}{y}_1-{\displaystyle \frac{1}{T_G}}{y}_4+{\displaystyle \frac{K_G}{T_G}}{w}_3$$

(4)

$${\displaystyle \frac{d}{dt}}{y}_5={\displaystyle \frac{2T_R{C}_{G1}}{T_G{S}_{gP}{T}_P}}{y}_1-{\displaystyle \frac{2}{T_W}}{y}_5+2({\displaystyle \frac{T_P+T_W}{T_W{T}_P}})y_6+{\displaystyle \frac{2}{T_P}}({\displaystyle \frac{T_R^2-T_G{T}_R}{T_G{T}_R}})y_7+{\displaystyle \frac{2T_R{C}_G}{T_P{T}_G}}{w}_4$$

(5)

$${\displaystyle \frac{d}{dt}}{y}_6=-{\displaystyle \frac{C_G{T}_R}{T_G{S}_{gP}{T}_P}}{y}_1-{\displaystyle \frac{1}{T_P}}{y}_6-({\displaystyle \frac{T_R-T_G}{T_G{T}_R}})y_7+{\displaystyle \frac{C_G{T}_R}{T_P{T}_G}}{w}_4$$

(6)

$${\displaystyle \frac{d}{dt}}{y}_7=-{\displaystyle \frac{C_G}{S_{gP}{T}_G}}{y}_1-{\displaystyle \frac{1}{T_G}}{y}_7+{\displaystyle \frac{C_G}{T_G}}{w}_4$$

(7)

$${\displaystyle \frac{d}{dt}}{y}_8=-{\displaystyle \frac{1}{M_I}}(C_D{y}_8+y_9-y_{13}-y_{15}-w_2)$$

(8)

$${\displaystyle \frac{d}{dt}}{y}_9=-{\displaystyle \frac{1}{T_D}}{y}_9+{\displaystyle \frac{1}{T_D}}{y}_{10}$$

(9)

$${\displaystyle \frac{d}{dt}}{y}_{10}={\displaystyle \frac{C_v{T}_S{T}_L}{T_F{T}_v{S}_{gG}{T}_G}}{y}_8-{\displaystyle \frac{1}{T_F}}{y}_{10}-{\displaystyle \frac{1}{T_F}}({\displaystyle \frac{T_S-T_G}{T_G}})y_{11}-{\displaystyle \frac{T_S}{T_G{T}_F}}({\displaystyle \frac{T_L+T_V}{T_V}})y_{12}$$

(10)

$${\displaystyle \frac{d}{dt}}{y}_{11}=-{\displaystyle \frac{C_v{T}_L}{T_v{T}_G{S}_{gG}}}{y}_1-{\displaystyle \frac{1}{T_G}}{y}_{11}+{\displaystyle \frac{1}{T_G}}({\displaystyle \frac{T_L+T_V}{T_V}})y_{12}+{\displaystyle \frac{C_V{T}_L}{T_G{T}_V}}{w}_5$$

(11)

$${\displaystyle \frac{d}{dt}}{y}_{12}=-{\displaystyle \frac{1}{T_V}}({\displaystyle \frac{C_V}{S_{gG}}}{y}_1+y_{12}-C_V{w}_5)$$

(12)

$${\displaystyle \frac{d}{dt}}{y}_{13}=T_S(y_1-y_8)$$

(13)

$${\displaystyle \frac{d}{dt}}{y}_{14}=-{\displaystyle \frac{1}{T_{SMES1}}}{y}_{14}+{\displaystyle \frac{K_{SMES1}}{T_{SMES1}}}{y}_1$$

(14)

$${\displaystyle \frac{d}{dt}}{y}_{15}=-{\displaystyle \frac{1}{T_{SMES2}}}{y}_{15}+{\displaystyle \frac{K_{SMES2}}{T_{SMES2}}}{y}_8$$

(15)

4.2. State-Space Equations Representation

The studied system is a multi-input, multi-output system. The equations of this system can be displayed as several sub-systems. Figure 6 shows the studied power system model implemented in the Simulink MATLAB environment. By using the first-order equations representing different parts and the block diagram, the dynamic equations of the studied multi-resource energy system can be displayed in the state space as follows:

$$\{\begin{array}{l}{\displaystyle \frac{d}{dt}}Y=AY+BW\\ Z=CY+DW\end{array}$$

(16)

where Y is the state variable vector W is the input or control vector, A is the system matrix, C is the control matrix, B is the input matrix, and D is the feedback matrix.

The vector of state variables is composed as follows:

$$Y=\left[\begin{array}{c}{Y}_{area1}\\ Y_{area2}\\ Y_{tie}\\ Y_{smes}\end{array}\right]$$

(17)

where the constituent vectors of the state variables based on system division are as follows:

$$Y_{area1}={\left[\begin{array}{ccccccc}{y}_1& y_2& y_3& y_4& y_5& y_6& y_7\end{array}\right]}^T$$

(18)

$$Y_{area2}={\left[\begin{array}{ccccc}{y}_8& y_9& y_{10}& y_{11}& y_{12}\end{array}\right]}^T$$

(19)

$$Y_{tie}=\left[y_{13}\right]$$

(20)

$$Y_{smes}={\left[\begin{array}{cc}{y}_{14}& y_{15}\end{array}\right]}^T$$

(21)

Therefore, the matrix representation of the system equations in the state space is expressed as follows:

$$A=\left[\begin{array}{cccc}{A}_{11}& A_{12}& A_{13}& A_{14}\\ A_{21}& A_{22}& A_{23}& A_{24}\\ A_{31}& A_{32}& A_{33}& A_{34}\\ A_{41}& A_{42}& A_{43}& A_{44}\end{array}\right]$$

(22)

where the submatrices are as follows:

$$A_{11}=\left[\begin{array}{ccccccc}-{\displaystyle \frac{C_D}{M_{I1}}}& {\displaystyle \frac{1}{M_{I1}}}& 0& 0& {\displaystyle \frac{1}{M_{I1}}}& 0& 0\\ 0& -{\displaystyle \frac{1}{T_H}}& {\displaystyle \frac{1-F_H}{T_H}}& {\displaystyle \frac{F_H}{T_H}}& 0& 0& 0\\ 0& 0& -{\displaystyle \frac{1}{T_T}}& {\displaystyle \frac{1}{T_T}}& 0& 0& 0\\ -{\displaystyle \frac{C_G}{T_G{S}_{gS}}}& 0& 0& -{\displaystyle \frac{1}{T_G}}& 0& 0& 0\\ {\displaystyle \frac{2C_G{T}_R}{S_{gP}{T}_G{T}_P}}& -{\displaystyle \frac{2}{T_W}}& {\displaystyle \frac{2(T_P+T_W)}{T_W{T}_P}}& {\displaystyle \frac{2}{T_P}}({\displaystyle \frac{T_R+T_G}{T_G}})& 0& 0& 0\\ -{\displaystyle \frac{C_G{T}_R}{S_{gP}{T}_G{T}_P}}& -{\displaystyle \frac{1}{T_P}}& 0& 0& 0& 0& -{\displaystyle \frac{1}{T_P}}({\displaystyle \frac{T_R+T_G}{T_G}})\\ -{\displaystyle \frac{C_G}{S_{gP}{T}_G}}& 0& 0& 0& 0& 0& -{\displaystyle \frac{1}{T_G}}\end{array}\right]$$

(23)

$$A_{22}=\left[\begin{array}{ccccc}-{\displaystyle \frac{C_{D2}}{M_{I2}}}& {\displaystyle \frac{1}{M_{I2}}}& 0& 0& 0\\ 0& -{\displaystyle \frac{1}{T_D}}& {\displaystyle \frac{1}{T_D}}& 0& 0\\ {\displaystyle \frac{T_L{C}_V{T}_S}{T_G{T}_W{S}_{gG}{T}_F}}& 0& -{\displaystyle \frac{1}{T_F}}& -{\displaystyle \frac{1}{T_F}}({\displaystyle \frac{T_G-T_S}{T_G}})& {\displaystyle \frac{-T_S}{T_G{T}_F}}({\displaystyle \frac{T_L+T_V}{T_V}})\\ -{\displaystyle \frac{C_V{T}_L}{T_V{S}_{gG}{T}_G}}& 0& 0& -{\displaystyle \frac{1}{T_G}}& {\displaystyle \frac{1}{T_G}}({\displaystyle \frac{T_V+T_L}{T_V}})\\ 0& 0& 0& 0& {\displaystyle \frac{1}{T_V}}\end{array}\right]$$

(24)

$$A_{33}=\left[0\right],A_{44}=\left[\begin{array}{cc}-{\displaystyle \frac{1}{T_{SMES1}}}& 0\\ 0& -{\displaystyle \frac{1}{T_{SMES2}}}\end{array}\right],A_{12}=A_{21}^T=0_{7\times 5}$$

(25)

$$A_{13}=\left[\begin{array}{c}-{\displaystyle \frac{1}{M_{IS}}}\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],A_{14}=\left[\begin{array}{cc}-{\displaystyle \frac{1}{M_{IS}}}& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right],A_{23}={\left[\begin{array}{ccccc}{\displaystyle \frac{1}{M_{IH}}}& 0& 0& 0& 0\end{array}\right]}^T$$

(26)

$$A_{24}=\left[\begin{array}{cc}0& -{\displaystyle \frac{1}{M_{IG}}}\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right],A_{31}=\left[\begin{array}{ccccccc}{T}_S& 0& 0& 0& 0& 0& 0\end{array}\right],A_{32}=\left[\begin{array}{ccccc}-T_S& 0& 0& 0& 0\end{array}\right]$$

(27)

$$A_{34}=A_{43}^T=o_{1\times 2},A_{41}=\left[\begin{array}{ccccccc}{\displaystyle \frac{K_{SMES1}}{T_{SMES1}}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right],A_{42}=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ {\displaystyle \frac{K_{SMES2}}{T_{SMES2}}}& 0& 0& 0& 0\end{array}\right]$$

(28)

5. Simulation Results

Simulation studies of a two-area energy system for different conditions are illustrated in this section. The frequency deviation and the rate of change of the transmission power of the connection line for each area are shown in two cases, with and without the power storage system. In this section, adverse scenarios are considered for simulation. The parameters of the studied power system are listed in Appendix A.

5.1. Multi-Resource Energy System Without SMES

Modes of this two-area energy system without SMES for three different modes, when each area works independently and when both areas are connected, are shown in

Table 4. Modes of two-area power systems without SMES.

Only Area 1	Only Area 2	Two-Area
−0.1190 −0.3682 −3.1655 −5.0000 −5.9078 −33.2930 −0.4908 ± j0.6379 (η = 0.6098)	−3.1422 −5.9026 −19.9651 −0.5954 ± j0.6424 (η = 0.6797)	−0.0248 −0.1194 −0.4254 −3.1431 −3.1665 −5.0000 −5.9016 −5.9024 −19.9651 −0.4397 ± j0.6082 (η = 0.5859) −0.5874 ± j0.6449 (η = 0.6734)

. As can be seen, the power system is stable in all three scenarios. The damping coefficient of oscillating modes for all three modes is more than 0.58.

The output mechanical power of three production units and frequency changes in two areas for step changes in demand load are shown in Figure 7 and Figure 8, respectively. Additionally, the transmission power between the two areas is shown in Figure 9. As can be seen, the frequency droop in two areas reaches 0.06 Hz in permanent mode. Moreover, the fluctuations in the mechanical power changes of the water unit are more than in the other two units, but in the end, the permanent state value of the mechanical power has reached 0.3 for all three production units.

5.2. Multi-Resource Energy System with SMES

Figure 10 and Figure 11 show the response of frequency changes and mechanical power output changes of three production units for step changes in load demand.

In this case, the power system is equipped with SMES. As can be seen, the controller has worked well, and the frequency changes in the two areas have tended to zero after changes in the load.

The changes in transmission power between two areas due to load changes in the first area are shown in Figure 12. As can be seen, the controller has caused the amount of transmission power to tend to zero in a permanent state.

Note that, for real-world implementation of SMES, key considerations include the large size of units, which may be constrained by available installation space; high initial costs due to superconducting materials and cooling systems; and the need for regular maintenance. Moreover, the integration of SMES into variable grid environments requires managing grid stability and the variability of renewable energy sources.

6. Conclusions

Over the past few years, there has been a substantial rise in the need for electricity, resulting in the growth of the electric energy-producing industry. As conventional energy sources have been running out, electric power networks have had to incorporate alternative renewable energy sources, such as solar panels, fuel cells, and wind turbines. The power system exhibits non-linearity in its behavior, and there exists a complex interconnection among its numerous components. The primary factors contributing to frequency challenges in the power system are typically variations in load and the sporadic power output of renewable energy sources. Ensuring a power equilibrium between the sources and the load is essential for maintaining a stable frequency in the network. Energy storage devices have gained significant importance in modern power systems because of their capacity to improve system stability and reliability. The objective of this study is to examine the influence of a superconducting magnetic energy storage system on load frequency regulation in a power system that is interconnected between two locations. This study incorporates findings from time-domain simulations of a power system that integrates hydroelectric, thermal, and gas power facilities spanning two regions. Future work will involve comparing the performance of various energy storage systems, including batteries, flywheels, and SMES, within the proposed energy system.