Introduction[*]

In recent years, the integration of renewable energy sources such as wind energy or PV in electrical power networks has been rapidly accelerated [1]. These types of electrical energy sources are environmentally friendly leading to less pollution concerning other resources such as fossil fuels or nuclear energy. On the other hand, limitations in building new transmission lines force to utilization of the maximum capacity of the transmission system optimally. This fact motivates the power engineers to use FACTs devices such as TCSC [2]. Meanwhile, DFIG-type wind turbines have been used to integrate wind turbines into power networks. The advantages of DFIGs are their simple control mechanism and long maintenance period [3, 4]. However, the major challenge in applying these elements in power networks is the possibility of losing transient stability under the effects of sudden fault disturbances.

Power Systems Stabilizers (PSS) devices are extensively employed in power networks to dampen unwanted oscillations [5, 6]. These control devices are traditionally designed by applying linear control methods such as PI control laws. However, the mentioned design method depends on the linearization of the power system equations around one specific operating point [7]. Applying nonlinear control strategies will naturally remove this limiting drawback. In general, major nonlinear controller design methods are based on applying the Lyapunov stabilization scheme. In this case, some transient stability assessment methods such as Transient Energy Function (TEF) [8-11], Multi-Input Backstepping [12], and SMC laws [13, 14] should be mentioned. The TEF design method is inherently complex. In Large-Scale power systems, the Multi-Input Backstepping method is faced with finding complex input control laws and extracting suitable control functions which is impossible in some cases [15,16]. Applying traditional SMC on power networks which include DFIG, PV, Synchronous Generators (SGs), plus TCSC may guarantee stability and also the robustness to unwanted changes. However, the traditional SMC method results in chattering in the overall control system and naturally in output signals [17]. The FTSMC method removes the above drawbacks and leads to more efficient canceling of the distortion in the outputs [18-19]. In comparison with SMC, the proposed method reaches the stable point of the system in a limited and adjustable time with less chattering.

In this paper, a nonlinear control design scheme is proposed to improve the stability and robustness of the power networks which include PV, DFIG, and TCSC elements. The contributions in this paper are as follows:

1) Applying FTSMC in a power system that includes PV, DFIG, and SG in the presence of TCSC.

2) Presenting a modified control law for input variables of DFIG, PV, and TCSC.

3) Robustness to changes in system parameters and topologies. In sliding mode control, the use of a fractional sliding surface and the optimal adjustment of its parameters, instead of the common sliding surfaces of the correct order, causes a more accurate adjustment of the sliding surface according to the increase in the degree of freedom, resulting in the improvement of tracking speed and accuracy.

The paper is organized as follows. In Section 2, power system modeling is presented. In the section, the modeling PV, SGs, DFIG, and the TCSC device model are discussed. The proposed control law in this paper is developed in Section 3. In Section 4 the time-simulation of the power system under the effect of this new control law is offered. A comprehensive comparison is added to show the benefits of the designed control law concerning the linear approach and traditional SMC design method. Section 5 is devoted to conclusions and final comparisons.

The variables used in the formulas and mathematic equations are introduced in Table 1.

Table (1): Description of Variables and their Equivalent Symbols

 Power system modeling

2.1. A PV Array Cells Model

Fig. 1 shows a PV system. The system contains a PV module, a boost DC-DC converter, a DC link, and an inverter connected to the network. The inverter controls the active and reactive powers to bind the injected current into the filter and to the grid as well. This is achieved by preserving a consistent DC link voltage via the DC-DC converter in case of a voltage decrease in the grid [20]. The penetration levels of renewable sources are considered to be 20%.

Fig. 1. PV System

To simply model a PV system in a widely spread power network, a current source with constant output power is introduced. This is depicted in Fig. 2. Here, the input power to the busses is considered as injected power. Besides, the use of the current model for solar cells makes the calculations easier. In this model, the current source with variable amplitude and phase is connected to a certain bus of the power grid. In this model, the effect of the inverter capacitor and the boost converter are neglected. This is because the behavior of the residual current is our main concern and the effect of the DC link is not significant. Moreover, the DC link on the boost converter has been kept constant. Considering that the solar radiation changes are slow and the studied problem is with grid frequency and constant sinusoidal mode, the above assumptions help to simplify the problem formulation and controller design technique. It should be also mentioned that the resulting changes are not noticeable with or without the capacitor effects. In a distributed power system, the connected bus to the PV cells is considered a PV-Bus. It is assumed that the injected active power of the PV-Bus to the power grid is constant. Moreover, it is also assumed that the reactive power may vary in a certain domain.

Fig. 2. The equivalent current source model of a PV array connected to the power network

 Therefore, the above property can be implemented to improve the voltage profile and power stability in the grid. Both active and reactive powers going to v_p from PV are defined as follows:

In Eqs. (1) and (2), , where,    و  are the amplitude and phase of the current-source model of PV cells, respectively. These quantities satisfy the following equations:

2.2.   DFIG Model

DFIG has two power electronics converters namely RSC and GSC. It is worthwhile to note that here the improvement of stability in the power system is considered and the RSC has the main effect on the stability; therefore, in the presented model, the inputs of the RSC are the control inputs. The considered model is known as the synchronous generator model for DFIG. In this model, the synchronous generator equivalent equations are employed to model the DFIG. In applying this model, one of the necessary conditions is to assume that the stator resistance is zero. Since the stability of the dynamics of mechanical variables is the main concern, the DC Link capacitor dynamic has been ignored. Hence, the following equations introduce a 3rd-order model for DFIG [21]:

where ,  and  with  are the amplitude and the phase of RSC, respectively. Therefore, the following equations can be obtained:

In Eq. (7), the variables , ,  and  are defined by Eq. (5).

• Synchronous Generators Model

Similar to Eq. (7), a third-degree corresponding model is assumed for synchronous generators. In this model,  is the excitation voltage of the generator which is assumed to be a controlled input [22]:

2.4.  Multi-Machine Power System Considering

TCSC

In this sub-section, the state-space equations of a multi-machine power system considering TCSC are introduced. Without loss of generality, it is assumed that the TCSC device is located between the i-th and j-th buses. In this case, algebraic equations of power balance of all the buses connected to the SGs, buses that are connected to TCSC and DFIG, and buses of PV cells arrays are introduced as follows [23]:

In Eq. (9), N is the total number of SGs buses, DFIG plus TCSC buses, and PV arrays buses. It is noted that N=n+3, where n is the collected number of SGs plus DFIG, n=n1+1 where n1 is the number of SGs.  is the element of the reduced-order impedance matrix of the whole grid. The considered model for the TCSC is a power injection model between s and r buses. In the equations, for simplicity, the controllable variable of TCSC, B_TCSC (variable susceptance) is considered instead of X_TCSC. Power balance algebraic equations of the r-th and s-th buses with added TCSC can be written in the following form:

Fig. 3. TCSC element connected to the power network

where  for the s-th bus. For the r-th bus, the corresponding equations are as follows:

The derivative of power balance equations leads to:

By separating the variables, it is obtained that:

The above equations can be rewritten in the following matrix form:  

Where,  ، ,  and . By using Eq. (15) with the definitions of  and  in Eqs. (5) and (6), the following equations will be obtained:

In Eq. (16), and  are related to the excitation circuit of SGs, and   is the controllable input of TCSC in the considering power system. Solving the matrix equations of Eq. (16) leads to:

Where

The parameters of the A matrix are calculated with load flow and in a similar fashion from the Jacobian. Hence, this matrix is non-singular. Furthermore, with the designed FTSMC or SMC control, the states of the system converge to the equilibrium point. Therefore, the A matrix will not be getting singularity. Finally, the state-space equations of the power system which includes DFIG, TCSC, and PV are derived in the following general form:

The first set of equations is related to the SGs, the second set is related to the DFIG, and the third set is about the grid buses' voltages. These obtained dynamic equations of the considered power system will be implemented in the design phase of proposed FTSMC laws which lead to the stability of the power grid. According to the mentioned references, the presented models for synchronous generators, DFIG, PV, and TCSC, help to implement the power system without decreasing the accuracy compared to the real model. In addition,  applying the above model makes it easier for controller design purposes. Here, the equations of the system states are expressed in a 3rd-order form. This fact leads to an acceptable level of accuracy and a simpler controller design procedure, in comparison with applying higher-order models which leads to the adjustment of many control parameters. It should be also mentioned that second-order equations need more assumptions on the system which naturally reduces the accuracy of the system behavior in practice. Therefore, the third-order model is an optimal one that can be used in the process of analysis and design of suitable control laws.

3.    Sliding Mode Controller Design

In the first stage of the design procedure, the following error variables are defined:

where is the internal angle of the i-th generator in the steady-state condition (before the occurrence of any possible fault), is the steady-state speed of the i-th generator, ( , and  is the active electrical power in the steady-state condition of the i-th generator, respectively.

3.1. Design of Traditional SMC

In the process of designing traditional SMC, the sliding surfaces are firstly defined as:

The i-th sliding surface is related to i-th generator. Derivation of (21) and substituting in (23) leads to:

The equivalent matrix form of Eq. (25) will be:

Hence,  results in:

The traditional SMC control law for the i-th generator is found to be:

3.2. Design of FTSMC

In this sub-section, the design procedure to obtain suitable nonlinear control laws in FTSMC formulation is developed. In this case, the sliding surface has a fractional order form for the i-th generator [25]:

The selected order in each sliding surface is a number between zero and one. Applying this method in the design process results in a faster control response and a high reduction of the chattering phenomenon in the outputs. In Eq. (29), in compliance with the FTSMC design method, the condition   should be observed.

Applying the same design procedure as in traditional SMC for this new case leads to the final control law of the form:

According to the comparison results of the input signal designed in Eq. (28) for SMC and Eq. (31)-(32) for FTSMC, it can be pointed out that in the SMC method, the sign function is used, while in FTSMC, the saturation function is implemented. This fact reduces a high amount of chattering in FTSMC. Secondly, in the FTSMC method, three parameters can be designed that are, γ₁, K₁, and K₂. These parameters guarantee the stability and convergence of the system in a limited and adjustable time. The details of designing the above control laws are presented in Appendix A. Fig. 4 depicts the block diagram of the proposed controllers. In fact, in this analysis, we have used the model close to the synchronous generator with its parameters for DFIG and the current source model for PV and its adaptation to the third-order model in SMC and FTSMC.

In Figure (4-a), the control system of the TCSC element is shown. The objective is to control the output active power of TCSC. This control scheme results in the appropriate reference value being produced in the output, taking into account the TCSC limitations. In Figure (4-b), the active and reactive output powers of PV are used to create the appropriate production flow in the PV. In Figure (4-c), the excitation inputs of synchronous generators are shown using the FTSMC method, considering the input limitations.

 (4-a)

(4-b)

(4-c)

Fig. 4. Proposed FTSMC block diagram for implementation on (4- a) TCSC (4-b) PV (4-c) DFIG, and SGs

 Fig. 5 The simulation flowchart of the power system including DFIG and PV in the presence of TCSC with applying FTSMC

In the following flowchart, the general configuration for the control of the power system with DFIG and PV in the presence of TCSC by applying the FTSMC is depicted.

4.    Simulation Results

To show the effectiveness of the proposed FTSMC law in stabilizing the whole power system in different operating conditions, a time simulation is carried out. The considered power system is assumed to be the standard 39-Bus New England network. The specifications of the considered NEW ENGLAND 39-bus power system are presented in Appendix B. The following figure shows the one-line diagram of the network. In this system, several TCSC elements are placed between buses No. 6 and 7. The reason for using TCSC in this place is to increase the active power transmission between buses 6 and 7. In the performed simulations, practical limits on the control inputs are assumed [23]:

In this power system under consideration, it is assumed that a 3-phase short-circuit fault occurs near generator No.4 and lasts for 100 milliseconds in time 5 sec.

This simulation has been done with  and damping = 0.7 for selected on K_1i, K_2i.

Fig. 7 shows the speed deviation of DFIG and several other SGs. In the simulation, the performance of the power system by applying PI linear control law, traditional nonlinear SMC law, and the proposed nonlinear FTSMC design-based are compared. Fig. 7 shows speed angle deviations of several machines. It can be seen that the performance of the power system by applying a linear PI controller has a high level of distortion in the outputs and in some cases the power system becomes unstable. Table 2 represents the resulting control characteristics by applying the proposed approach in comparison with applying linear control technique.

Fig. 6. The New England 39-Bus power system includes a DFIG and a PV array.

Table (2): A comparison between applying the proposed approach with applying the linear control method in the final values of control characteristics

This table also shows the promised improvement in the stability characteristics by applying the proposed controller concerning the results of applying linear control laws.

For the nonlinear controller design technique, the comparison shows the improved performance of applying FTSMC concerning traditional SMC. For instance, for Generator No.7 the amplitude of speed deviation is about 0.005 Pu by applying FTSMC, in comparison with 0.03 Pu as a result of applying traditional SMC. Another comparison is related to the speed profile of DFIG, in which, applying FTSMC leads to 1.2 seconds to reach the initial values after the removal of fault occurrence, however, this value by applying traditional SMC takes about 2.4 seconds to result.

In Fig. 8, the amplitudes of internal angle deviation of several SGs are depicted. For instance, the above value is significantly reduced by applying FTSMC concerning traditional SMC for Generators No. 6 & 7. Moreover, the above value for DFIG, whenever the FTSMC is applied, is about 0.36 Pu, and transient response lasts about 0.5 seconds, while similar values for DFIG in applying traditional SMC is 0.6 Pu for the amplitude of the first oscillation with 2 seconds for settling time represented in Table.3.

 Fig. 7. Speed deviations of the following, after occurring the fault: (a) DFIG and SGs, (b) DFIG, (c) Generator No. 6, (d) Generator No. 4, and (e) Generator No. 7.

Table (3): A comparison between the final control characteristics in applying the proposed approach concerning the SMC method

 Fig. 8. Plots of internal angle after the occurred fault (a) DFIG and SGs (b) Generator No.6 (c) Generator No.5 (e) Generator No.7

 Furthermore, in order to compare FTSMC with traditional SMC in damping performance, the phase diagrams of Generator No. 6 and DFIG are depicted in Fig. 9. It can be seen that controller design by applying FTSMC results in a faster response in returning to the initial value (the value before the occurrence of fault) concerning the same response whenever the traditional SMC is applied.

Fig. 9. Phase diagram of a) an SG and b) DFIG after the occurred fault

 Fig. 10 shows the variation in the output power of PV. It is assumed that a 3-phase short-circuit fault occurs near generator No.4 and lasts for 100 milliseconds in time 5 sec. It can be seen that in applying SMC control law, the resulting settling time is about 2.5 seconds while a similar value for FTSMC is 1 second. The finite time for system stabilization is assumed to be 1.1 seconds. This time, with attention in which there is a limit on system inputs are considered. It is also observed that the amplitude of chattering phenomena is highly reduced whenever the FTSMC is implemented concerning SMC traditional case. In Fig. 10 considering that the variations in active power are insignificant, it is acceptable to ignore the inverter capacitor in PV modeling.

Fig. 10. Plots of active power injected from PV generators after the occurred fault

Fig. 11 shows the TCSC's active power flow by applying the above three controller schemes. Implementing while applying nonlinear FTSMC and traditional SMC leads to improved performance. FTSMC in comparison to traditional SMC results in a high reduction of chattering amplitudes.

Fig. 11. Plots of power flow in TCSC after the occurred fault

 Fig. 12 depicts the voltage changes in the DC link of the grid-side converter. The required equations are given in Appendix C. According to the figure, it can be seen that the designed controller for the grid-side converter can adjust the voltage to the desired value of pre-fault values.

To demonstrate the efficiency of the designed controller and its robustness to the location of the fault occurring, a short-circuit fault is considered. The occurred fault is assumed to last for 100 milliseconds on bus No. 22 near generator No. 6. In Figure 13, the angular speed changes for generators No. 6 and No. 7, which are closer to the fault, are shown. According to these figures, both generators become stable after a short period and the closed-loop power system acts well against the occurring fault.

In practical cases, the mechanical input power to the generators may also vary with time. This phenomenon usually happens in wind-turbine ones. Hence, the robustness of the closed-loop control system in applying traditional SMC concerning FTSMC should be also compared. In the simulation, a 10% change in mechanical input power to the DFIG is assumed. The comparison results are depicted in Fig. 14. It can be observed that the robustness quality of applying FTSMC is highly improved concerning applying the traditional SMC.

Fig. 14. Plots of (a) input power, and (b) speed deviation of DFIG under the effect of a sudden change in the mechanical.

 Conclusions

In this study, it is assumed that both DFIG and PV are implemented in the power system network. The first motivation is the growing attention to use these renewable energy sources which have less pollution effects. The second motivation is their resulting inherent stability effect in the power system due to their low inertial values in comparison with synchronous machines. Therefore, for DFIG, 4 number is assumed concerning synchronous machines inertias. This effect can be observed in Tables 4 and 5. These two tables are added to show their importance in improving stability. It should be also noticed that in comparison with other machines, the angular velocity of this machine is higher. Hence, the above fact reveals that the proposed control law can prevent high angular oscillations.

PVs produce around 100 MW of electrical power, DFIGs produce about 200 MW. In comparison with 5000 MW production of synchronous machines, their contributions are approximately very low, however, the suitable control effect of TCSC leads to making a whole stable closed-loop power system network with nice controlling measures.

A modified suitable control law for a power system which includes PV, TCSC, DFIG, and several SGs, is developed by applying the FTSMC design technique. The proposed nonlinear control law improves the stability and robustness characteristics of the power system. The effectiveness of the proposed control scheme is validated on the 39-Bus New England power network. Also, a comparison is made with PI linear control law and traditional nonlinear SMC design techniques. The comparison shows the improvements and effectiveness of the presented method concerning the above control schemes. According to the results, it can be concluded that the FTSMC method has a better controlling performance compared to other methods in different aspects. For instance, it has been shown in the results that the amplitude of the first oscillation is reduced to 0.36 Pu, while this value in applying the traditional SMC design technique is about 0.6 Pu. The settling time is reduced from 2 seconds in applying traditional SMC to 0.5 seconds for FTSMC. As far as the robustness performance of the power system is concerned, simulation results show a high reduction in the chattering phenomenon by implementing the FTSMC design method in comparison with the SMC design.

Appendix A

Finite Time Sliding Mode Control (FTSMC):

Let's consider the nonlinear 3rd-order strict feedback systems under the effect of disturbance d, in the following form:

In this system,  are the state-variables for i=1, 2,3,  represents the uncertainty in the system model with the upper bound D˃0  , d( , which represents the uncertainties and the disturbances, is assumed to satisfy |d( )| ≤ ld, where ld > 0 is a constant  and  are differentiable functions from 3rd-order with . The u∈ R is the control input. The sliding surface is defined as follows [25-26]:

In the above equation, and  are constant design parameters. The variables and  should satisfy the following constraint: = ,

If both p and q are positive odd integers and satisfy the following condition: 0.5 <q/p < 1.

TSM manifold (A-2) can be modified to the following form:

Whenever the system reaches the sliding surfaces, we have, . Therefore:

which leads to

According to (A.4) with the selection of positive and , such that the equation becomes Hurwitz, hence, it is guaranteed that the equilibrium point is stable and the state variables of the system converge in a limited time. The complete proof is presented in [25]. By differentiating (A.5) with applying (A-1), the control input can be found from:

With

In (A.7),  is the saturation function which is depicted in Fig.A.1. In this function, , where  is a designed constant, and   is the threshold value for saturation function.

Control (A-7) has no singularity in the equilibrium point of the system, [x1, x2, x3], T= [0, 0, 0], because of which guarantees that the system converges to the origin in finite time.

Fig. A.1. Saturation Function

Appendix B

The specification of considered NEW ENGLAND 39-bus power system is presented here.

Table (4): Equivalent specifications of wind-turbine-based DFIG

 Table (5):  Equivalent specifications of synchronous generators

Appendix C

The dynamics of DC link capacitor:By suitably controlling the grid-side converter, it is possible to control the amount of DC link voltage ripple and the amount of output reactive power. The following relationship can be used to obtain the dynamics of the capacitor voltage:

Considering the output voltage of the grid-side converter in polar coordinates, that is, , the output power from the grid-side converter to the generator terminal is obtained as follows:

As a result, the voltage dynamics of the DC link capacitor is obtained as follows:

[*] Submission date:23, 05, 2024

Acceptance date: 29, 10, 2024

Corresponding author: Saeed Abazari, Department of Engineering, Shahrekord University, Shahrekord, Iran