Load shedding scheme for the two-area system with linear quadratic regulator

Abstract

The power system is prone to many emergency conditions which may lead to emergency state of operation with decay in the system frequency. The dramatic change in the frequency can result in cascaded failure of the system. In order to avoid power system collapse, load shedding (LS) schemes are adopted with the optimal amount of load shed. This paper proposed a methodology in a two-area thermal-thermal system for finding the required amount of load to be shed for setting the frequency of the system within minimum allowable limits. The LS steps have been obtained based on the rate of change of frequency with the increase in load in steps. A systematic study has been conducted for three scenarios: the scheme with a conventional integral controller; the scheme with a linear quadratic regulator (LQR); and the scheme with an LQR and superconducting magnetic energy storage devices (SMES). A comparison of the results has been presented on the two-area system.

Keywords

critical load / frequency response / load shedding (LS) / multi-area system / rate of change of frequency / linear quadratic regulator (LQR) / superconducting magnetic energy storage devices (SMES)

Introduction

The frequency is an important indicator of balance between load and generation. Any deviation in the system frequency due to excess load or occurrence of unpredictable incidents, e.g. faults, loss of generation, and loss of lines etc. can cause the system to enter into an emergency state. The effect of decrease in frequency may result in cascaded failure of the system components which may lead to the loss of synchronicity, thereby, the collapse of the system. Therefore, the frequency of the system must be controlled to a certain extent so that the system remains in synchronicity. Under extreme conditions of load generation mismatch, the frequency is restored by shedding the load in appropriate amount. However, the load shed must be economical without loss of economy to the system. Several studies have been proposed to restore the system operation frequency after serious disturbances. The demand side management (DSM) with load-shaping as objectives may help to restore the system frequency. The broad categories of load shaping are peak clipping, valley filling, load shifting, strategic conservation, strategic load growth, and flexible load shape. But in a critical situation, when the load demand is higher than generation, the strategy of load shedding (LS) can be adopted to maintain the system frequency. Several methods are described in the literature for generation rescheduling and LS to alleviate line overloading [1,2], in which attention has been focused on the problem of line over-flows. Many researchers have conducted extensive research into the problem related to under-frequency LS [3-5]. The decrease in system frequency has been taken as the criteria for load shedding in suitable amount so as to maintain system frequency within its normal range. LS was suggested to overcome the voltage stability problem [6], where application factors such as under-voltage relay setting and time delay are discussed.

Many researchers have proposed LS schemes with improved under-frequency-based approaches based on the rate of change of frequency in the system. Efforts to improve under-frequency LS using the rate of change of frequency as additional control variables have been suggested [7-13]. Other studies have been conducted on the adaptive scheme considering the voltage variation in order to identify and shed the sensitive load busses [14], a combination of frequency, df/dt and voltage changes [5,11] and the initial slope of df/dt for setting the under-frequency relays [8]. There are other LS schemes which take into account the interruptible part of loads [15], the frequency second derivatives [11], the cost of event based customers interruption [16], and adaptive LS scheme in the isolated system [17]. For the islanded distribution system, not much work has been conducted. The developed LS schemes so far have focused on the approaches for obtaining the optimal LS in the islanded system. Among the strategies are the scheme based on the frequency and df/dt information, the willingness of customers to pay and load histories [18], and the best time to shed the loads [19]. Karimi et al. have proposed a new conceptual strategy of a LS scheme to provide a solution for two scenarios in an islanded distribution system i.e. the moment the island is disconnected from the grid and during islanded system [20]. Sanaye-Pasand et al have utilized a strategy of combination of adaptive and intelligent under-frequency-load shedding scheme (UFLS) adopting the event-based and response-based method proposed [21,22] to intelligently and optimally shed the load in accordance to the load priority, and have implemented an intentional islanding operation considering Malaysia’s existing distribution network. Based on literature survey, Giroletti et al. have worked on the LS scheme for the isolated area system. A hybrid method based on frequency and power for industrial LS has been presented [23]. Tyagi et al. have suggested an LS scheme for multi-area system based on the concept put forward in Ref. [8] and studied the impact of SMES on the LS scheme [24]. The SMES under the excursions may provide effective countermeasure and the device has been utilized for the load frequency control of the multi-area systems [25]. The storage devices can play an important role in minimum load shedding to provide active power during the overloaded conditions and, thereby, improve the frequency.

In this paper, an LS scheme has been devised based on the same concept as utilized in Ref. [23] with the incorporation of linear quadratic regulator (LQR) based optimal control and the presence of SMES. The frequency response of a system with load variation has been determined without and with the LQR-SMES and results have also been obtained with the conventional integral controller and the LQR based optimal controller. The load disturbance increase has been taken in steps and the change in frequency of a system has been obtained. The comparison of the results obtained for the required amount of load to be shed with the conventional integral controller, the LQR based optimal controller, and the LQR and SMES having been provided. The Laplace equations for both areas and the corresponding polynomial of Laplace inverse have been solved using MATLAB 7.04 [26].

Mathematical model

In this section, a two-area interconnected power system has been considered for the LS taking load disturbance in Area 1. The impact of the LQR without and with SMES on the LS has been modeled for the two-area system and comparison has been provided for the LS steps to maintain frequency in the allowable range. Three scenarios, the scenario with the conventional integral controller, the scenario with the LQR based optimal controller, and the scenario with the LQR and SMES, have been considered for the LS scheme in the two-area system under study. A general diagram of the two-area interconnected thermal power system is shown in Fig. 1.

Model of the two-area system with the conventional integral controller

The block diagram of the two-area thermal system with the conventional integral controller is shown in Fig. 1. The change in demand as a step function is ΔP_d1 & ΔP_d2 in both areas. The sign of ΔP_d for a sudden increase in generation demand, ΔP_d>0, for a sudden decrease in load, ΔP_d<0. The step load change is expresses as

(1)\mathrm{\Delta }{P}_{\mathrm{d}}\left(t\right)=\mathrm{\Delta }{P}_{\mathrm{L}}u\left(t\right),

where ΔP_L is the disturbance magnitude in per unit based on the system volt-ampere base S_SB and u(t)is the unit step function. In the Laplace domain, the equation can be written as

(2)\mathrm{\Delta }{P}_{\mathrm{d}}\left(\mathrm{s}\right)=\frac{\mathrm{\Delta }{P}_L}{S},

The two-area interconnected thermal power system can be described by the standard state space equation described as

(3)\dot{\mathit{x}}\mathit{=}\mathit{A}\mathit{x}\mathit{+}\mathit{B}\mathit{u}\mathit{+}\mathit{\varsigma }\mathit{P}

and

(4)\mathit{y}=\mathit{C}\mathit{x}+\mathit{D}\mathit{u}\mathit{,}

where BoldItalic, BoldItalic and BoldItalic are the state, control, and disturbance vectors respectively; BoldItalic, BoldItalic, BoldItalic, BoldItalic and □ are real constant matrices of appropriate dimensions which in turn depend on the system parameters; and y is the output vector. The vectors in Eqs. (3) and (4) can be represented as

(5)\mathit{x}={\left[\mathrm{\Delta }{f}_1,\mathrm{\Delta }{f}_2,\mathrm{\Delta }{P}_{tie12},\mathrm{\Delta }{P}_{g1},\mathrm{\Delta }{P}_{g2},\mathrm{\Delta }{P}_{r1},\mathrm{\Delta }{P}_{t1},\mathrm{\Delta }{P}_{t12}\right]}^T,

(6)\mathit{u}\mathit{=}{\left[u_1,u_2\right]}^T,

(7)\mathit{P}={\left[\mathrm{\Delta }{P}_{d1},\mathrm{\Delta }{P}_{d2}\right]}^T.

By writing all equations for the two-area system, the matrices A, B and P can be obtained. With the help of these matrices, the transfer function of the system can be obtained as

(8)\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}=C{\left(SI-A\right)}^{-1}B+D.

For Area 1, the transfer function can be calculated for Area 1 and Area 2 corresponding to the change in demand as explained in Ref. [23]. The transfer function with only conventional integral controller is given as

(9){\displaystyle \frac{\mathrm{\Delta }{F}_1\left(s\right)}{\mathrm{\Delta }{P}_{d1}\left(s\right)}}=\left(5s^8+45.25s^7+154.75s^6+285.06s^5+364.47s^4+290.83s^3+57.69s^2+2.81s\right)/\left(s^9+9.1s^8+31.65s^7+69.07s^6+121.91s^5+125.13s^4+115.93s^3+40.83s^2+5.22s+0.2152\right).

With the presence of the LQR, the transfer function can be expressed as

(10){\displaystyle \frac{\mathrm{\Delta }{F}_1\left(s\right)}{\mathrm{\Delta }{P}_{d1}\left(s\right)}}=\left(5s^8+62.3s^7+337.9s^6+1015.1s^5+1837.6s^4+2090.7s^3+972.6s^2+87.2s\right)/\left(s^9+12.51s^8+68.45s^7+230.98s^6+538.42s^5+863.33s^4+962.88s^3+656.17s^2+224.04s+26.56\right).

Including the impact of the SMES along with the optimal controller, the transfer function can be expressed as

(11){\displaystyle \frac{\mathrm{\Delta }{F}_1\left(s\right)}{\mathrm{\Delta }{P}_{d1}\left(s\right)}}=\left(0.0001s^{10}+0.021s^9+0.275s^8+1.55s^7+4.48s^6+6.79s^5+5.87s^4+4.47s^3+0.74s^2+0.033s\right)/\left(0.01s^{10}+0.11s^9+16.34s^8+133.12s^7+362.71s^6+489.21s^5+671.65s^4+416.13s^3+309.77s^2+60.16s+0.02\right).

The Laplace inverse can be represented as a polynomial function of time t and is solved in MATLAB. It can be represented as

(12)\mathrm{\Delta }f1\left(t\right)=\left[\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{o}\mathrm{f}t\right]*\mathrm{\Delta }{P}_{\mathrm{d}}.

Since both areas are identical, the transfer function for Area 2 with the conventional integral controller, the LQR and the LQR and SMES are the same as given by Eqs. (9)-(11). Putting values of all constants and differentiating Eq. (11) with respect to time variable tand let the post-differentiated result be equal to zero, the result of t_mreveals the time when the variation of frequency is maximum. Substituting the value of time t_1m in Eq. (11), the maximum change in the frequency can be represented as

(13)\mathrm{\Delta }{f}_{1\max }=k_1\mathrm{\Delta }{P}_{\mathrm{L}}.

Similarly, for Area 2, the Laplace equation is solved and putting values of all constants, and differentiating the polynomial in time domain with respect to time variable tand let the post-differentiated result be equal to zero, the result of t_mcan be expressed and the maximum change in the frequency in Area 2 can be represented as

(14)\mathrm{\Delta }{f}_{2\max }=k_2\mathrm{\Delta }{P}_{\mathrm{L}},

k₁ and k₂ can be calculated based on the slope of the frequency deviations corresponding to the load change in both areas. With TPC limits on system frequency deviation within±4% (i.e.,±2.4 Hz for 60 Hz base), the minimum allowable frequency of f_min is equal to 57.6 Hz. Thus, the value of f_1min can be found by using

(15)f_{1\min }=60+\mathrm{\Delta }{f}_{1\max }.

Model with the LQR controller without and with the SMES for the two-area system

The two-area thermal system with the implementation of the LQR-based optimal controller is illustrated in Fig. 2. In this section, an application of modern control theory is applied to design an optimal LQR-based controller for the two-area system. State variables are defined as the outputs of the entire block having either an integrator or a time constant.

(16)\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}\mathit{u}+\mathit{F}\mathit{w},

where

\mathit{x}={\left[x_1,x_2,\mathrm{...},x_9\right]}^T=state\; vector,

\mathit{u}={\left[u_1,u_2\right]}^T=\; LQR\; control\; vector,

\mathit{w}={\left[w_1,w_2\right]}^T=\; disturbance\; vector,

BoldItalic, BoldItalic, BoldItalic are the real constant matrices of appropriate dimensions which, in turn, depend on the system parameters. In the optimal control scheme, the control inputs u₁ and u₂ are generated by means of the feedbacks from all the nine states with feedback constants to be determined in accordance with an optimality criterion.

The standard form employed in optimal control theory is

(17)\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}\mathit{u}.

For full state feedback, the control vector u is constructed by a linear combination of all the states, i.e.

(18)\mathit{u}=-\mathit{K}\mathit{x}\mathit{,}

where BoldItalic is the feedback matrix. The feedback matrix BoldItalic is determined so that a certain performance index (PI) is minimized in transferring the system from an arbitrary initial state x\prime \left(0\right) origin in infinite time (i.e x\prime (\propto ) = 0). A convenient performance index (PI) has the quadratic form expressed as

(19)PI\mathit{=}{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^{\infty }\left({{\mathit{x}}^{\prime }}^T\mathit{Q}{\mathit{x}}^{\prime }\mathit{+}{{\mathit{u}}^{\prime }}^T\mathit{R}{\mathit{u}}^{\prime }\right)}dt.

The matrices Q and R are defined with the following design considerations:

1) Excursing of ACEs about the steady values (x₇+B₁x₁, -a₁₂x₇+B₂x₄) are minimized. The steady state values of ACEs are of course zero.

2) Excursions of \int ACE₁dt about steady values (x₈ and x₉) are minimized.

3) Excursions of the control vector (BoldItalic₁ and BoldItalic₂) about steady value are minimized. The steady value of the control vector is a constant. The minimization is intended to indirectly limit the control effort within the physical capability of components.

With the above considerations, the performance index (PI) can be expressed as

(20)PI\mathit{=}{\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^{\infty }\left[{\left(x_7+B_1{x}_1\right)}^2+{\left(-a_{12}{x}_7+B_2{x}_4\right)}^2+\left(x_8^2+x_9^2\right)+k\left(u_1^2+u_2^2\right)\right]}dt,

where k= 1.

From the PI equation, BoldItalicand R can be obtained, BoldItalic = BoldItalic is symmetric matrix. The determination of the feedback matrix BoldItalic, which minimizes the above PI is the standard optimal regulator problem. BoldItalic is obtained from the solution of the reduced matrix Riccati equation as

(21){\mathit{A}}^T\mathit{S}+\mathit{S}\mathit{A}-\mathit{S}\mathit{B}{\mathit{R}}^{-1}{\mathit{B}}^T\mathit{S}\mathit{+}\mathit{Q}\mathit{=}0\mathit{.}

The acceptable solution of BoldItalic is that for which the system remains stable. For stability, all the eigen values of the matrix (BoldItalic-BoldItalic) should have negative real parts. The feedback matrix obtained for the two-area system is

\mathit{k}=\left[0.52861.14190.6813-0.0046-0.0211-0.0100-0.74370.99990.0000-0.0046-0.0211-0.01000.52861.14190.68130.74370.00000.9999\right]

The obtained eigen values of (BoldItalic-BoldItalic) are -3.8068, -0.8223+ 1.8150i, -0.8223–1.8150i, -2.4869+ 0.7552i, -2.4869 -0.7552i, -0.6952+ 0.8442i, -0.6952-0.8442i, -0.6952-0.8442i, -0.6909, and -0.0001. Each value contains negative real parts, thus the corresponding system is stable.

The model of the system with the LQR controlled SMES is depicted in Fig. 3. A thyristor controlled SMES consisting of a DC magnetic coil connected to the AC grid with its schematic diagram is well documented in Ref. [24]. The model has been utilized for the LS in a hybrid system and is described in Ref. [23] for observing the impact of SMES on the LS scheme. From above SIMULINK diagram, the arrangement of the LQR optimal controller along with SMES in an interconnected two-area thermal-thermal system can be observed.

With the LQR only and the LQR and SMES in the system, the similar procedure as described in section 2.1 is developed in this model for the two-area system to decide the steps to control the frequency deviations adopting appropriate steps to shed load. After deriving equations for change in frequency and the corresponding polynomial in time domain, putting the values of all parameters, the corresponding frequency response can be derived. After repeating the same procedure as in the section without SMES, t_m can be obtained when the variation of frequency is maximum. Substituting the value of tm in Δ f₁ (t) , the maximum change in the frequency in Area 1 can be obtained as

(22)\mathrm{\Delta }{f}_{1\max }\left(t\right)=k_3\mathrm{\Delta }{P}_{\mathrm{L}}.

Similarly, for Area 2, the value of time t_1m can be obtained as well as the change in the frequency

(23)\mathrm{\Delta }{f}_{2\max }\left(t\right)=k_4\mathrm{\Delta }{P}_{\mathrm{L}}

Now the value of f_1min can be found by using

(24)f_{1\min }=60+\mathrm{\Delta }{f}_{1\max }.

The amount of load to be shed for obtaining tolerable range of frequency can be calculated by using

(25)P_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{d}}=\left|\mathrm{\Delta }{P}_{\mathrm{L}}\right|-\mathrm{\Delta }{P}_{\mathrm{L}}\mathrm{a}\mathrm{t}57.6 \mathrm{H}\mathrm{z},

where P_shed is the amount of load to be shed.

Resuls and discussion

The amount of load to be shed has been calculated with the conventional integral controller, the LQR based optimal controller, and the LQR based SMES in the system. The results have been described for all the cases calculating the values of time t_m, constant, and change in the frequency corresponding to the amount of load shed.

General system with the conventional Integral controller

For observing the system, load disturbance is considered in Area 1. By applying the procedure, t_m = 1.4626 s, and k = 0.0706 are obtained. Thus, the maximum value of change in frequency is represented as

(26)\mathrm{\Delta }{f}_{\max }=0.0706 \mathrm{\Delta }{P}_{\mathrm{L}}.

According to Eq. (26), Table 1 has been formulated for frequency deviation at particular step load disturbances. Six cases with different load change have been given in Table 1 along with a critical load at which the frequency obtained is at tolerable value. From Table 1, it can be seen that for Cases 4, 5, and 6, the minimum frequency is less than the minimum allowable frequency limit (57.6 Hz). Thus, the LS is required for these cases.

The load at which frequency reaches at its minimum allowable limit is known as Critical Load. In this case, critical load is observed for Case 3 , which is 0.574 pu. For Cases 4, 5, and 6, LS is needed as the frequency is less than 57.6 Hz. The change in frequecy of Area 1 and Area 2 for all six cases corresponding to load disturbance in Area 1 before LS is displayed in Figs. 4 and 5 respectively.

For different amount of step loads the frequency responses are taken and the corresponding amount of loads to shed is calculated. For Cases 1, 2, and 3, LS is not needed as the minimum frequencies in these cases are already under the minimum allowable limit of 57.6 Hz. But for other remaining cases, the load needed to shed. Thus, LS is required for Cases 4, 5, and 6. The cases are presented below for the required amount of load to be shed.

For Case 4, the amount of load to be shed is 0.026 pu.

For Case 5, the amount of load to be shed is 0.226 pu.

For Case 6, the amount of load to be shed is 0.426 pu.

The frequency responses of both areas are demonstrated in Figs. 6 and 7 after the required amount of load is shed. It can be noticed from Figs. 6 and 7 that for Cases 4, 5, and 6, after LS, the frequency is obtained within the allowable limits and the frequency curves coincide with these cases, thus only one curve is visible.

General System with the LQR controller

In this section, the LS has been obtained with an application of the LQR. With the LQR and following the same procedure as described in the previous section, the values of time and k are obtained to decide the amount of load to be shed. The load disturbance is considered in Area 1. By applying the procedure explained in the previous section, t_m = 1.1626 s, and k=0.0538 is obtained. Thus, the maximum value of change in frequency is represented as

(27)\mathrm{\Delta }{f}_{\max }=\mathrm{0.0538}\mathrm{\Delta }{P}_{\mathrm{L}}.

According to Eq. (27), Table 2 has been formulated and the frequency deviation at particular step load disturbances and change in load steps are given. With the LQR in the system, the frequency deviations in Areas 1 and 2 are obtained corresponding to the load steps and for six different cases of load change. The frequency responses are exhibited in Figs. 8 and 9. The critical load obtained is at Case 6 with a critical load of 0.743 pu. It is observed that with the LQR in the system, the critical load is higher than that of the case without the LQR. With the LQR, the load step reduces and the system can take more load change.

From Table 2, it can be seen that for Cases 5 and 6, the minimum frequency is less than the minimum allowable frequency (57.6 Hz). Thus, LS is required for these cases. Cases 5 and 6 are given below with the required amount of load to be shed.

For Case 5, the amount of load to be shed is 0.057 pu.

For Case 6, the amount of load to be shed is 0.257 pu.

The frequency responses obtained after the required amount of load shed is shown in Figs. 10 and 11. As can be seen from Figs. 10 and 11, the frequency responses of Cases 5 and 6 coincide with Case 4 after LS and the frequency is within the allowable range of the frequency.

General System with LQR controller and SMES

In this section, the results are obtained with the LQR controlled SMES for observing the behavior of the system. By applying the procedure for LS, t_m = 0.0299 s and k = 0.0012446 are obtained. Thus, the maximum change in the frequency of the system corresponding to the amount of load shed is calculated as

(28)\mathrm{\Delta }{f}_{\max }=\mathrm{0.0012446}\mathrm{\Delta }{P}_{\mathrm{L}}.

According to Eq. (28), Table 3 has been formulated and the frequency deviation at particular step load disturbances and change in load steps are given. From Table 3, it is observed that LS is not needed due to the power supplied from SMES during load increase, thereby increasing the loadability of the system. Thus, LS is not needed for these cases. The frequency responses for Area 1 and Area 2 are shown in Figs. 12 and 13 for all five cases of step load change. It is observed that the frequency is within the limits for both areas.

The amount of load increase in the system is supplied by SMES optimally controlled with the LQR. Thus, LS is not needed as frequency deviations are observed in the tolerable band. For the purpose of differentiating all cases from each other, the figures have been obtained for narrow time scales. Thus, from Figs. 12 and 13, it can be seen that using the LQR controlled SMES, the frequency deviation in both areas are very low as compared to that of two other cases. Therefore, LS is not required in this system, either.

Comparisons of strategies

For comparing the given strategies, results are compared for equal load disturbance of 0.2 pu in Area 1 and the results are obtained with the conventional integral controller, the LQR, and the LQR-SMES. The frequency responses in Area 1 and Area 2 are presented in Figs. 14 and 15. The tie line deviations are given in Fig. 16. The deviations in the frequency and tie line power are observed for all cases. It is observed that the results obtained with the LQR-SMES are better than those of other cases. The frequency and tie line power deviations with the LQR-SMES are minimal compared to the cases with the conventional integral and the LQR based optimal conventional controller.

As the amount of load to be shed depends on the value of constant k (the load to be shed is less for less value of constant k) because the deviation in frequency increases as the value of constant k increases.

In the first case with only the conventional controller, the value of k is k₁=0.0706.

In the second case with the LQR based conventional integral controller, the value of k is k₂=0.0538.

In the third case with the LQR-SMES, the value of k is k₃ = 0.0012446.

With the LQR-SMES, the loadability of the system increases by a large value. This is due to the ability of the SMES to provide extra power to the system. Thus, the frequency deviation in the third strategy is minimal. Hence, there is no load to be shed for getting the acceptable frequency. Hence, the LQR based method with the SMES provides better results compared to other cases. The loadability of the system increases greatly with the LQR-SMES due to the additional capability of the SMES to supply load during the frequency dips in the system with load growth. Figure 17 is the minimum frequency for different loads. It can be seen from Fig. 17 that on using the LQR-SMES, the frequency for different load disturbances is more acceptable as compared to other cases, which in turn decreases the value of LS on using the LQR-SMES. With the LQR based LS scheme, the results are better compared to the conventional integral controller. With SMES as an additional source of power, the frequency deviations are minimal with different loads and there is no need to shed load with additional availability of power in the system.

Conclusions

In this paper, a study of LS scheme has been conducted with the conventional integral controller, the LQR, and the LQR and SMES. The results of all cases are obtained and compared for different load steps. It is observed that the LQR-SMES based LS gives a better performance compared to other cases. With the LQR-SMES, no load shed is needed for the allowable frequency and the settling time for frequency and tie line power flows are very small compared to other cases. The loadability of the system increases with the LQR-SMES due to the additional capability of the SMES to supply load during the frequency dips in the system with load increase. It can be seen that by using the LQR-SMES, the minimum frequency is more acceptable as compared to the system without the SMES and with the conventional integral controller for the same loads.

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