Load shedding scheme for an interconnected hydro-thermal hybrid system with SMES

Abstract

The frequency of the power system varies based on the load pattern of the consumers. With continuous increase in the load, the frequency of the system keeps decreasing and may reach its minimum allowable limits. Further increase in the load will result in more frequency drop leading to the need of load shedding, if excess generation is not available to cater the need. This paper proposed a methodology in a hybrid thermal-hydro system for finding the required amount of load to be shed for setting the frequency of the system within its minimum allowable limits. The load shedding steps were obtained based on the rate of change of frequency with the increase in the load in both areas. The impact of superconducting magnetic energy storage (SMES) was obtained on load shedding scheme. The comparison of the results was presented on the two-area system.

Keywords

critical load / frequency response / load shedding / multi-area system / rate of change of frequency / superconducting magnetic energy storage (SMES) device

Introduction

The frequency deviation is caused by the imbalance between load and generation. This effect is most serious in the system that has excess load or occurrence of unpredictable incidents like faults, loss of generation, and loss of lines. These events may result in the cascaded effects of reduction in the frequency, causing the loss of synchronous thereby the collapse of the system. Therefore, the frequency of the system must remain in the tolerable range without causing any stress to the turbine of the system. Since there is no direct control of utility load, the primary method of restoring frequency is to shed the load in appropriate amount. This must be done with considerate planning, since there are no benefits in shedding excessive amount of loads which may lead to the loss of economy to the system. Several studies were conducted to restore the system operation frequency after serious disturbances. One of these strategies is the adoption of the under-frequency load shedding design [1-5]. Many authors proposed load shedding schemes with improved under-frequency-based-approaches based on the rate of change of frequency in the system. Efforts to improve under-frequency load shedding using the rate of change of frequency as additional control variables were suggested in Refs. [6-11]. Shilling proposed the detection of the initial rate of change of frequency with predetermined time intervals to formulate an appropriate load shedding strategy in Ref. [10]. The approach was made adaptive to avoid load-shedding failures considering the maximum change of frequency at the last step in order to adjust the required total amount of shed load. On account of tedious computations required for determining the amount of overload and the load shed per step, a neural network approach was employed to calculate the minimum frequency during a forced outage of a generating unit [12].

Giroletti et al. provided a systematic review of the existing of load shedding schemes [13] and proposed a hybrid frequency/power-based approach based on the information provided by the measurement of frequency and power available from generators for industrial load shedding.

To compensate for the sudden load changes, an active power source with fast response such as superconducting magnetic energy storage (SMES) unit is expected to be the most effective countermeasure. Abraham utilized SMES for the load frequency control of the multi-area systems [14]. The frequency response of the system varies with the load and parameters. Thus, by using suitable values of these parameters, the frequency response of the system can be controlled. The SMES can play an important role in the load shedding scheme providing active power during the overload conditions and thereby improving the frequency of the system.

In this paper, load shedding scheme in an interconnected two-area thermal-hydro hybrid system was proposed. Two cases of load shedding scheme for thermal-hydro system without SMES device, and with SMES were presented. Moreover, the results were obtained for step load change in both the areas. The transfer function of both the areas was developed and steps for load shedding scheme in both the areas were derived. Furthermore, the results obtained for the required amount of load to be shed and the impact of SMES on the amount of load to be shed were compared. And finally, the Laplace equations for both the areas and the corresponding polynomial of Laplace inverse were solved using MATLAB 7.04<FootNote>

The MATLAB by Mathworks Corporation, SIMULINK toolbox of MATLAB version 7.6, 2008

</FootNote>.

Mathematical model

In this section, a two-area interconnected hydrothermal power system is considered for load shedding, taking load disturbances in both areas into consideration. The impact of SMES on load shedding is modeled for hydro-thermal system, and a comparison is made for load shedding steps to maintain frequency in the allowable range. Two scenarios under observation are presented, one without using SMES, and one with SMES in the thermal area. A general diagram of interconnected hydrothermal power plant is given in Fig. 1.

Model without SMES

The block diagram of the two-area hydro-thermal system without using SMES is illustrated in Fig. 1. The change in demand is considered as a step function, ΔP_d1 & ΔP_d2 in both the areas. The sign of ΔP_d is such that, 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}}(t)=\mathrm{\Delta }{P}_{\mathrm{L}}u(t)

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

(2)\mathrm{\Delta }{P}_{\mathrm{d}}(s)={\displaystyle \frac{\mathrm{\Delta }{P}_L}{s}}\mathrm{.}

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

(3)\stackrel{·}{\mathit{X}}=A\mathit{X}+\mathit{B}U+s\mathit{P},

and

(4)\mathit{Y}=\mathit{C}\mathit{X}+\mathit{D}\mathit{U},

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

(5)\mathit{X}={[\mathrm{\Delta }{f}_{\mathrm{1}},\mathrm{\Delta }{f}_{\mathrm{2}},\mathrm{\Delta }{P}_{\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{12}},\mathrm{\Delta }{P}_{\mathrm{g}\mathrm{1}},\mathrm{\Delta }{P}_{\mathrm{g}\mathrm{2}},\mathrm{\Delta }{P}_{\mathrm{r}\mathrm{1}},\mathrm{\Delta }{P}_{\mathrm{t}\mathrm{1}},\mathrm{\Delta }{P}_{\mathrm{t}\mathrm{2}}]}^{\mathrm{T}},

(6)\mathit{U}={[u_{\mathrm{1}},u_{\mathrm{2}}]}^{\mathrm{T}},

(7)\mathit{P}={[\mathrm{\Delta }{P}_{\mathrm{d}\mathrm{1}},\mathrm{\Delta }{P}_{\mathrm{d}\mathrm{2}}]}^{\mathrm{T}}\mathrm{.}

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

Transfer function= BoldItalic(BoldItalic-BoldItalic) ^-1+BoldItalic.

For Area 1, the transfer function is calculated as

(8)$$\begin{gathered} {\displaystyle \frac{\mathrm{\Delta }{f}_{\mathrm{1}}(s)}{\mathrm{\Delta }{P}_{\mathrm{d}\mathrm{1}}(s)}}=(\mathrm{0.0006}{s}^{\mathrm{11}}+\mathrm{0.014}{s}^{\mathrm{10}}+\mathrm{0.107}{s}^{\mathrm{9}}+\mathrm{0.4}{s}^{\mathrm{8}}+\mathrm{0.87}{s}^{\mathrm{7}}+\mathrm{1.5}{s}^{\mathrm{6}} \\ +\mathrm{1.98}{s}^{\mathrm{5}}+\mathrm{1.24}{s}^{\mathrm{4}}+\mathrm{0.13}{s}^{\mathrm{3}}+\mathrm{0.008}{s}^{\mathrm{2}}+\mathrm{0.0003}s)/ \\ (\mathrm{0.0001}{s}^{\mathrm{12}}+\mathrm{0.002}{s}^{\mathrm{11}}+\mathrm{0.018}{s}^{\mathrm{10}}+\mathrm{0.08}{s}^{\mathrm{9}}+\mathrm{0.28}{s}^{\mathrm{8}}+\mathrm{0.68}{s}^{\mathrm{7}} \\ +\mathrm{1.09}{s}^{\mathrm{6}}+\mathrm{1.22}{s}^{\mathrm{5}}+\mathrm{0.84}{s}^{\mathrm{4}}+\mathrm{0.25}{s}^{\mathrm{3}}+\mathrm{0.03}{s}^{\mathrm{2}}+\mathrm{0.002}s+\mathrm{0.0001})\mathrm{.} \end{gathered}$$

Similarly for Area 2, the transfer function can be obtained as

(9)$$\begin{gathered} {\displaystyle \frac{\mathrm{\Delta }{f}_{\mathrm{1}}(s)}{\mathrm{\Delta }{P}_{\mathrm{d}\mathrm{2}}(s)}}=(\mathrm{0.002}{s}^{\mathrm{9}}+\mathrm{0.045}{s}^{\mathrm{8}}+\mathrm{0.34}{s}^{\mathrm{7}}+\mathrm{1.18}{s}^{\mathrm{6}}+\mathrm{2.04}{s}^{\mathrm{5}}+\mathrm{1.63}{s}^{\mathrm{4}} \\ +\mathrm{0.457}{s}^{\mathrm{3}}+\mathrm{0.034}{s}^{\mathrm{2}}+\mathrm{0.0006}s)/ \\ (\mathrm{0.0001}{s}^{\mathrm{12}}+\mathrm{0.002}{s}^{\mathrm{11}}+\mathrm{0.018}{s}^{\mathrm{10}}+\mathrm{0.08}{s}^{\mathrm{9}}+\mathrm{0.28}{s}^{\mathrm{8}}+\mathrm{0.68}{s}^{\mathrm{7}}+\mathrm{1.09}{s}^{\mathrm{6}}+\mathrm{1.22}{s}^{\mathrm{5}} \\ +\mathrm{0.84}{s}^{\mathrm{4}}+\mathrm{0.25}{s}^{\mathrm{3}}+\mathrm{0.03}{s}^{\mathrm{2}}+\mathrm{0.002}s+\mathrm{0.0001})\mathrm{.} \end{gathered}$$

On adding Eqs. (8) and (9), and representing

\mathrm{\Delta }{P}_{\mathrm{d}\mathrm{1}}(s)=\mathrm{\Delta }{P}_{\mathrm{d}\mathrm{2}}(s)=\mathrm{\Delta }{P}_{\mathrm{d}}(s),

Equation (10) can be obtained

(10)$$\begin{gathered} {\displaystyle \frac{\mathrm{2}\mathrm{\Delta }{f}_{\mathrm{1}}(s)}{\mathrm{\Delta }{P}_{\mathrm{d}}(s)}}=(\mathrm{0.0006}{s}^{\mathrm{11}}+\mathrm{0.014}{s}^{\mathrm{10}}+\mathrm{0.1087}{s}^{\mathrm{9}}+\mathrm{0.4397}{s}^{\mathrm{8}}+\mathrm{1.21}{s}^{\mathrm{7}} \\ +\mathrm{2.685}{s}^{\mathrm{6}}+\mathrm{4.02}{s}^{\mathrm{5}}+\mathrm{2.87}{s}^{\mathrm{4}}+\mathrm{0.59}{s}^{\mathrm{3}}+\mathrm{0.042}{s}^{\mathrm{2}}+\mathrm{0.009}s)/ \\ (\mathrm{0.0001}{s}^{\mathrm{12}}+\mathrm{0.002}{s}^{\mathrm{11}}+\mathrm{0.018}{s}^{\mathrm{10}}+\mathrm{0.08}{s}^{\mathrm{9}}+\mathrm{0.28}{s}^{\mathrm{8}}+\mathrm{0.68}{s}^{\mathrm{7}} \\ +\mathrm{1.09}{s}^{\mathrm{6}}+\mathrm{1.22}{s}^{\mathrm{5}}+\mathrm{0.84}{s}^{\mathrm{4}}+\mathrm{0.25}{s}^{\mathrm{3}}+\mathrm{0.03}{s}^{\mathrm{2}}+\mathrm{0.002}s+\mathrm{0.0001})\mathrm{.} \end{gathered}$$

The Laplace inverse of Eq. (10) represented as a polynomial function of time t, is solved in MATLAB that can be represented as

(11)\mathrm{\Delta }{f}_{\mathrm{1}}(t)=[\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]\mathrm{\Delta }{P}_{\mathrm{d}}\mathrm{.}

Putting the values of all constants and differentiating Eq. (11) with respect to time variable t and let the post-differentiated result be equal to zero, the result of t_m reveals the time when the variation of frequency is maximum. This value of time obtained is t_1m = 2.21 s. Substituting the value of time t_1m for the value in Eq. (11), the maximum change in the frequency is represented as

(12)\mathrm{\Delta }{f}_{\mathrm{1}\max }=\mathrm{5.1}\mathrm{\Delta }{P}_{\mathrm{L}}\mathrm{.}

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 t and let the post-differentiated result be equal to zero, the result of t_m can be expressed as t_2m = 1.5 s, and the maximum change in frequency in Area 2 can be represented as

(13)\mathrm{\Delta }{f}_{\mathrm{2}\max }=\mathrm{6.5}\mathrm{\Delta }{P}_{\mathrm{L}}\mathrm{.}

As for the selection of the minimum allowable frequency, because the TPC limits the 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. Now the value of f_1min can be found by using

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

Configuration of the SMES in the power system

A thyristor controlled SMES consisting of DC magnetic coil connected to the AC grid with its schematic diagram is displayed in Fig. 2 [13]. The superconducting coil contained in a helium vessel and the heat generated is removed by means of a low-temperature refrigerator. Helium is used as the working fluid in the refrigerator and the current in the superconducting coil can be tens of thousands or hundreds of thousands of amperes. A transformer is mounted on each side of the converter unit to convert the high voltage and low current of the AC system to the low voltage and high current required by the coil as no AC power system normally operates at these current levels. The energy exchange between the superconducting coil and the electric power system is controlled by a line commutated converter. To reduce the harmonics produced on the AC bus and in the output voltage to the coil, a 12-pulse converter is preferred [13]. The superconducting coil can be charged to a set value from the grid during normal operation of the power system. Once the superconducting coil is charged, it conducts current without any losses as the coil is maintained at extremely low temperatures. With a sudden rise in the load demand, the stored energy is released to the power system as alternating current. As the governor and other control mechanisms start working to set the power system to the new equilibrium condition, the coil current changes back to its initial value.

The DC voltage appearing across the inductor is continuously varying within a certain range of positive and negative values which is controlled by the converter firing angle α control. The inductor is initially charged to its rated current I_d0 by applying a small positive voltage.

Once the current reaches its rated value, it is maintained constant by reducing the voltage across the inductor to zero since the coil is superconducting. Neglecting the transformer and the converter losses, the DC voltage is given by

(15)E_{\mathrm{d}}=\mathrm{2}{V}_d\cos \alpha -\mathrm{2}{I}_{\mathrm{d}}{R}_{\mathrm{C}},

where E_d is the DC voltage applied to the inductor in kV, α is the firing angle in degrees, I_d is the current flowing through the inductor in kA, R_C is the equivalent commutating resistance in kΩ, and V_d0 is the maximum circuit bridge voltage in kV. Charging and discharging of the SMES unit is controlled by the change of commutation angle α. If α is less than 90°, the converter acts in the converter mode (charging mode), and if α is greater than 90°, the converter acts in the inverter mode (discharging mode).

Control of SMES unit

SMES block diagram with negative inductor current deviation feedback is shown in Fig. 3. In the case of a drop in the frequency due to the sudden loading in the area, the power is to be pumped back into the grid. The control voltage E_d is negative since the current in the inductor and the thyristors cannot change its direction. The incremental change in the voltage applied to the inductor can be expressed as [13]

(16)\mathrm{\Delta }{E}_{\mathrm{d}}={\displaystyle \frac{K_{\mathrm{S}\mathrm{M}\mathrm{E}\mathrm{S}}}{\mathrm{1}+s{T}_{\mathrm{d}\mathrm{c}}}}\mathrm{\Delta }\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}{\mathrm{r}}_{\mathrm{1}},

where ΔE_d is the incremental change in converter voltage, T_dc is the converter time delay, K_SMES is the gain of the control loop, and ΔError is the input signal to the SMES control logic. The inductor current deviation is given by

(17)\mathrm{\Delta }{I}_{\mathrm{d}}={\displaystyle \frac{\mathrm{\Delta }{E}_{\mathrm{d}}}{sL}}\mathrm{.}

The area control error (ACE) of Area 1 is considered as the input signal to the SMES control logic (i.e., ΔError₁ = ACE₁). The ACE of the two-areas are defined as

(18)\mathrm{A}\mathrm{C}{\mathrm{E}}_i=B_i\mathrm{\Delta }{f}_i+\mathrm{\Delta }{P}_{\mathrm{t}\mathrm{i}\mathrm{e}ij},i,j=\mathrm{1},\mathrm{2},

where Δf_i is the change in the frequency of Area i, ΔP_ij is the change in tie-line power flown out of Area i-j, and B_i is the frequency bias factor. Thus, from Eqs. (15) and (16), an incremental change in the voltage applied to the inductor can be expressed as

(19)\mathrm{\Delta }{E}_{\mathrm{d}}={\displaystyle \frac{K_{\mathrm{S}\mathrm{M}\mathrm{E}\mathrm{S}}(B_{\mathrm{1}}\mathrm{\Delta }{f}_{\mathrm{1}}+\mathrm{\Delta }{P}_{\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{12}})}{\mathrm{1}+s{T}_{\mathrm{d}\mathrm{c}}}}\mathrm{.}

The inductor current in the SMES unit will return to its nominal value very slowly using Eq. (19). But, the inductor current must be restored to its nominal value very quickly after occurrence of a disturbance in a system to respond to the next load perturbation immediately. Hence, the inductor current deviation can be sensed and used as a negative feedback signal in the SMES control loop, so that the current restoration to its nominal value can be enhanced. Thus, the dynamic equations for the inductor voltage deviation and current deviation of the SMES unit area is given as

(20)\mathrm{\Delta }{E}_{\mathrm{d}}={\displaystyle \frac{K_{\mathrm{S}\mathrm{M}\mathrm{E}\mathrm{S}}(B_{\mathrm{1}}\mathrm{\Delta }{f}_{\mathrm{1}}+\mathrm{\Delta }{P}_{\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{12}})-K_{\mathrm{i}\mathrm{d}}\mathrm{\Delta }{I}_{\mathrm{d}}}{\mathrm{1}+s{T}_{\mathrm{d}\mathrm{c}}}}\mathrm{.}

Model with SMES

With SMES in the system, the similar procedure as described in Section 2.1 is developed in this model for the two-area system for deciding the steps to control the frequency deviations adopting appropriate steps to shed loads. SIMULINK diagram of hydro-thermal plant with SMES in thermal area is shown in Fig. 4. After deriving equations for change in frequency and the corresponding polynomial in time domain, and putting the values of all parameters, the corresponding frequency response is derived. After repeating the same procedure as described in Section 2.1, model without SMES, t_m can be obtained when the variation of frequency is the maximum. The value of the time obtained is t_1m = 1.82 s. Substituting the value of t_m in Δf₁ (t), Eq. (21) can be obtained.

(21)\mathrm{\Delta }{f}_{\mathrm{1}\max }(t)=\mathrm{3.14}\mathrm{\Delta }{P}_L\mathrm{.}

Similarly, for Area 2, the value of time obtained is

(22)t_{\mathrm{1}\mathrm{m}}=\mathrm{1.21}\mathrm{s},

and

(23)\mathrm{\Delta }{f}_{\mathrm{2}\max }(t)=\mathrm{4.13}{P}_{\mathrm{L}}\mathrm{.}

Now, the value of f_1min can be found by using

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

Now, the amount of load to be shed for obtaining tolerable range of frequency can be found 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}\mathrm{57.6}\mathrm{H}\mathrm{z},

where P_shed is the amount of load to be shed.

Results and discussion

Effect of change in load demand without SMES

With continuous increase in load, the frequency of the system keeps decreasing and it reaches its minimum allowable value of 57.6 Hz. Further increase in load will result in more frequency drop. Frequency response in Area 1 with the increase in load is depicted in Fig. 5. With the increase in loads, the frequency drops below the minimum allowable limits as can be observed from the figure.

Considering equally varying loads in both areas, Table 1 gives the values of frequency variation for seven different cases with change in frequency as well as frequency in both areas. For Area 1, a load of up to the limit of 0.46 pu can be increased. But if the load is further increased, the frequency of Area 1 decreases beyond 57.6 Hz, which is the minimum allowable frequency. Thus, load shedding is required. When the load is 0.3692 pu, the frequency of Area 2 is 57.6 Hz. The load at which the frequency reaches its minimum allowable limit is termed as Load_critical (Critical load). For Area 1, the critical load is 0.46 pu, while for Area 2, the critical load is 0.3692 pu. The frequency response corresponding to the equally varying loads in both areas, is also demonstrated in Fig. 6 for Area 2. With the increase in load, the frequency drops below the minimum allowable limits as can be observed from the figure. For the hydro area, the value of the critical load is lower than that of the thermal area.

Calculation of the amount of load to be shed to maintain frequency:

1) For Area 1 Cases 5-7 need load shedding as in these cases the minimum value of frequency is less than 57.6 Hz as observed from Table 1 and Figs. 5 and 6. Based on the methodology discussed in the previous section, the amount of load to be shed for these cases are calculated.

For Case 5, the amount of load to be shed is 0.14 pu;

For Case 6, the amount of load to be shed is 0.34 pu;

For Case 7, the amount of load to be shed is 0.54 pu.

After the required amount of the shed load for the cases, the frequency response in Area 1 is obtained and exhibited in Fig. 7. It is observed that the frequency is within the allowable limit after load shedding.

2) For Area 2 Cases 3-7 need load shedding as in these cases the minimum value of frequency is less than 57.6 Hz as observed from Table 1 and Fig. 6. The amount of load shed is calculated based on the strategy discussed in the previous section. The amount of laod shed is calculated as:

For Case 3, the amount of load to be shed is 0.0308 pu;

For Case 4, the amount of load to be shed is 0.0908 pu;

For Case 5, the amount of load to be shed is 0.2308 pu;

For Case 6, the amount of load to be shed is 0.4308 pu;

For Case 7, the amount of load to be shed is 0.6308 pu.

There are more steps for load shed in the case of Area 2 than in the case of Area 1. Based on the load shed, the frequency response for Area 2 is presented in Fig. 8 in which only two curves are visible as the frequency response curves coincide. The frequency is obtained within the tolerable limit after load shedding.

Effect of change in load demand with SMES

With continuous increase in load with SMES in the hybrid area system, the frequency of the system keeps decreasing and it reaches its minimum allowable value of 57.6 Hz as shown in Fig. 9. Considering equally changing loads in both areas, Table 2 lists the values of frequency variation for seven different cases with change in frequency as well as frequency in both areas. For Area 1, a load of up to the limit of 0.764 pu can be increased. But if the load is further increased, the frequency of Area 1 decreases beyond 57.6 Hz, which is the minimum allowable frequency. Thus, for Area 1, the critical load is 0.764 pu, and load shedding is required. For Area 2, the frequency response with the equal increase in loads in both areas is also shown in Fig. 10. It is observed that when the load is 0.581 pu, the frequency of Area 2 is 57.6 Hz which is the minimum allowable limit. Thus, for Area 2, the critical load is 0.581 pu. It is observed from Table 2 that there is more load increment with SMES in both areas than without SMES. The values of critical loads are higher for both areas with SMES than without SMES. This is due to the effect of additional power available from the SMES unit installed in the thermal area. Thus, the critical values of loads are also higher.

Calculation of the amount of load to be shed to maintain frequency:

3) For Area 1 Cases 6 and 7 need load shedding as in these cases the minimum value of frequency is less than 57.6 Hz. as observed from Table 2 and Fig. 9. Based on the methodology discussed in the previous section, the amount of load to be shed for these cases are calculated.

For Case 6, the amount of load to be shed is 0.036 pu;

For Case 7, the amount of load to be shed is 0.236 pu.

After the required amount of the shed load for the cases, the frequency response in Area 1 is obtained and is shown in Fig. 11. It is observed that the frequency is within the allowable limit after load shedding.

4) For Area 2 Cases 4-7 needs load shedding as in these cases the minimum value of frequency is less than 57.6 Hz as observed from Table 2 and Fig. 10. The amount of load shed is calculated based on the strategy discussed in the previous section.

For case 4, the amount of load to be shed is 0.019 pu;

For case 5, the amount of load to be shed is 0.183 pu;

For case 6, the amount of load to be shed is 0.219 pu;

For case 7, the amount of load to be shed is 0.419 pu.

In this case, more steps are required to shed load than those in Area 1 with SMES. After the required amount of the shed load for the cases, the frequency response in Area 2 is obtained and shown in Fig. 12. It is observed that the frequency is within the allowable limit after load shedding. In Fig. 12, only three curves are visible due to the coincidence of the curves.

Comparison of the two strategies

1) In the first case, for Area 1, when the load was 0.46 pu, the system frequency reaches its minimum allowable limit. But in the second case with SMES, the frequency reaches its minimum allowable limit at the load of 0.764 pu. Hence, the system loadability increases up to 1.66 times, taking into account the minimum allowable frequency.

2) Similarly, in the first case, for Area 2, when the load was 0.3692 pu, the system frequency reaches its minimum allowable limit. But in the second case with SMES, the frequency reaches its minimum allowable limit at the load of 0.581 pu. Hence, the system loadability increases up to 1.66 times, taking into account the minimum allowable frequency.

3) For Area 1, in the first case, Cases 5-7 need load shedding, but in the second case only, Cases 6 and 7 need load shedding. Thus, the load shedding for Case 5 is avoided. For Cases 6 and 7, the amount of load which is shed also reduces. Thus, the load is saved. The saved amount of load in Case 6 as well as in Case 7 is 0.304 pu.

4) For Area 2, in the first case, Cases 3-7 need load shedding, but in the second case, only Cases 4-7 need load shedding. Thus, the load shedding for Case 3 is avoided. For Cases 4-7, the amount of load which is shed also reduces. Thus, the load is saved. The saved amount of load in Case 4 is 0.0718 pu, in Case 5 is 0.0478 pu, in Case 6 as well as in Case 7 is 0.2118 pu.

The frequency for different loads without and with SMES for Areas 1 and 2 are shown in Figs. 13 and 14. It is observed that with SMES, the minimum frequency is more acceptable as compared to the system without SMES for the same loads. Thus, SMES reduces the load amount to be shed for the frequency to reach in allowable limits.

Conclusions

Load shedding plays an important role in maintaining the frequency in overloaded conditions, but the objective of this paper is to minimize the required amount of load shedding. From the above two systems, it can be observed that in the second system, the required amount of load to be shed is minimized. So both of the two aims are achieved, first finding the required amount of load to be shed, and second, minimizing that amount. From Figs. 13 and 14, it can be seen conveniently that on using SMES the dip in frequency is less as compared to that of the system without SMES. It can also be seen that by using SMES, the minimum frequency is more acceptable as compared to the system without SMES for the same loads.

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