# Frequency Response Characteristics and Dynamic Performance

Chapter
Part of the Power Electronics and Power Systems book series (PEPS)

## Abstract

This chapter describes frequency control characteristics and dynamic performance of a power system with primary and secondary control loops. An overview of frequency response model for primary, secondary, tertiary, and emergency controls is presented. Static and dynamic performances are explained and the effects of physical constraints (generation rate, dead band, time delays, and uncertainties) on power system frequency control performance are emphasized.

## Keywords

Dynamic performance Frequency deviation LFC Primary frequency control Secondary frequency control Frequency response State-space model Physical constraint Linearized model Time delay Dead-bound GRC Uncertainty Frequency control loops Droop characteristic Droop control

This chapter describes frequency control characteristics and dynamic performance of a power system with primary and secondary control loops. An overview of frequency response model for primary, secondary, tertiary, and emergency controls is presented. Static and dynamic performances are explained and the effects of physical constraints (generation rate, dead band, time delays, and uncertainties) on power system frequency control performance are emphasized.

## 3.1 Frequency Response Analysis

A linear dynamical model is useful for secondary frequency control (LFC) analysis and synthesis was described in the previous chapter. Figure 3.1 shows the block diagram of typical control area i with n generator units in an N-multi-area power system. The blocks and parameters are defined as follows: Fig. 3.1 LFC system model
$$\varDelta f$$

frequency deviation

$$\varDelta P_{m}$$

governor valve position

$$\varDelta P_{C}$$

secondary control action

$$\varDelta P_{P}$$

primary control action

$$\varDelta P_{tie}$$

net tie-line power flow

$$H$$

equivalent inertia constant

$$D$$

equivalent damping coefficient

$$T_{ij}$$

tie-line synchronizing coefficient with area j

Β

frequency bias

$$v$$

area interface

$$R$$

droop characteristic

$$ACE$$

area control error

$$\alpha$$

participation factor

$$M\text{(}s\text{)}$$

governor-turbine dynamic model

$$K\text{(}s\text{)}$$

secondary frequency controller

Several low-order models for representing turbine-governor dynamics $$M_{i} (s)$$ for use in power system frequency analysis and control design are introduced in . Here, it is assumed that all generators are non-reheat steam units, therefore
$$M_{ki} (s) = \frac{1}{{(1 + T_{gk} s)}}\frac{1}{{(1 + T_{tk} s)}}$$
(3.1)
where $$T_{gk}$$ and $$T_{tk}$$ are governor and turbine time constants, respectively. The balance between connected control areas is achieved by detecting the frequency and tie-line power deviations to generate the ACE signal, which is in turn utilized in a dynamic controller. In Fig. 3.1, the $$v_{i}$$ is the area interface and can be defined as follows.
$$v_{i} = \sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{N} {T_{ij} } \varDelta f_{j}$$
(3.2)
Considering the effect of primary and secondary controls, the system frequency can be obtained as
$$\varDelta f_{i} (s) = \frac{1}{{2H_{i} s + D_{i} }}[\sum\limits_{k = 1}^{n} {\varDelta P_{{m_{ki} }} (s)} - \varDelta P_{tie, i} (s) - \varDelta P_{Li} (s)]$$
(3.3)
where
$$\varDelta P_{{m_{ki} }} (s) = M_{ki} (s)[\varDelta P_{{C_{ki} }} (s) - \varDelta P_{{P_{ki} }} (s)]$$
(3.4)
and
$$\varDelta P_{{P_{ki} }} (s) = \frac{{\varDelta f_{i} (s)}}{{R_{ki} }}$$
(3.5)
Here $$\varDelta P_{P}$$ and $$\varDelta P_{C}$$ are primary (governor natural response) and secondary (LFC) control actions. The expressions (3.4) and (3.5) can be substituted into (3.3) with the result
$$\varDelta f_{i} (s) = \frac{1}{{2H_{i} s + D_{i} }}\left( {\sum\limits_{k = 1}^{n} {M_{ki} (s)} [\varDelta P_{{C_{ki} }} (s) - \frac{1}{{R_{ki} }}\varDelta f_{i} (s)] - \varDelta P_{tie, i} (s) - \varDelta P_{Li} (s)} \right)$$
(3.6)
For the sake of load disturbance analysis, $$\varDelta P_{Li} (s)$$ to be usually considered in the form of a step function, i.e.,
$$\varDelta P_{Li} (s) = \frac{{\varDelta P_{Li} }}{s}$$
(3.7)
Substituting $$\varDelta P_{Li} (s)$$ in (3.6) and summarizing the result yields
$$\varDelta f_{i} (s) = \frac{1}{{g_{i} (s)}}[\sum\limits_{k = 1}^{n} {M_{ki} (s)} \varDelta P_{{C_{ki} }} (s) - \varDelta P_{tie, i} (s)] - \frac{1}{{sg_{i} (s)}}\varDelta P_{Li}$$
(3.8)
where
$$g_{i} (s) = 2H_{i} s + D_{i} + \sum\limits_{k = 1}^{n} {\frac{{M_{ki} (s)}}{{R_{ki} }}}$$
(3.9)
Substituting $$M_{ki} (s)$$ from (3.1) in (3.8) and (3.9), and using the final value theorem, the frequency deviation in steady state $$\varDelta f_{ss, \, i}$$ can be obtained from (3.8).
$$\varDelta f_{ss, i} = \mathop {Lim}\limits_{s \to 0} s\varDelta f_{i} (s) = \frac{1}{{g_{i} (0)}}\varDelta P_{Ci} - \frac{1}{{g_{i} (0)}}\varDelta P_{Li}$$
(3.10)
It is assumed that $$\varDelta P_{{tie,{{i}}}}$$ approaches zero at steady state, and
$$\varDelta P_{Ci} = \mathop {Lim}\limits_{s \to 0} s\sum\limits_{k = 1}^{n} {M_{ki} (s)} \varDelta P_{{C_{ki} }} (s)$$
(3.11)
$$g_{i} (0) = D_{i} + \sum\limits_{k = 1}^{n} {\frac{1}{{R_{ki} }}} = D_{i} + \frac{1}{{R_{{sys,{{i}}}} }}$$
(3.12)
Here, $$R_{sys, i}$$ is the equivalent system drooping characteristic, and
$$\frac{1}{{R_{{sys,{{i}}}} }} = \sum\limits_{k = 1}^{n} {\frac{1}{{R_{ki} }}}$$
(3.13)
By definition (), $$g_{i} (0)$$ is equivalent to the system’s frequency response characteristic ($$\beta_{i}$$).
$$\beta_{i} = D_{i} + \frac{1}{{R_{{sys,{{i}}}} }}$$
(3.14)
Using (3.12), the Eq. (3.10) can be rewritten into the following form
$$\varDelta f_{ss, i} = \frac{{\varDelta P_{Ci} - \varDelta P_{Li} }}{{D_{i} + 1/R_{{sys,{{i}}}} }}$$
(3.15)
Equation (3.15) shows that if the disturbance magnitude matches with the available power reserve (secondary control) $$\varDelta P_{Ci} = \varDelta P_{Li}$$, the frequency deviation converges to zero in steady state. Since the value of a droop characteristic $$R_{ki}$$ is bounded between about 0.05 and 0.1 for most bulk generator units ($$0.05 \le R_{ki} \le 0.1$$) , for a given control system according to (3.13) we can write $$R_{sys, i} \le R_{min}$$. For a small enough $$DR_{{sys,{{i}}}}$$, (3.15) can be reduced to
$$\varDelta f_{ss, i} = \frac{{R_{{sys,{{i}}}} (\varDelta P_{Ci} - \varDelta P_{Li} )}}{{(D_{i} R_{{sys,{{i}}}} + 1)}} \cong R_{{sys,{{i}}}} (\varDelta P_{Ci} - \varDelta P_{Li} )$$
(3.16)
Without a secondary control signal ($$\varDelta P_{Ci} = 0$$), the steady state frequency deviation will be proportional to disturbance magnitude as follows.
$$\varDelta f_{ss, i} = - \frac{{R_{{sys, i}} \varDelta P_{Li} }}{{(D_{i} R_{{sys,{{i}}}} + 1)}}$$
(3.17)

It is noteworthy that the above result is obtained by assuming no secondary control and tie-line variations. However, practically tie-line deviation is not zero, and hence to achieve an exact result, it should be properly reflected in the steady state frequency deviation (3.17).

In an interconnected power system, the frequency deviation following a load variation becomes zero, after all tie-line flow changes (and ACE signals) have been zeroed. Without the intervention of the secondary control, the steady state frequency deviation would depend on the equivalent drooping characteristic of all the system, namely of all generators in all areas, as well as from the damping factors of all areas.

The time constants of governor-turbine units are smaller than the time constant of an overall power system (rotating mass and load) . Hence, for the purpose of simplification in dynamic frequency analysis, it is reasonable to assume that $$T_{gk} = 0$$, $$T_{tk} = 0$$. With this assumption, the frequency response (3.8) for a generator unit with primary control loop ($$\varDelta P_{Ci} (s) = 0$$, $$\varDelta P_{tie, i} (s) = 0$$) can be reduced to
$$\varDelta f_{i} (s) \cong \frac{1}{{2H_{i} s + D_{i} + 1/R_{{sys,{{i}}}} }}( - \frac{{\varDelta P_{Li} }}{s})$$
(3.18)
Simplification of (3.18) and resolving into partial fractions yields
$$\varDelta f_{i} (s) \cong - \frac{{R_{{sys,{{i}}}} \varDelta P_{Li} }}{{(D_{i} R_{{sys,{{i}}}} + 1)}}\left( {\frac{1}{s} - \frac{1}{{s + \frac{{D_{i} R_{{sys,{{i}}}} + 1}}{{2H_{i} R_{{sys,{{i}}}} }}}}} \right)$$
(3.19)
Considering (3.17), the Eq. (3.19) becomes
$$\varDelta f_{i} (s) \cong \varDelta f_{{ss,{{i}}}} \left( {\frac{1}{s} - \frac{1}{{s + \tau_{i} }}} \right)$$
(3.20)
where $$\tau_{i}$$ is time constant of the closed loop system
$$\tau_{i} = \frac{{D_{i} R_{{sys,{{i}}}} + 1}}{{2H_{i} R_{{sys,{{i}}}} }}$$
(3.21)
and, inverse Laplace transformation of (3.20) gives
$$\varDelta f_{i} (t) \cong \varDelta f_{{ss,{{i}}}} \left( {1 - \mathop e\nolimits^{{ - \tau_{i} t}} } \right)$$
(3.22)

As shown in the next chapters, the frequency dynamic behaviour of an LFC system with a secondary control loop is more complex than the described dynamic given in (3.22).

## 3.2 State-Space Dynamic Model

State-space model of an LFC dynamical system is a useful representation for the application of the modern/robust control theory. Using appropriate definitions and state variables, as given in (3.24)–(3.28), the state-space realization of control area i shown in Fig. 3.1 can be easily obtained as (3.23) .
\begin{aligned} \dot{x}_{i} & = A_{i} x_{i} + B_{1i} w_{i} + B_{2i} u_{i} \\ y_{i} & = C_{yi} x_{i} \\ \end{aligned}
(3.23)
$$x_{{^{i} }}^{T} = \left[ {\begin{array}{*{20}c} {\varDelta f_{i} } & {\varDelta P_{tie - i} } & {x_{mi} } & {x_{gi} } \\ \end{array} } \right]$$
(3.24)
$$x_{{^{mi} }} = \left[ {\begin{array}{*{20}c} {\varDelta P_{m1i} } & {\varDelta P_{m2i} } & \cdots & {\varDelta P_{mni} } \\ \end{array} } \right],\;x_{gi} = \left[ {\begin{array}{*{20}c} {\varDelta P_{g1i} } & {\varDelta P_{g2i} } & \cdots & {\varDelta P_{gni} } \\ \end{array} } \right]$$
(3.25)
$$u_{i} = \varDelta P_{Ci}$$
(3.26)
$$y_{i} = ACE_{i} = \beta_{i} \varDelta f_{i} + \varDelta P_{tie, i}$$
(3.27)
$$w_{i}^{T} = \left[ {\begin{array}{*{20}c} {\varDelta P_{Li} } & {v_{i} } \\ \end{array} } \right]$$
(3.28)
where $$\varDelta P_{gi}$$ denotes the governor valve position change, and
\begin{aligned} A_{i} & = \left[ {\begin{array}{*{20}c} {A_{i11} } & {A_{i12} } & {A_{i13} } \\ {A_{i21} } & {A_{i22} } & {A_{i23} } \\ {A_{i31} } & {A_{i32} } & {A_{i33} } \\ \end{array} } \right],\;B_{1i} = \left[ {\begin{array}{*{20}c} {B_{1i1} } \\ {B_{1i2} } \\ {B_{1i3} } \\ \end{array} } \right],\;B_{2i} = \left[ {\begin{array}{*{20}c} {B_{2i1} } \\ {B_{2i2} } \\ {B_{2i3} } \\ \end{array} } \right] \\ A_{i11} & = \left[ {\begin{array}{*{20}c} { - D_{i} /2H_{i} } & { - 1/2H_{i} } \\ {2\pi \sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{N} {T_{ij} } } & 0 \\ \end{array} } \right],\;A_{i12} = \left[ {\begin{array}{*{20}c} {1/2H_{i} } & \cdots & {1/2H_{i} } \\ 0 & \cdots & 0 \\ \end{array} } \right]_{2 \times n} \\ A_{i22} & = - A_{i23} = diag\left[ {\begin{array}{*{20}c} { - 1/T_{t1i} } & { - 1/T_{t2i} } & \cdots & { - 1/T_{tni} } \\ \end{array} } \right] \\ A_{i33} & = diag\left[ {\begin{array}{*{20}c} { - 1/T_{g1i} } & { - 1/T_{g2i} } & \cdots & { - 1/T_{gni} } \\ \end{array} } \right] \\ A_{i31} & = \left[ {\begin{array}{*{20}c} { - 1/(T_{g1i} R_{1i} )} & 0 \\ \vdots & \vdots \\ { - 1/(T_{gni} R_{ni} )} & 0 \\ \end{array} } \right],\;A_{i13} = A_{i21}^{\text{T}} = 0_{2 \times n} ,\;A_{i32} = 0_{n \times n} \\ B_{1i1} & = \left[ {\begin{array}{*{20}c} { - 1/2H_{i} } & 0 \\ 0 & { - 2\pi } \\ \end{array} } \right],\;B_{1i2} = B_{1i3} = 0_{n \times 2} \\ B_{2i1} & = 0_{2 \times 1} , \, B_{2i2} = 0_{n \times 1} ,\;B_{2i3}^{T} = \left[ {\begin{array}{*{20}c} {\alpha_{1i} /T_{g1i} } & {\alpha_{2i} /T_{g2i} } & \cdots & {\alpha_{ni} /T_{gni} } \\ \end{array} } \right] \\ C_{yi} & = \left[ {\begin{array}{*{20}c} {\beta_{i} } & 1 & {0_{1 \times n} } & {0_{1 \times n} } \\ \end{array} } \right] \\ \end{aligned}
According to Fig. 3.1, in each control area the control input is performed by the ACE signal.
$$u_{i} = \varDelta P_{Ci} = f(ACE_{i} )$$
(3.29)
where $$f( \cdot )$$ is a function which identifies the dynamics of the controller.
To illustrate the system frequency response in a multi-area power system based on the model described in Fig. 3.1, consider three identical interconnected control areas as shown in Fig. 3.2. Figure 3.3 shows a realized model of the three-interconnected control areas, which are presented in MATLAB/SIMULINK environment. This 3-control area power system is used in [3, 4]. The power system parameters are given in Appendix A. Each area has three generating units that are owned by different generation companies. Here, the MVA base is 1000, and each control area uses proportional integral (PI) controller in its secondary frequency control loop. Fig. 3.2 3-Control area power system Fig. 3.3 Building the LFC model in SIMULINK environment; a 3-control area and b detailed model of Area 1
The frequency response model (shown in Fig. 3.1) is implemented for each control area in the SIMULINK environment of the MATLAB software. The system response following a simultaneous 0.1 pu load step (disturbance) increase at 2 s in control areas 1 and 2 is shown in Fig. 3.4. The PI control parameters are considered as those given in . Although, the load disturbances are happened in areas 1 and 2, however, area 3 also participates to restore the system frequency and minimize the tie-line power fluctuation using generating units $$G_{8}$$ and $$G_{\text{9}}$$. Fig. 3.4 System response; a Area 1, b Area 2, c Area 3

Figure 3.4 shows the frequency deviation, ACE, and control action signal for control areas 1, 2, and 3, respectively. The proposed simulation shows the secondary frequency control loops properly act to maintain system frequency and exchange powers close to the scheduled values by sending corrective signal to the generating units in proportion to their participation in the AGC system.

The active power to compensate the frequency deviation and tie-line power change initially comes from all generating units to respond to the step load increase in areas 1 and 2, and results in a frequency drop sensed by the speed governors of all generators. However, after few seconds (at steady state), the additional powers against the local load changes only come from generating units that are participating in the secondary control issue. The amount of additional generated power by each unit is proportional to the related participation factor.

## 3.3 Physical Constraints

The performed studies in the previous sections on frequency control dynamic performance have been made based upon a linearized analysis. The described frequency response model so far does not consider the effects of the physical constraints. Although considering all dynamics in frequency control synthesis and analysis may be difficult and not useful , it should be noted that to get an accurate perception of the frequency subject it is necessary to consider the important inherent requirement and the basic constraints imposed by the physical system dynamics, and model them for the sake of performance evaluation.

### 3.3.1 Generation Rate and Dead Band

An important physical constraint is on the rate of change of power generation due to the limitation of thermal and mechanical movements. The frequency control studies that do not take into account the delays caused by the crossover elements in a thermal unit, or the behavior of the penstocks in a hydraulic installation, in addition to the sampling interval of the data acquisition system, results in a situation where frequency and tie-line power could be returned to their scheduled value within one second.

In a real LFC system, rapidly varying components of system signals are almost unobservable due to various filters involved in the process. Hence, the performance of a designed control system is dependent on how generation units respond to the control signal. A very fast response for a secondary system is neither possible nor desirable . A useful control strategy must be able to maintain sufficient levels of reserved control range and control rate.

The generation rate for non-reheat thermal units is usually higher than the generation rate for reheat units [7, 8]. The reheat units have a generation rate about of 3–10 % puMW per minute. For hydro units, the rate is on the order of 100 % maximum continuous rating per minute . Results of investigations of the impacts of generation rate constraint (GRC) on the performance of secondary systems are reported in [10, 11, 12].

Speed governor dead band is known as another important issue in power system performance. By changing the input signal, the speed governor may not immediately react until the input reaches a specified value. All governors have a dead band in response which are important for power system frequency control in the presence of disturbances. Governor dead band is defined as the total magnitude of a sustained speed change, within which there is no resulting change in valve position.

The effect of the governor dead band is to increase the apparent steady state speed regulation . The maximum value of dead band for governors of conventional large steam turbines is specified as 0.06 % (0.036 Hz) . The effects of the governor dead band on power system dynamics and frequency control performance were studied in the last five decades [6, 13, 15, 16, 17, 18]. The results indicate that for a wide dead band the frequency control performance can be significantly degraded.

Several methods have been developed to consider the GRC and speed governor dead band for the analysis/synthesis of frequency control systems. For the frequency response analysis and simulations, these nonlinear dynamics can be usually considered by adding a limiter and a hysteresis pattern to the governor-turbine system model, as shown for a non-reheat steam turbine in Fig. 3.5. The $$V_{U}$$ and $$V_{L}$$ are the maximum and minimum limits that restrict the rate of valve (gate) closing (opening) speeds. Fig. 3.5 Non-reheat generator unit model with GRC and dead band

### 3.3.2 Time Delay

In LFC practice, rapid responses and varying components of frequency are almost unobservable due to various filters and delays involved in the LFC process. Any signal processing and filtering introduces delays that should be considered. Typical filters on tie-line metering and ACE signal (with the response characteristics of generator units) use about 2 s or more for the data acquisition and decision cycles of the LFC systems.

In a new environment, the communication delays in the LFC synthesis/analysis are becoming a more significant challenge due to the restructuring, expanding of physical setups, functionality, and complexity of power systems. Most published research works on the frequency control design during the last decades have neglected problems associated with the communication network. Although, under the traditional dedicated communication links, this was a valid assumption, the use of an open communication infrastructure to support the ancillary services in deregulated environments raises concerns about problems that may arise in the communication system. In the control systems, it is well known that time delays can degrade the system’s performance and even causes system instability [19, 20, 21].

The time delays in a secondary control system mainly exist on the communication channels between the control center and operating stations; specifically on the measured frequency and power tie-line flow from remote terminal units (RTUs) to the control center and the delay on the produced rise/lower signal from control center to individual generation units [22, 23]. These delays are schematically shown in Fig. 3.6. Here, the delay is expressed by an exponential function $$e^{ - s\tau }$$ where $$\tau$$ gives the communication delay time. Fig. 3.6 Time delays representation in a secondary control system

The introduction of time delays in the secondary control loop reduces the effectiveness of controlled system performance. It is shown that the frequency control performance declines when the delay time increases . In order to satisfy the desired performance for a multi-area power system, the design of a controller should take into account these delays.

Recently, several papers have been published to address the LFC modeling/synthesis in the presence of communication delays [18, 22, 23, 24, 25, 26, 27, 28, 29]. The effects of signal delays on the load following task have been discussed. Reference  is focused on the network delay models and communication network requirement for a third party LFC service. A compensation method for communication time delay in the LFC systems is addressed in Ref. , and some control design methods based on linear matrix inequalities (LMI) are proposed for the LFC system with communication delays in [23, 25, 26, 27, 29].

In frequency control practice, to remove the fast changes and probable added noises, system frequency gradient and ACE signals must be filtered before being used. Considering the total effects of generating rate, dead band, filters and delays for primary and secondary control performance analysis gives an appropriate model, which is shown in Fig. 3.7. This model is useful for digital simulations. Fig. 3.7 A Frequency response model for primary and secondary controls

The ACE signal is filtered and if exceeds a threshold at interval $$T_{W}$$, it will be applied to a PI control block. The controller can be activated to send higher/lower pulses to generating plants if its input ACE signal exceeds a standard limits. Delays, ramping rate, and range limits are different for various plants.

In the presence of GRC and dead band and time delays, the LFC system model becomes highly nonlinear; hence, it will be difficult to use the linear control theory for performance optimization and control design. In the next chapters it is shown that the above constraints can be easily considered through the robust synthesis procedure using appropriate fictitious weights on the controlled signals.

### 3.3.3 Uncertainties

Investigation of power system behavior typically involves numerous uncertainties. With ongoing system restructuring, continuous change of dynamics/load and operating conditions, the uncertainty issue in power system operation and control has increasingly become a challenge.

The uncertainty reflects the lack of complete knowledge of the exact value of parameters, components, and quantities being measured. Generally, continuous parameters variation, unmodeled dynamics, inexact definition/measurement, and consequent approximations are the main sources of power system uncertainties. Recently, in order to address uncertainties and their formulation in a power system control, various approaches have been proposed [30, 31, 32, 33, 34, 35].

The linearized frequency control system model may only be valid for a narrow band around a particular operating condition. To deal with this problem, for frequency control design it is important to formulate/model the effects of uncertainties on the system dynamics and performance .

For the application of the robust control theory in LFC synthesis, the control area uncertainties can be represented using appropriate modeling techniques. For instance, the uncertainties due to unmodeled dynamics and parameter variations can be modeled by an unstructured multiplicative uncertainty block as shown in Fig. 3.8. Let $$\hat{G}_{i} \text{(}s\text{)}$$ denotes the transfer function from the control input $$u_{i}$$($$\varDelta P_{Ci}$$) to the control output $$y_{i}$$($$ACE_{i}$$) at operating points other than nominal point. Then, following a practice common in robust control, the uncertainties transfer function is represented as
$$\varDelta_{i} \text{(}s\text{)} = \text{[}\hat{G}_{i} \text{(}s\text{)} - G_{i} \text{(}s\text{)]}G_{i} \text{(}s\text{)}^{ - 1} ,\quad G_{i} \text{(}s\text{)} \ne 0$$
(3.30)
where, $$\varDelta_{i} \text{(}s\text{)}$$ shows the uncertainty block corresponding to the uncertainties and $$G_{i} \text{(}s\text{)}$$ is the nominal transfer function model. Fig. 3.8 Block diagram representation of multiplicative uncertainty
As an example, consider the power system example described in , and assume that the rotating mass and load pattern parameters have uncertain values in each control area. The variation range for $$D_{i}$$ and $$H_{i}$$ parameters in each control area is assumed $$\pm \text{20}\,{\% }$$. The resulting uncertainty in each control area can be modeled as a multiplicative uncertainty. Using (3.30), some sample uncertainties corresponding to different values of $$D_{i}$$ and $$M_{i}$$ for control area 1 are obtained, as shown in Fig. 3.9. Fig. 3.9 Uncertainty plots due to parameters changes in area 1; $$D_{i}$$ (dotted) and $$M_{i}$$ (dash-dotted)

Since the frequency responses of the above uncertainties are close to each other, using a single norm bounded transfer function to cover all possible perturbed plants reduces the complexity of the frequency control synthesis procedure .

## 3.4 Overall Frequency Response Model

Considering four existing control loops (primary, secondary, tertiary, and emergency controls), according to Figs.  and 3.7, an overall frequency response model for control area i in a multi-area power system is shown in Fig. 3.10. The synchronous generators are equipped with primary and secondary frequency control loops. The secondary loop performs a feedback via the frequency deviation and adds it to the primary control loop through a dynamic controller. The resulting signal ($$\varDelta P_{C}$$) is used to regulate the system frequency. In real world power systems, usually the dynamic controller is a simple integral (I) or PI controller. Following a change in the load, the feedback mechanism provides an appropriate signal for the turbine to make generation ($$\varDelta P_{m}$$) track the load and restore the system frequency. Fig. 3.10 Overall frequency response model for dynamic performance analysis

In addition to the area frequency regulation, the secondary control loop should control the net interchange power with neighbouring areas at scheduled values. This is generally accomplished by feeding a linear combination of tie-line flow and frequency deviations, known as area control error (ACE), via secondary feedback to the dynamic controller. The block diagram shown in Fig. 3.10 illustrates how the ACE is implemented in the secondary frequency control loop.

As mentioned, the secondary loop performance is highly dependent on how the participant generating units would respond to the control action signals. The NERC classifies generator actions into two groups. First group is associated with large frequency deviations where generators would respond through governor action and then the second group is associated with a continuous regulation process in response to the secondary frequency control signals only. During a sudden increase in area load, the area frequency experiences a transient drop. At the transient state, there would be flows of power from other areas to supply the excess load in that area. Usually, certain generating units within each area would be on regulation to meet this load change. At steady state, the generation would be closely matched with the load, causing tie-line power and frequency deviations to drop to zero . The frequency is assumed to be same in all points of a control area.

In a multi-area power system, the trend of frequency measured in each control area is an indicator of the trend of the mismatch power in the interconnection and not in the control area alone. Therefore, the power interchange should be properly considered in the frequency response model. In , it is shown that in an interconnected power system with N control areas, the tie-line power change between area i and other areas can be represented as,
$$\varDelta P_{tie,i} = \sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{N} {\varDelta P_{tie,ij} } = \frac{2\pi }{s}\left[ {\sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{N} {T_{ij} \varDelta f_{i} } - \sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{N} {T_{ij} \varDelta f_{j} } } \right]$$
(3.31)

Equation (3.31) is realized in the right side of frequency response block diagram in Fig. 3.10. The effect of changing the tie-line power for an area is equivalent to changing the load of that area. That is why, the $$\varDelta P_{tie,i}$$ has been added to the mechanical power change ($$\varDelta P_{m}$$) and area load change ($$\varDelta P_{L}$$) using an appropriate sign in Fig. 3.10.

The participation factor indicates an amount of participation of a generator unit in the secondary frequency control system. Following a load disturbance within the control area, the produced appropriate control signal is distributed among generator units in proportion to their participation, to make generation follow the load. As discussed in , in a competitive environment, the participation factors are actually time dependent variables and must be computed dynamically based on bid prices, availability, congestion problems, costs and other related issues [38, 39].

The balance between generation and load can be achieved by detecting frequency and tie-line flow deviations via ACE signal through a PI feedback control mechanism. If supply and demand do not match in the long run as well as in the short run, the market will fail. The supply of regulation power services is mostly ensured by conventional generating units. Marginally, other participants also provide regulation services, such as storage devices that smooth either consumption or generation, consumers that can modulate their consumption upon request or automatically, and to some extent the renewable energy sources (RESs). The demand for frequency regulation services is defined by the market operator and depends on the power system structure .

As mentioned, in the case of a large generation loss disturbance, the scheduled power reserve may not be enough to restore the system frequency and the power system operators may follow an emergency control plan such as UFLS. The UFLS strategy is designed so as to rapidly balance the demand of electricity with the supply and to avoid a rapidly cascading power system failure. Allowing normal frequency variations within expanded limits will require the coordination of primary control and scheduled reserves with generator load set points; for example under-frequency generation trip (UFGT), over-frequency generation trips (OFGT), or over-frequency generator shedding (OFGS), and other frequency-controlled protection devices.

In the case of contingency analysis, the emergency protection and control dynamics must be adequately modeled in the frequency response model. Since they influence the power generation/load balance, the mentioned emergency control dynamics can be directly included to the system frequency response model. This is made by adding an emergency protection/control loop to the primary and secondary frequency control loops, as shown in Fig. 3.10. The $$\varDelta P_{UFLS} (s)$$, $$P_{UFGT} (s)$$, and $$\varDelta P_{OFGT} (s)$$ represent the dynamics effects of the UFLS, UFGT, and OFGT actions, respectively.

The emergency control schemes and protection devices dynamics are usually represented using incremented/decremented step behaviour. Thus, in Fig. 3.10, for simplicity, the related blocks can be represented as a sum of incremental (decremental) step functions. For instance, as explained in , for a fixed UFLS scheme, the function of $$\varDelta P_{UFLS}$$ in the time domain could be considered as a sum of the incremental step functions of $$\varDelta P_{j} u(t - t_{j} )$$. Therefore, for L load shedding steps:
$$\varDelta P_{UFLS} (t) = \sum\limits_{j = 0}^{L} {\varDelta P_{j} \, } u(t - t_{j} )$$
(3.32)
where $$\varDelta P_{j}$$ and $$t_{j}$$ denote the incremental amount of load shed and time instant of the jth load shedding step, respectively. Similarly, to formulate the $$\varDelta P_{OFGT}$$, $$\varDelta P_{UFGT}$$, and other emergency control schemes, appropriate step functions can be used. Therefore, using the Laplace transformation, it is possible to represent $$\varDelta P_{E} (s)$$ in the following summarized form:
$$\varDelta P_{E} (s) = \sum\limits_{l = 0}^{N} {\frac{{\varDelta P_{l} }}{s} \, } e^{{ - t_{l} s}}$$
(3.33)
where $$\varDelta P_{l}$$ is the size of equivalent step load/power changes due to a generation/load event or a load shedding scheme at $$t_{l}$$. The effect of tertiary control loop in a control unit (e.g., SCADA/dispatching center) organized by market operator in relation with Gencos and security plans is also shown in Fig. 3.10.

As explained, the generating units could respond to fast load fluctuations, in time scale of 1–3 s, depending on the droop characteristics of governors in the primary frequency control loop. The generating units could respond to slower disturbance dynamics in range of few seconds measuring ACE signal via secondary frequency control loop. The longer term load changes in timescale of ten seconds to several minutes could be responded based on economic dispatch plans and special control actions that would utilize the economics of the AGC system to minimize operating costs.

The deviations in load and power could be procured by market operator on purpose, because of planned line and unit outages. This kind of deviation may produced by market operator as a control plan in response to energy imbalances following unpredicted disturbances. These deviations are basically different than unpredicted frequency/tie-line deviations that usually occur by variations of load and generation from scheduled levels following a fault such as unplanned line and unit outages. The participant generating units in frequency regulation market could respond to unpredicted frequency/tie-line deviations proportion to the assigned participation factors from their schedules within few seconds .

## 3.5 Droop Characteristic

For understanding droop characteristic concept, consider a single machine infinite bus system which is shown in Fig. 3.11. As explained in , the swing equation of a synchronous generator is given by:
$$M\frac{d}{dt}\omega + D\left( {\omega - 1} \right) = P_{m} - P_{e}$$
(3.34)
where
$$P_{e} = \frac{{V_{g} V_{\infty } \sin \delta }}{{x^{\prime}_{d} + x_{l} }}$$
(3.35)
here, ω is the angular velocity, M is the inertia constant, D is the damping coefficient, P m is the mechanical input to the generator, P e is the electrical output, V g is the generator voltage, V is the voltage of the infinite bus, δ is the rotor angle of the generator, x’ d is the transient reactance of the generator, and x l is the line reactance. Note that the resistance of the generator and the transmission line is not considered for the simplicity. In the swing Eq. (3.34), assuming that the voltages V g and V are constant, and $$\cos \varDelta \delta \cong 1$$, $$\sin \varDelta \delta \cong \varDelta \delta$$, the deviation of the output power of the generator is given by:
$$\varDelta P_{e} \cong \frac{{V_{g} V_{\infty } \cos \delta_{0} }}{{x^{\prime}_{d} + x_{l} }}\varDelta \delta$$
(3.36)
where, the subscript 0 is used to denote the initial value of the equilibrium point and
$$\varDelta \delta = \delta_{g} - \delta_{\infty }$$
(3.37)
The V is the voltage at infinite-bus side of the connecting line; the $$\delta_{g}$$ and $$\delta_{\infty }$$ are the angles of V g and V , respectively. For small $$\varDelta \delta$$, the ∆P mainly depends on the $$\varDelta \delta$$. It means that the generator can determine the transferring real power P e flows from itself to the grid considering the phase (and therefore frequency) of its output voltage. Above relationship show if the active power increases, the voltage angle must decrease, and vice versa. This relationship which is formulated in (3.38) allows us to establish a feedback loop in order to control generator’s real power and frequency.
$$f - f_{0} = - R(P_{g} - P_{g0} )$$
(3.38)
The f 0 and p g0 are the nominal values (references) of frequency and active power, respectively. The ratio of frequency change ($$\varDelta f$$) to change in output generated power ($$\varDelta P_{g}$$) is known as droop or frequency regulation, and can be expressed as:
$$R \, \left( {{\raise0.7ex\hbox{{Hz}} \!\mathord{\left/ {\vphantom {{Hz} {pu.MW}}}\right.\kern-0pt} \!\lower0.7ex\hbox{{pu.MW}}}} \right) = \frac{\varDelta f}{{\varDelta P_{g} }}$$
(3.39)
For example, a 5 % droop means that a 5 % deviation in nominal frequency (from 60 to 57 Hz) causes 100 % change in output power. In Fig. 3.12, the droop characteristics for the generating units ($$R_{ki}$$) are properly shown in the primary frequency control loop. This figure shows a graphical representation for (3.38). Fig. 3.12 Load tracking by generators with different droops
The interconnected generating units with different droop characteristics can jointly track the load change to restore the nominal system frequency. This is illustrated in Fig. 3.12, representing two units with different droop characteristics connected to a common load. Two generating units are operating at a unique nominal frequency with different output powers. Any change in the network load causes the units to decrease their speed, and the governors increase the outputs until they reach a new common operating frequency. As expressed in (3.40), the amount of produced power by each generating unit to compensate the network load change depends on the unit’s droop characteristic.
$$\varDelta P_{gi} = \frac{\varDelta f}{{R_{i} }}$$
(3.40)
hence,
$$\frac{{\varDelta P_{g1} }}{{\varDelta P_{g2} }} = \frac{{R_{2} }}{{R_{1} }}$$
(3.41)
It is noteworthy that the described droop controls characteristic in (3.38), and Fig. 3.12 have been obtained for electrical grids with inductive impedance (X ≫ R) and a great amount of inertia.
$$X_{l} = R_{l} + jx_{l} \, ;\quad \quad x_{l} > > R_{l}$$
(3.42)

In such a power system, immediately following a power imbalance due to a disturbance, the power is going to be balanced by natural response generators using rotating inertia in the system via the primary frequency control loop. In microgrids with low inertia renewable energy sources and distributed generators, there is no significant inertia and if an unbalance occurs between the generated power and the absorbed power, the voltages of the power sources change. Therefore in this case, voltage may be also triggered by the power changes. In fact, for medium and low-voltage lines which the microgrids are working with, the impedance is not dominantly inductive ($$x_{l} \cong R_{l}$$). This fact suggests different droop control characteristics .

## 3.6 Summary

The frequency control characteristics and dynamic performance are described. The effects of physical constraints (generation rate, dead band, time delays, and uncertainties) on power system frequency control performance are emphasized. An overall frequency response model for the existing frequency control loops (primary, secondary, tertiary, and emergency controls) is presented; and finally the droop characteristic is explained.

## References

1. 1.
P.M. Anderson, M. Mirheydar, A low-order system frequency response model. IEEE Trans. Power Syst. 5(3), 720–729 (1990)
2. 2.
T.K. Nagsarkar, M.S. Sukhija, Power System Analysis (Oxford University Press, New Delhi, 2007)Google Scholar
3. 3.
H. Bevrani, Y. Mitani, K. Tsuji, Robust decentralized load-frequency control using an iterative linear matrix inequalities algorithm. IEE Proc. Gener. Transm. Distrib. 150(3), 347–354 (2004)
4. 4.
D. Rerkpreedapong, A. Hasanovic, A. Feliachi, Robust load frequency control using genetic algorithms and linear matrix inequalities. IEEE Trans. Power Syst. 18(2), 855–861 (2003)
5. 5.
H. Bevrani, Decentralized Robust Load-Frequency Control Synthesis in Restructured Power Systems. PhD dissertation, Osaka University, Japan, 2004Google Scholar
6. 6.
N. Jaleeli, D.N. Ewart, L.H. Fink, Understanding automatic generation control. IEEE Trans. Power Syst. 7(3), 1106–1112 (1992)
7. 7.
IEEE Committee Report, Power plant response. IEEE Trans. Power App. Syst. 86, 399–484 (1967)Google Scholar
8. 8.
IEEE Committee Report, Dynamic models for steam and hydro turbines in power system studies. IEEE Trans. Power App. Syst. 92, 1904–1915 (1973)
9. 9.
P. Kundur, Power System Stability and Control (McGraw-Hill, New York, 1994)Google Scholar
10. 10.
J. Nanda, M.L. Kothari, P.S. Satsangi, Automatic generation control of an interconnected hydro-thermal system in continuous and discrete modes considering generation rate constraints. IEE Proc. D 130(1), 455–460 (1983)Google Scholar
11. 11.
T. Hiyama, Optimisation of discrete-type load-frequency regulators considering generation rate constraints. IEE Proc. C 129(6), 285–289 (1982)Google Scholar
12. 12.
M.L. Kothari, P.S. Satsangi, J. Nanda, Sampled data automatic generation control of interconnected reheat thermal systems considering generation rate constraints. IEEE Trans. Power App. Syst. 100, 2334–2342 (1981)
13. 13.
C. Concordia, L.K. Kirchmayer, E.A. Szymanski, Effect of speed governor dead-band on tie-line power and frequency control performance. Am. Inst. Electr. Eng. (AIEE) Trans. 76, 429–435 (1957)Google Scholar
14. 14.
IEEE Standard 122–1991, Recommended practice for functional and performance characteristics of control systems for steam turbine–generator units, 1992Google Scholar
15. 15.
C.W. Taylor, K.Y. Lee, D.P. Dave, Automatic generation control analysis with governor dead band effects. IEEE Trans. Power App. Syst. 98, 2030–2036 (1979)
16. 16.
S.C. Tripathy, G.S. Hope, O.P. Malik, Optimization of load frequency control parameters with reheat steam turbines and governors dead-band nonlinearity. IEE Proc. Gener. Transm. Distrib. 129(1), 10–16 (1982)
17. 17.
S.C. Tripathy, T.S. Bhatti, C.S. Jha, O.P. Malik, G.S. Hope, Sampled data automatic generation control analysis with reheat steam turbines and governor dead-band effects. IEEE Trans. Power App. Syst. 103(5), 1045–1051 (1984)
18. 18.
T. Sasaki, K. Enomoto, Dynamic analysis of generation control performance standards. IEEE Trans. Power Syst. 17(3), 806–811 (2002)
19. 19.
M.S. Mahmoud, Robust Control and Filtering for Time-Delay Systems (Marcel Dekker, New York, 2000)
20. 20.
J. Aweya, D.Y. Montuno, M. Ouellette, Effects of control loop delay on the stability of a rate control algorithm. Int. J. Commun. Syst. 17, 833–850 (2004)
21. 21.
S.I. Niculescu, Delay Effects on Stability: A Robust Control Approach (Springer, Berlin, 2001)Google Scholar
22. 22.
S. Bhowmik, K. Tomosovic, A. Bose, Communication models for third party load frequency control. IEEE Trans. Power Syst. 19(1), 543–548 (2004)
23. 23.
H. Bevrani, T. Hiyama, Robust load-frequency regulation: A real-time laboratory experiment. Optim. Control Appl. Methods 28(6), 419–433 (2007)
24. 24.
T. Hiyama, T. Nagata, T. Funabashi, Multi-agent based automatic generation control of isolated stand alone power system. Proc. Int. Conf. Power Syst. Technol. 1, 139–143 (2004)Google Scholar
25. 25.
H. Bevrani, T. Hiyama, A robust solution for PI-based LFC problem with communication delays. IEEJ Trans. Power Energy 25(12), 1188–1193 (2005)
26. 26.
X. Yu, K. Tomosovic, Application of linear matrix inequalities for load frequency control with communication delays. IEEE Trans. Power Syst. 19(3), 1508–1515 (2004)
27. 27.
H. Bevrani, T. Hiyama, Robust decentralized PI based LFC design for time-delay power systems. Energy Convers. Manag. 49, 193–204 (2007)
28. 28.
S. Fukushima, T. Sasaki, S. Ihara, et al., Dynamic analysis of power system frequency control, in Proceedings of the CIGRE 2000 Session, No. 38–240, Paris, 2000Google Scholar
29. 29.
H. Bevrani, T. Hiyama, On load–frequency regulation with time delays: Design and realtime implementation. IEEE Trans. Energy Convers. 24(1), 292–300 (2009)
30. 30.
I.A. Hiskens, Jassim Alseddiqui, Sensitivity, approximation, and uncertainty in power system dynamic simulation. IEEE Trans. Power Syst. 21(4), 1808–1820 (2006)
31. 31.
A.K. Al-Othman, M.R. Irving, A comparative study of two methods for uncertainty analysis in power system state estimation. IEEE Trans. Power Syst. 20(2), 1181–1182 (2005)
32. 32.
J.R. Hockenberry, B.C. Lesieutre, Evaluation of uncertainty in dynamic simulations of power system models: The probabilistic collocation method. IEEE Trans. Power Syst. 19(3), 1483–1491 (2004)
33. 33.
A.K. Al-Othman, M.R. Irving, Uncertainty modeling in power system state estimation. IEE Proc. Gener. Transm. Distrib. 152(2), 233–239 (2005)
34. 34.
V.A. Maslennikov, S.M. Ustinov, J.V. Milanovic, Method for considering uncertainties for robust tuning of PSS and evaluation of stability limits. IEE Proc. Gener. Transm. Distrib. 149(3), 295–299 (2002)
35. 35.
A.T. Saric, A.M. Stankovic, Model uncertainty in security assessment of power systems. IEEE Trans. Power Syst. 20(3), 1398–1407 (2005)
36. 36.
H. Bevrani, Y. Mitani, K. Tsuji, On robust load–frequency regulation in a restructured power system. IEEJ Trans. Power Energy 124-B(2), 190–198 (2004)
37. 37.
H. Bevrani, M. Watanabe, Y. Mitani, Power System Monitoring and Control (Wiley-IEEE Press, New York, 2014)
38. 38.
H. Bevrani, Robust Power System Frequency Control, 1st edn. (Springer, New York, 2009)
39. 39.
H. Bevrani, T. Hiyama, Intelligent Automatic Generation Control (CRC Press, New York, 2011)Google Scholar
40. 40.
UCPTE Doc. UCPTE rules for the co-ordination of the accounting and the organization of the load-frequency control, 1999Google Scholar
41. 41.
H. Bevrani, S. Shokoohi, An intelligent droop control for simultaneous voltage and frequency regulation in islanded Microgrids. IEEE Trans. Smart Grid 4(3), 1505–1513 (2013)