### DFR model

Full time-domain simulations can simulate frequency response of power systems in detail. However, due to the coupled active power-frequency dynamics and reactive power-voltage dynamics, both frequency and voltage need to be considered when studying power system dynamic behaviour. The influence of frequency and voltage can hardly be distinguished. Consequently, the DFR model is proposed in this paper to decouple frequency and voltage dynamics and to consider the influence of the network. In the DFR model, system network is simulated by direct current power flow so the redistribution of imbalanced power between different generators and the space-time distribution characteristics of frequency can be considered. To focus on active power-frequency dynamics, some assumptions are made as follows. (1) Excitation and regulation system is strong enough to hold the generator terminal voltage and thus, the dynamics of the excitation and regulation systems and the PSS can be eliminated for its negligible influence on active power-frequency dynamics. (2) Generators swing equations are reserved while the influence of transient process of the internal windings on system frequency change can be neglected due to the constant generator terminal voltage. Since turbine-governors have significant effect on power system frequency dynamics, details of the turbine-governor are modelled in the DFR model.

In the DFR model, the dynamic behaviour of frequency is only influenced by active power change. With constant bus voltages, direct current power flow is introduced to simulate the network when calculating active power flow under initial operating condition [15]:

$$ \mathbf{P}=\mathbf{B}\boldsymbol{\uptheta } $$

(1)

where **P** is the active power injection, **θ** is the voltage angle of all buses except the slack bus, and **B** is the network susceptance matrix.

The assumption of constant voltage leads to the simplification of load models. With constant terminal voltage, reactive power of loads can be ignored and the polynomial load model [16] can be reduced as a static active power load with frequency dependency:

$$ {P}_L={P}_0\left(1+{K}_{pf}\varDelta f\right) $$

(2)

where *P*
_{
L
} is the actual load, *P*
_{0} is active power of the load under initial condition, *K*
_{
pf
} is the load regulation coefficient, and Δ*f* is frequency deviation.

Other models can also be simplified with appropriate assumptions. For example, high voltage direct current links (HVDC) can be represented as loads for sending and receiving ends with or without frequency dependency.

With the simplifications of the generating units, network, loads and other equipment, the DFR model can be shown in Fig. 1. Quantities in Fig. 1 are listed as follows. *ω*
_{
i
}, *δ*
_{
i
}, *P*
_{
mi
}, and *P*
_{
ei
} are rotor speed, rotor angle, mechanical power, and electrical power of generating unit *i*. Δ*f*
_{
j
} and *P*
_{
j
} are the bus frequency and active power of load *j*.

Similar to full time-domain simulation, the DFR model can be expressed in terms of differential-algebraic equations (DAEs), and can be solved by step-by-step integration such as implicit trapezoid integration. Comparing with full time-domain simulation, the computational burden of the DFR model is greatly reduced and it achieves a better computational efficiency with acceptable accuracy. The DFR model can be used to analyse events of load change, generator tripping, etc. It can be also applied to fast frequency response calculation for active power disturbances and event screening.

With the introduction of direct current power flow, the DFR model is applicable to systems in which the network reactance is significantly greater than the resistance, e.g., high voltage transmission systems. The DFR model is primarily useful for cases where frequency stability is the main concern and angle stability and voltage stability can be maintained.

### ASF model

In real systems, the frequency difference among buses is trivial if generators remain in synchronism during transient process [13]. Thus frequency at different buses can be treated as uniform and space-time distribution of frequency can be neglected. By neglecting the network, the DFR model reduces to ASF model from which uniform frequency can be achieved. The general diagram of the ASF model is shown in Fig. 2(a) where turbine-governors and loads are modelled explicitly. *P*
_{
mΣ} and *P*
_{
eΣ} are total mechanical power and total active power load of the system. Δ*ω* is the uniform frequency of the system which is generated from the equivalent swing equation. In addition, all loads can be aggregated into an equivalent load model to simplify the ASF model. It can be applied in applications such as spinning reserve allocation, load frequency control, etc [17, 18].

The ASF model can be modelled with DAEs and solved with step-by-step integration. With network neglected, the computational burden of the ASF model is much less than that of the DFR model.

### Single machine model

Single machine model can be treated as a special case of the ASF model, as shown in Fig. 2(b). It is obtained by further aggregating all turbine-governors and loads in the ASF model. The nonlinearity of the turbine-governors, such as the valve limits and dead bands, is reserved. The structure of aggregated turbine-governors is usually the same as normal turbine-governors. For example, for stand-alone system with most of electricity generated by thermal generating units, steam turbine-governor is preferred for the aggregated model. Step-by-step integration is also used to solve the nonlinear single machine model.

### SFR model

Nonlinearity of the turbine-governors is considered in the DFR model, ASF model and the single machine model. No analytical expression can be directly obtained and step-by-step integration is the most popular method to get discrete response. By neglecting the nonlinear blocks and small time constants, SFR model was proposed in [14] to derive an analytical expression of frequency dynamics for stand-alone systems, in which the generators are dominated by reheat steam turbines. The block diagram of the SFR model is shown in Fig. 3 where *P*
_{
d
}, *P*
_{
m
}, *H*, *D*, *R*, *F*
_{
H
}, *T*
_{
R
}, and *K*
_{
m
} are disturbance, mechanical power, inertia, damping, droop, fraction of total power generated by high-pressure turbine, time constant of reheater, and mechanical power gain factor of the aggregated system. Using the analytical expression given in [14], the largest frequency deviation, its corresponding time, and steady frequency under a given active power disturbance can be calculated. Several research adopts SFR model for adjusting UFLS [19, 20].

### Discussion

The frequency dynamic characteristics can be categorized in different ways. For applications depending on the overall dynamic characteristics of frequency, e.g., frequency regulation, uniform frequency is usually assumed and the frequency at different locations is treated as the same. In this case, network can be neglected, and ASF model, single machine model, and SFR model are appropriate. The space-time distribution feature of frequency during event is of most interest for applications such as event location and oscillation detection where the difference between the generators at different locations should be taken into account. In this case, the influence of network should be retained to get the space-time characteristics, and the DFR model is suitable for such applications.

For detailed study of power system dynamic characteristics, the coupling between active power-frequency dynamics and reactive power-voltage dynamics should be included, resulting in the complex full time-domain simulation. However, for cases where frequency dynamic characteristic is of most concern and voltage dynamic is of little interest or voltage can be held at desired levels, the active power-frequency dynamics can be decoupled from the reactive power-voltage dynamics for simplification. It makes active-power the only factor affecting frequency, and the reduced models introduced above are suitable to examine the key impact of active power on frequency.