System inertia is used to describe all forms of inertia in a power system. For synchronous generators, the inertia can be described by the moment of inertia, and the inertia is reflected in the degree of resistance to changes in rotational speed, which is expressed as: J = ∫ r 2 d m (1) where r is the radius of rotation. m is the mass of the rigid body. For a generator i, its moment of inertia is constant, and the kinetic energy stored in its rotor rotation is: E i = 1 2 J i ω i 2 (2) where ω _i is the mechanical angular velocity of a single machine i. The generator kinetic energy E _i depends on the moment of inertia and the rotational speed, and is not related to the actual output power. For a generator operating at rated power, the kinetic energy depends only on its moment of inertia. The total generator inertia H _g can also be expressed as follow: H g = ∑ E i f 0 = ∑ H i S i I i f 0 (3) where S _i is the rated capacity of a single machine i. H _i is the inertia constant of a single machine (seconds) with base of machine’s rated capacity. I _i is the operation status of generator i with 0/1 indicating generator being offline/online. For power electronic interface sources, its inertia forms are various and its inertia characteristics are no longer limited to the moment of inertia, but depend on the control mode of the converter and its operating state. The inertia provided by the power electronics interface sources is called virtual inertia.

In this paper, we define the system inertia as the sum of the inertia of the synchronous generators and the virtual inertia of the power electronics interface sources. The system inertia is defined as follows: H s y s = H g + H e (4) where H _e is the virtual inertia provided by power electronics interface sources.

When an active imbalance disturbance occurs in the system, each synchronous generator instantaneously shares the disturbance power according to its synchronous power coefficient. After that, the electromagnetic power of the i-th generator changes suddenly, while the mechanical power remains unchanged. The rotor motion state will change according to the rotor motion equation under the action of unbalanced torque: 2 H i d ∆ f i 0 + d t = − ∆ P i L 0 + (5) where H _i represents the inertia of generator i. ∆ f i 0 + represents the frequency deviation of generator i at time 0⁺. t represents time. ∆ P i L 0 + represents the power change of generator i at time 0⁺. The difference among the inertia of the generators leads to different rotor speeds of the generators, but the effect of the synchronous torque will make the rotor speeds tend to be consistent. Therefore, each generator will re-share the disturbance power according to the generator inertia: ∆ P i L t = ∆ P L ⋅ H i ∑ i = 1 N H i (6) where N represents the number of generators. ∆ P L represents the disturbance power. The inertia response begins when the electromagnetic power changes suddenly due to the disturbance power ∆ P L , and ends when the frequency nadir, i.e., the balance between the electromagnetic power and the mechanical power, is reached. As shown in Figure 1, the overall process can be divided into two stages:

(1) t₀₊ ∼ t_gov: The moment of distribution of the disturbance power to the moment when the governor starts to act. The inertia of synchronous generators responds to support system power balance.

Note that this paper only describes synchronous generator inertia supporting power balance, but virtual inertia can also play a similar role. The virtual inertia can be divided into current-sourced virtual inertia and voltage-sourced virtual inertia. The current-sourced virtual inertia controls the active power by measuring the system frequency and feeding it back to the converter. The voltage-sourced virtual inertia mainly refers to virtual synchronous generator technology (VSG). The VSG, also known as the synchronous converter, refers to the introduction of synchronous generator rotor motion and electromagnetic transient equations in the control link of the converter so that it can simulate the voltage-sourced characteristics of the synchronous generator.

The frequency response of the power system in the actual operation process actually reflects the physical process of the frequency transition from the initial steady state to the new frequency steady state after the system active power balance equation is not satisfied. The role of the frequency security constraints is to guarantee the security, and stability of the system frequency during the above process (Muzhikyan et al., 2018).

As shown in Figure 2, the inertia response occurs first to ensure the balance of electromagnetic power and mechanical power. Then, the primary frequency regulation starts to restore the frequency from the frequency nadir to the quasi-steady state. In the process of frequency response, three key metrics are used to evaluate the stability of system frequency, i.e., the rate of change of frequency (RoCoF), the frequency nadir (f_nadir), and the steady-state frequency deviation (f_ss). These three key metrics need to meet the frequency limits set by the National Grid.

(1) Rate of change of frequency ( Chu et al., 2020 ): ROCOF indicates the slope (Hz/s) of frequency drop, and its value is related to the system inertia and power deficit. The mathematical formulation can be expressed as follow:

The UC model combined with frequency security constraints is as follows: min P i t , I i t ∑ t = 1 T ∑ i = 1 N C i P i t I i t + S U i I i t 1 − I i t − 1 (10) subject to ∑ i = 1 N I i t P i t = D t (11) ∑ i = 1 N I i t P i , max t − D t ≥ R t (12) P i , min t I i t ≤ P i t ≤ P i , max t I i t (13) X i , o n t − 1 − T i , o n ⋅ I i t − 1 − I i t ≥ 0 (14) X i , o f f t − 1 − T i , o f f ⋅ I i t − I i t − 1 ≥ 0 (15) 0 ≤ P i t − P i t − 1 ≤ U R i (16) 0 ≤ P i t − 1 − P i t ≤ D R i (17) − ∆ P L 2 H s y s ≤ ∆ f ˙ lim (18) ∆ P L 2 T d 4 H s y s R ≤ ∆ f lim (19) ∆ P L − R D P D ≤ ∆ f lim s s (20) where C _i is the marginal cost of generator i. P i t and I i t are the output power and on-off state of generator i at time t, respectively. SU _i is the startup cost of generator i. D ^t and R ^t are load demand and reserve demand at time t, respectively. P i , min t and P i , max t are the minimum and maximum active output of generator i at time t, respectively. UR _i and DR _i are the ramping-up rate and ramping-down rate of generator i, respectively. T i , o n and T i , o f f are the minimum on and off requirements of generator i, respectively. X i , o n t − 1 and X i , o f f t − 1 are the continuous running time and downtime of the generator i, respectively. ∆ f ˙ lim , ∆ f lim and ∆ f lim s s are the maximum permissible RoCoF, maximum permissible frequency deviation and maximum permissible steady-state frequency deviation, respectively.

The objective (10) is to minimize the total operation cost including startup and generation costs. Constraint (11) ensures active power balance. Constraint (12) ensures sufficient reserve capacity. The generator constraints include generation limits (13), minimum on and off requirements (14)-(15), and ramping limits (16)-(17). Constraint (18) ensures that the RoCoF is within the allowed deviation range. Constraint (19) ensures that the frequency deviation is within the allowed deviation range. Constraint (21) ensures that the steady-state frequency deviation is within the allowed deviation range.

In this paper, we defined the dual multipliers of constraint (11) as the energy clearing price, which can be obtained by solving the dual problem of the above model.

The system operation cost is obtained by solving day-ahead UC and real-time economic dispatch (ED). Specifically, since H _e remains unknown in the day-ahead stage, UC is first conducted to clear the day-ahead market, providing the actual UC cost, i.e., start-up cost. After H _e is realized in the real-time stage, ED is executed to clear the real-time market, providing the actual ED cost, i.e., generation cost for all units. Therefore, the system operation cost is equal to the sum of the actual UC cost and the actual ED cost. The simplified ED model is as follows: min P i t ∑ t = 1 T ∑ i = 1 N C i P i t I i t ∗ (21) subject to ∑ i = 1 N I i t ∗ P i t = D t (22) ∑ i = 1 N I i t ∗ P i , max t − D t ≥ R t (23) P i , min t I i t ∗ ≤ P i t ≤ P i , max t I i t ∗ (24) c o n s t r a i n s 16 − 20 (25) where I i t * is the solution of the UC model. Therefore, the system operation cost can be calculated as follows: c o s t = ∑ t = 1 T ∑ i = 1 N C i P i t * I i t ∗ + S U i I i t ∗ 1 − I i t − 1 (26) where P i t * is the solution of the real time ED model.

Several case studies of the modified GB 2030 power system are carried out to analyze the impact of system inertia on system operation costs (Teng et al., 2016). The characteristics of generation plants are included in Table 1. The system includes nuclear power generators, open-cycle gas turbines (OCGTs), and combined-cycle gas turbines (CCGTs). OCGTs can be regarded as slow synchronous generators with low marginal cost and CCGTs can be regarded as faster synchronous generators with higher marginal cost. The system parameters are set as follows: load demand P _D ∈ [20, 60] × 10³ MW, damping D = 0.5% P _D /1 Hz, frequency response delivery time T _d = 10 s and maximum power loss ΔP _L = 2100 MW. The frequency limits set by National Grid are: Δ f lim = 0.8 Hz, Δ f lim s s = 0.5 Hz, and Δ f ˙ lim = 0.5 Hz/s. The simulation adopts day-ahead scheduling, with a total duration of 24 h and a time interval of 1 h.

We first set the system inertia at each moment to be the same, and then gradually increase the system inertia to observe the change in system operation cost. As shown in Figure 3, with the gradual increase of inertia, the overall system operation cost presents a downward trend. This is because the frequency security constraints will be easily satisfied when the system inertia is sufficient, resulting in fewer CCGTs being turned on in the day-ahead decisions to provide inertia and frequency regulating reserve. It is also worth noting that there is not a linear relationship between system inertia and system operation costs. It can be seen from Figure 4 that with the increase of the system inertia volume, the energy clearing price decreases, and the decrease is particularly obvious in the shaded area. This is because without additional inertia support, satisfying the system frequency security constraints would require OCGTs with higher marginal costs to be online to provide inertia and frequency regulating reserves support. And when the system inertia volume increases, OCGTs do not need to be scheduled.

We first set the system inertia at each moment to be a normal distribution with a mean of 50 (MWs/Hz), and then gradually increase the standard deviation of the system inertia to observe the change of system operation costs. As shown in Figure 5, as the volatility of inertia gradually increases, the system operation cost shows an upward trend. This is because the greater the volatility of the system inertia, the more CCGTs are required to be online for day-ahead scheduling decisions, thus resulting in expensive start-up costs. It can be seen from Figure 6 that the energy clearing price increases in some time periods as the standard deviation of inertia increases. This is because due to the constraints of ramping and minimum on and off requirements, the units cannot instantly provide inertia support when the system inertia volume is insufficient, and therefore more units are required online to cope with the volatility of the system inertia, resulting in increased energy clearing prices.

When making day-ahead decision-making, the virtual inertia in the system inertia needs to be forecasted. This is because virtual inertia depends on multiple factors, such as weather-dependent non-synchronous sources, load, market forces, etc. The system operation cost caused by the inertia forecasting error may be asymmetric, so it is necessary to analyze the impact of the inertia forecasting errors on the system operation costs.

We set the true value of the system inertia to 250 (MWs/Hz), and the system operation cost caused by the forecasted error is shown in Figure 7. Figure 7A shows the impact of system inertia forecasting errors on system operation costs for the high average load scenario. The under-forecasting of inertia in this scenario does not result in an increase in system operation costs, as the units being dispatched satisfy the power demand while also satisfying the day-head and real time frequency security constraints. However, over-forecasting the inertia will lead to higher system operation costs, because the real-time frequency security constraints may not be satisfied, so high-cost OCGTs with fast responding capability have to be dispatched online for inertia provision, resulting in a significant increase in system operation costs. Figure 7B shows the impact of system inertia forecasting errors on system operation costs for the low average load scenario, the under-forecasting of inertia results in more CCGTs being prepared during day-ahead scheduling, leading to less efficient part-loading operation. Figures 7C, D show the impact of the system inertia forecasting errors on the system operation costs when the marginal cost of OCGT is 75£/MWh and 300£/MWh, respectively. It can be seen that an increase in the marginal price of OCGT causes an increase in the cost of ED, thereby further increasing the system operation costs.