Error Analysis Due to the Transient Reactance of the Generator in Simulation Verification of Power System Inertia Evaluation

In the simulation validation process of the inertia evaluation model, it is necessary to collect the power and frequency data of a node accurately after the disturbance and use a reasonable data processing algorithm to reduce the error. Given that the power and frequency characteristics of the generator outlet side bus are similar to those of the generator, some research work selects the generator outlet side bus as the test object of the power and frequency data acquisition device. However, the generator outlet side bus inertia and the theoretical generator inertia are not equivalent because of the generator transient reactance. In this paper, we first establish a 3-machine 6-node inertia simulation model using a constant power load connected via a circuit breaker to provide system unbalanced power and then simulate the generator outlet side bus inertia under two conditions of the transient reactance set to zero and non-zero. Through comparison with the theoretical generator inertia, it is concluded that the generator outlet side bus inertia represents the theoretical generator inertia only when the transient reactance is set to zero. This conclusion applies to complex power systems as well.


Introduction
Inertia is an inherent property of a power system that reflects the ability to suppress frequency changes and reduce the impact of disturbances [1], [2].The level of inertia is typically measured by the inertia time constant [3], [4].The power system inertia is mostly stored in the synchronous generator rotors and turbines, and it has a fixed value.However, with the gradual increase in the share of new energy sources, there is a significant trend of weak inertia in power systems [5]- [7], which greatly weakens the power system stability [8], [9].As a result, accurate evaluation of the system inertia is critical for studying the dynamic characteristics of the system after perturbation and improving system stability.
There are two types of inertia assessment methods: the system-level power system equivalent inertia assessment methods and the node-level power system node inertia assessment methods.These two methods have different research objects but similar research processes in that they both propose a model to analyse its superiority and validate it in simulation software.During the simulation verification process, we need to inject the unbalanced power into the model and calculate the inertia based on the node power and frequency variations measured by the theoretical model at each node of the system.So, it is critical to accurately construct the simulation model to collect the power and frequency data, as well as to use data processing algorithms to eliminate singular data and reduce errors after calculating the results.Because the power and frequency change detected at the generator outlet bus after the disturbance is similar to that of the connected generator, some researchers select the generator outlet bus as the test object for the power and frequency data collection device.The generator outlet side bus inertia is calculated from test data and compared to the theoretical generator inertia, and the error is reduced by improving the measurement device and using the data processing algorithm.Finally, the test-data processing process with lower error is applied to the data acquisition and post-processing of the system inertia evaluation simulation [10], [11].However, there is an electrical distance between the generator outlet side bus and the generator internal potential node because of the generator transient reactance, which leads to the inequality of the generator outlet side bus inertia and the theoretical generator inertia.
This paper constructs a 3-machine 6-node inertia simulation model using a constant power load connected via a circuit breaker to provide system unbalanced power.The generator outlet side bus inertia is simulated and compared with the theoretical generator inertia under two conditions of zero and non-zero generator transient reactance, to verify that the generator outlet side bus inertia represents the theoretical generator inertia only when the transient reactance is set to zero.

The inertia theory
The inertia theory distinguishes between two kinds of inertia: theoretical inertia and detected inertia.The theoretical inertia is based on the equation of motion of the synchronous generator rotor, which reveals the overall inertia level of the system.The detected inertia is the current inertia of the system calculated by measuring various electrical parameters, such as frequency, active power, etc. Calculating the current inertia of the system allows us to evaluate the current inertia level of the system.The theoretical inertia of a generator is represented by the kinetic energy of the generator rotor at rated angular velocity, which is expressed as follows: where E i , H i , S i , J i and ω n respectively represent the theoretical inertia, the inertia time constant, the rated capacity, the rotational inertia and the rated angular velocity of the generator i.
Without considering the inertia provided by the energy storage element and the load, the equivalent inertia expression for the multi-machine power system is as follows: where E s is the theoretical inertia value of the system; and G is the set of generators in the system.With the gradual increase in new energy penetration of the power system, the trend of lowering power system inertia is significant, and virtual inertia technology is rapidly developing to improve the stability of the new energy power system.The equivalent inertia of the new energy power system is provided by conventional generating units and new energy units using virtual control technology.The theoretical expression for the equivalent inertia of the new energy power system is as follows: where E s is the theoretical inertia of the conventional generator set; E r is the theoretical inertia of the new energy unit using virtual control technology; and G s and G r represent the conventional generator and the new energy generator set, respectively.When the power system is disturbed, the detected inertia expressions are mostly used to assess the current inertia level of the system.For a single generator, the system equivalent detected inertia is represented by the generator rotor equation of motion as: where H COI is the system equivalent inertia time constant; ∆P is the disturbed power of the system; and f COI is the system inertia centre node frequency.

The inertia evaluation simulation test process and methods to reduce measurement errors
The inertia evaluation follows the process of model proposal, theoretical analysis, and simulation verification.For inertia simulation verification, Equation ( 4) is primarily used to build simulation models in various power system simulation software.The disturbance power of the power system can be introduced in two ways.The first method, using the cutter event as an example, involves controlling the circuit breaker to remove one generator from the system at a specific time, and the active power output of this generator is the system disturbance power at that time.The second method involves connecting a constant power load to a system node via a circuit breaker and introducing the system power imbalance by controlling the circuit breaker, at which point the system power imbalance is the active power of the constant power load.Both two disturbance power introduction methods can be applied to the inertia detection simulation scenario, but the first method is less likely to occur because of the difficulty of the cutter fault simulation, so the second method is more in line with the disturbed situation in the operation process of the actual power system.Due to the difficulty in measuring the frequency change rate of the system equivalent inertia center, discrete models are generally used in practical modelling to reduce the difficulty in collecting simulation data.Using the constant power load approach to introduce perturbations, the expression for evaluating the equivalent inertia of the power system simulated by the discrete model is as follows: where H COI t is the system equivalent inertia at moment t during the simulation; ∆P is the disturbance power of the system, which is numerically equal to the active power of the constant power load; and ∆t is the system sampling interval in the simulation.
In fact, the constant power load is connected to one node.Because working at the rated voltage of the load is difficult, the system disturbance power ∆P can be expressed as follows: where k is the node connected to the constant power load; P k t is the power measured at the node k at moment t after the occurrence of the disturbance; and P k t 0 is the power measured at the node k at moment t 0 before the disturbance.Therefore, the discrete model power system equivalent inertia evaluation equation is as follows: As shown in Equation ( 7), if we need to assess the system inertia level, the simulation modelling should accurately measure the power change of the node connected to the constant power load and the frequency of the system inertia centre node.To reduce the error in the calculation of the H COI t , the average value within a period after the occurrence of the disturbance (generally taken 0-0.2 s or less than 0.2 s) can be selected, and other numerical calculation methods can also be used.
To verify the accuracy of the power and frequency devices in the simulation model, the active power change and frequency change detected at the generator outlet side bus after the disturbance should be similar to the connected generator power and frequency change considering that the generator outlet side bus is connected to the generator.Many researchers calculate the inertia time constant of the bus by measuring the power change and frequency change rate at the bus side and then compared them with the theoretical inertia of the synchronous generator.The error between the two can be reduced using numerical calculation methods such as the averaging method and the sliding window method, demonstrating the accuracy of the power and frequency measurement device.These two devices are finally used to measure the system disturbance power and the inertia centre node frequency.However, this method is seriously flawed and the reasons are discussed below.

The inertia evaluation simulation test error generation causes analysis
The generator model is equated as a voltage source plus transient reactance in the power system transient analysis.The power and frequency detected on the generator outlet bus side are not equal to the power and frequency of the generator itself in a complex multi-machine power system, and there is a generator transient reactance between these nodes.The specific structure is shown in Figure 1.As shown in Figure 1, the theoretical inertia of the generator G1, G2, ..., Gm can be calculated by the power change and frequency change rate at the nodes n+1, n+2, ..., n+m.The nodal inertia is calculated by the power and frequency variation at the generator-connected bus at nodes 1, 2, ..., m in Figure 1, which does not represent the theoretical generator inertia.The reason is that the nodes n+1, 1; n+2, 2; ...; n+m, m are separated by the generator transient reactance, and an electrical distance exists between the generator outlet bus and the generator internal potential nodes.When the generator transient reactance is set to zero, the generator potential node and the generator-connected bus coincide, and the node inertia calculated by Equation ( 7) represents the theoretical generator inertia.Because the two are not equivalent in the case of non-zero generator transient reactance, it is incorrect to detect the data of the generator outlet side bus and use the data processing algorithm to reduce the error of the theoretical generator inertia.In this paper, we prove the accuracy of the above discussion by building the 3-machine 6-node inertia simulation model and comparing the generator outlet bus inertia with the theoretical generator inertia under zero and non-zero generator transient reactance conditions, as well as extending the model to complex power systems.

The 3-machine 6-node Simulink model inertia simulation
The inertia simulation employs a 3-machine 6-node MATLAB/Simulink model.The disturbance is provided by connecting a 0.1 (p.u.) constant power load at bus 6 via the circuit breaker and the circuit breaker is closed at the moment t=0.2 s.The system frequency is 60 Hz and the system wiring diagram is shown in Figure 2. As shown in Figure 2, the system nodes mostly use the V-I measurement module instead of the bus, and the bus inertia during the simulation is calculated by measuring the power and frequency at the bus with a power measurement device and a frequency measurement device.The power and frequency detected devices based on Simulink are respectively shown in Figures 3 and 4  As shown in Figures 3 and 4, the bus power is detected by the three-phase power measurement device that extracts the bus voltage and current.The bus frequency is calculated by detecting the voltage phase angle at the bus and differentiating it.The time interval between each sampling point is ∆t and the inertia time constant of node k at moment t after the disturbance occurs is as follows: where ∆P k t is the active power change measured at the node k at moment t after the occurrence of the disturbance; and ∆P k t 0 is the active power change measured at the node k at moment t 0 before the occurrence of the disturbance.
To reduce the measurement error, the average value of the inertia time constant within 0.2 s of the disturbance is typically used instead of the bus inertia at the time of the disturbance.Combined with the 3-machine 6-node wiring diagram shown in Figure 1, the detection method of the Equation ( 8) was used to detect buses 1, 2, and 3 in Figure 1 respectively, and the bus inertia time constants and the inertia time constants of the generator G1, G2 and G3 are compared and shown in Table 1 1, the inertia time constant of each generator detected by simulation is similar to the theoretical generator inertia time constant, which proves that the detected power and frequency discrete data can be used to calculate the inertia accurately using Equation (8).However, the inertia detected by generator connected bus in the fifth column is in great error with the generator inertia.Many papers use various data processing algorithms such as the sliding window method to make the data in the fifth column close to the data in the fourth column, thus proving that they can reduce the measurement error by using numerical processing methods.But this statement is flawed.In Figure 1, there is no impedance between the generator and the bus, and the power frequency measured from the generator and the power frequency measured from the bus should be highly compatible theoretically.But in fact, these two elements are not directly connected and there is a generator transient reactance in the middle.The 3-machine 6-node electrical wiring diagram is shown in Figure 5.The multi-machine system transient model in Figure 5 shows that there is a transient reactance between the generator's internal potential node and the generator outlet bus that causes an electrical distance.Thus, the detection of the generator internal potential node reactance at the generator outlet bus 1, 2 and 3 is wrong.To verify this conclusion, the transient reactance of the generator model in Figure 1 is set to zero and the rest of the conditions remain unchanged.The experiments in Table 1 were repeated to obtain the comparison of the bus inertia time constant detected when the potential reactance inside the generator is zero and the generator G1, G2 and G3 inertia time constant.From columns 3 and 5 of Table 2, after setting the transient reactance of the three generator sets to zero, the inertia detected by the generator-connected bus is approximately equal to the generator inertia.Comparing the data in columns 4 and 5, there is a great difference in the inertia detected by the generator-connected bus before and after setting the transient reactance to zero.The reason is that the generator-connected bus does not coincide with the potential node in the generator when the transient reactance is non-zero and the generator outlet side bus inertia cannot represent the generator inertia.Only when the generator transient reactance is set to zero, the generator outlet side bus inertia can be used instead of the generator inertia, and the accuracy of the measurement module detection can be measured by comparing the error between the generator inertia and the generator outlet side bus inertia.

The 10-machine 39-node Simulink model inertia simulation
Extending the above theory to a multi-machine system, the IEEE 10-machine 39-node New England system in Figure 6 is used to verify the relationship between the bus side detected inertia and the theoretical generator inertia.The system disturbance power is added to any of the 39 nodes selected.A constant power load of 0.1 (p.u.) is provided through the circuit breaker connection and the circuit breaker is closed at the moment t=0.2 s.The theoretical inertia time constant of the 39 nodes generator and the measured inertia time constant on the bus side are shown in Table 3.
From Table 3, it can be seen that the conclusion drawn from the 10-machine model is consistent with the 3-machine model, indicating that the theory of detecting generator inertia at the generator outlet side bus is inappropriate for the complex power system model, and it is wrong to reduce the error between the generator outlet side bus inertia and the generator inertia by a series of data processing methods.For a multi-machine power system, the generator outlet bus inertia is the generator inertia only when the generator transient reactance is zero.To verify the superiority of the numerical calculation method in reducing the measurement error, the generator transient reactance should be set to zero first.

Conclusions
This paper establishes the 3-machine 6-node and 10-machine 39-node inertia simulation model through MATLAB/Simulink simulation environment, introduces disturbance to the system in the form of a circuit breaker connected to a constant power load, calculates the generator outlet bus inertia with the power change and frequency change rate of each generator outlet bus, and compared the calculated inertia with the theoretical generator inertia.The main conclusions of this paper are summarized as follows.
1) The proposed method for the power and frequency measurement device can measure the power and frequency data of the measured node accurately and calculate the generator outlet side bus inertia.This method is suitable for complex multi-machine power systems.
2) It is wrong to reduce the error between the generator outlet side bus inertia and the theoretical generator inertia by the averaging method, the sliding window method and other data processing methods.The generator outlet side bus inertia and the theoretical generator inertia are not equivalent because there is an electrical distance between these two nodes provided by the generator transient reactance.Only when the generator transient reactance is set to zero, the generator outlet side bus inertia is equivalent to the theoretical generator inertia and it is feasible to use data processing methods to reduce the error between the two in this scenario.
3) In order to reduce the inertia error in the power system simulation evaluation process, we can experiment with the detection device and data processing method that can reduce the error between the generator outlet side bus inertia and the theoretical generator inertia in the circumstances that the generator transient reactance is set to zero.

Figure 1 .
Figure 1.The equivalent diagram of multimachine power system transient analysis.

Table 1 .
. Comparison of the inertia detected by generator and the inertia detected by generator connected bus

Table 2 .
Comparison of the detected inertia of generator-connected bus before and after setting zero for transient reactance

Table 3 .
Comparison of the detected generator inertia and the detected generator outlet bus inertia