Frequency Oscillation Analysis of Power System With Multiple Governor Deadbands

In recent years, the ultra-low frequency oscillation of power system is one of the most concerned frequency stability problem. The governor deadband are considered as one of the main influences. However, the existing methods can only deal with single deadband segment. As for a multi-machine system with multiple deadband segments, it lacks the analysis method for frequency oscillation. To solve this problem, this paper first define the extended description function applied to nonlinear composite system according to the principle of traditional description function method. The extended description function can quantify the relationship of input and output of a system with multiple nonlinear segments. Then, based on the uniform frequency model of multiple governors, the turbines and governors of hydro and thermal power units with different deadband widths are equivalent to one extended description function. The locus of minus reciprocal of the extended description function with oscillation amplitude and the Nyquist curve of the generator are plotted on the complex plane for the analysis of system stability. Next, the critical oscillation amplitude of disturbance to trigger the system frequency oscillation is calculated according to the point of intersection of both curves. Finally, the simulation results show that the critical oscillation amplitude can reflect the stability of system. The change of deadband width is positively correlated with the critical oscillation amplitude. The appropriate setting of deadband and configuration of hydro-thermal generation are effective measures to suppress the ultra-low frequency oscillation while retain the ability of frequency regulation.


I. INTRODUCTION
The development of direct current (DC) transmission project has promoted the transition from synchronous interconnection to asynchronous interconnection of regional power grids in China. One of the main problems from the transition is the prominent frequency instability, especially the ultra-low frequency oscillation [1]- [3]. Take the Southwest Grid in China as an example [4]. After the back-to-back DC project for the Southwest and Central China Connection began operation, the Southwest Grid is asynchronously isolated. The capacity of generation is only 1/5 as that in synchronous interconnection. The decline of frequency due to disturbance is fast so the parameters of all governors are adjusted for fast The associate editor coordinating the review of this manuscript and approving it for publication was Yunfeng Wen . regulation. As a result, an ultra-low frequency oscillation of 0.05 Hz occurs in the quasi steady state after the governor regulation.
The common method applied in the mechanism study of ultra-low frequency oscillation is to use a low-order system frequency response (SFR) model as the equivalent model to the whole power system [5], [6]. Reference [7] makes a regional power grid become equivalent to a SFR model of single-machine with multiple governors. The equivalent SFR model consists of main governors and prime movers connected to the regional grid and is a high-order linear system. By calculation the eigenvalue of the equivalent model, the influence of hydro-thermal power ratio and hydro governor parameters on the ultra-low frequency oscillation is analyzed. However, the eigenvalue computation of a high-order system is already difficult, especially considering the eigenvalue VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ change caused by the variation of governor parameters. In order to derive the analytical solution of oscillation frequency and damping of a single-machine system with load model, Reference [8] simplifies the governor to a regulation coefficient. It focuses on how the turbine parameters and the power-frequency characteristic of load affect the damping ratio. Reference [9] considers the power-voltage characteristic of load and system power loss in the established singlemachine single-load SFR model of multiple governors. The eigenvalue analysis of the improved model indicates that the consideration of the two conditions produces a better stability of power system. So it follows that the ignorance of the two conditions brings forth more conservative but secure result. Reference [10] establishes the SFR model consisting of hydro and thermal power governors. It defines the minimum distance from the system Nyquist curve to the point (−1, 0) in the complex plane as the stability margin. By analyzing the influence of hydro and thermal power respectively on the minimum distance, the contribution of each to the system stability is presented. Reference [11] proves that the parameters of hydro governor leading to system negative damping are the main cause of ultra-low frequency oscillation. It specifically puts forward a method of withdrawing a portion of hydropower units and optimizing the governor deadbands to suppress the ultra-low frequency oscillation. The idea of applying the describing function method to overcome nonlinearity is proposed in the oscillation analysis of power system [12], [13]. The accuracy and completeness of stability analysis is proven to be enhanced. Regarding the influence of deadband, Reference [14] firstly introduce the governor deadband into the ultra-low frequency oscillation analysis. The ultra-low frequency oscillation occurs when the system is subjected to a certain magnitude of power disturbance according to the Nyquist criterion. But this method is only applicable to a system with one single deadband due to the restriction of description function method. Reference [15] proposes further on a numerical analysis method to deal with a system with multiple deadbands. The main idea is to combine the locus of eigenvalue with the describing function method. In each step of locus, the describing function of deadband is a constant real number. The computation of multiple nonlinearities are thus avoided. However, the difficulty of computing the eigenvalue of high-order system repeatedly is still a problem. Besides, the accuracy depends on the step of frequency amplitude.
Besides the SFR based method, the state-space method and the energy function method are available. The state-space method is based on the linear model and fully involves the correlation of generation-transmission-load, while the difficulty of modeling and computation is high when applicable in a complex system [16]. The energy function method is in consistent with the damping torque method. Combined with the measurements from PMU equipment, it is feasible to evaluate the damping characteristic of governor online [17].
Based on the existing researches, the following conclusions can be drawn. Firstly, all generators are synchronous in the ultra-low frequency oscillation. Secondly, the hydropower unit deteriorates the frequency stability of power system while the thermal power enhances it. Thirdly, the governor deadband make it complicated to analyze the system stability. The governor deadband exists extensively in the practical power grid and the deadband width is different. Therefore, on the basis of existing methods, this paper adopts the extended describing function method and the Nyquist stability criterion to calculate the critical magnitude of frequency disturbance to cause oscillation. The critical magnitude is an indicator of system frequency stability. The influence of multiple governor deadbands on frequency oscillation is quantitatively evaluated.

II. NONLINEAR COMPOSITE SYSTEM
Reference [14] studies the influence of step, non-step and hysteresis deadband on the frequency oscillation based on the single-machine system with single governor. The main kind of governor deadband in both the practical system and the simulation model of power company is non-step deadband. Given the situation, this paper focuses on the non-step kind of deadband as shown in Fig. 1.
The hydro and thermal power generations are the main kind of sources in the practical power grid. The settings of governor deadband for the two kinds are quite different. Fig. 2 shows the diagram of a SFR model. It contains a hydro turbine-governor and a thermal turbine-governor along with each deadband segment. The generators all maintain synchronous in the ultra-low frequency oscillation, so an equivalent machine of inertial center is adopted.
Ignore the system power loss and consider the powerfrequency characteristic of load and generator damping, the transfer function of the equivalent machine is where T J is the rotational inertia, and D S is the sum of generator damping coefficient and load power-frequency coefficient.
The hydro turbine-governor model is referred from Reference [14], and is expressed as Equation (2) and Equation (3).
where T W is the time constant of water hammer effect, K D , K P and K I are the differential, proportional and integral coefficients respectively, B P is the regulation coefficient, T G is the time constant of servosystem.
The steam turbine-governor model is referred from Reference [10], and is expressed as Equation (4) and Equation (5).
where F HP is the ratio of high pressure cylinder power in the total turbine power, T RH is the reheat time constant, T CH is the time constant of air intake volume. The system shown in Fig. 2 is nonlinear. The hydro and thermal governor contain different widths of deadband, so they should be regarded as two individual nonlinear segments. Except that, the turbine-governor and generator are all linear. The whole system is a typical nonlinear composite system. The conventional describing function method can only deal with a system with one single nonlinear segment as shown in Fig. 3, or a system of multiple nonlinear segments with no linear segment interleave.

III. EXTENDED DESCRIBING FUNCTION METHOD
The describing function method uses the fundamental harmonic to approximate the nonlinear segment output given the input of a sinusoidal signal [18]. Given a sinusoidal input, the output of the nonlinear segment is expanded by the Fourier series. The complex-valued ratio of the output fundamental component to the sinusoidal input is the describing function of the nonlinear segment. It can be regarded as a variable gain of its input as a substitution of nonlinear segment [19]. It is applicable to analyze the oscillation in the critically stable state. For a nonlinear composite system, the multiple nonlinear segments make it difficult to apply the describing function method directly. In order to ascertain the input-output relation of a nonlinear composite system, the describing function method is improved. Firstly, each nonlinear segment is expressed by a conventional describing function. Secondly, the whole system input-output relation is obtained through the vector calculation algorithm. Finally, an extended describing function to substitute the whole nonlinear composite system is obtained [20].
Take the simple nonlinear composite system in Fig. 4 as an example. N 1 ∼ N m are describing functions for each individual nonlinear segment. G 1 ∼ G m are transfer functions of linear parts. The starting input signal is expressed as The input for each nonlinear segment can be expressed as So the input for each linear segment, which is also the output of each nonlinear segment, is expressed as The amplitude and phase angle of the extended describing function is defined as |G e (A 0 , ω)| and G e (A 0 , ω) respectively.
For a nonlinear composite system shown in Fig. 4, the extended describing function G e (A 0 , ω) represents the input-output relation. Use G e (A 0 , ω) to substitute the whole VOLUME 8, 2020 FIGURE 5. Transfer function graph of system with multiple deadbands.
nonlinear composite system, the stability analysis can be implemented.

IV. ANALYSIS OF MULTIPLE DEADBAND SEGMENTS IN MULTI_MACHINE SYSTEM
Different deadband widths, governor models and parameters make it a nonlinear composite system in Fig. 2, on which the conventional describing function method cannot be applied. Fig. 5(a) shows the transfer function graph of the example system. NL h and NL s are the describing functions of hydro and thermal governors respectively. By using the method proposed in Section 3, each branch in Fig. 5 (a) is equivalent to an extended describing function as shown in Fig. 5(b). G he and G se represent the extended describing functions of hydro and thermal power. Then, using the idea of combination algorithm from the conventional describing function method, the multi-branch system is equivalent to the form shown in Fig. 3. The stability analysis is implemented based on its Nyquist curve. All the curves depicted in this section are based on the parameters from Section V.A.

A. EXTENDED DESCRIBGING FUNCTION OF SINGLE BRANCH
First analyze the hydro governor branch. The describing function of the non-step deadband segment is where b 1 is the deadband width, A 0 is the disturbance magnitude.
The deadband segment and the governor are connected in series. According to the method proposed in Section 3, the extended describing function amplitude and phase angle of the hydro governor with deadband are  Likewise, the extended describing function G se of thermal governor with deadband can be obtained.

B. COMBINATION OF MULTIPLE BRANCHES
Similar to the conventional describing function method, the two paralleled branches of nonlinear system share the same input and their output are added together according to the combination algorithm. So the entire nonlinear system is equivalent to the algebraic addition of the two extended describing functions.
G e is the function of disturbance magnitude A 0 and rotation speed ω. Given different frequency f (ω = 2πf ), the curves of amplitude and phase angle changing with A 0 are presented in Fig. 6 and Fig. 7.

C. DEADBAND INFLUENCE ANALYSIS
After the extended describing function of multi-deadband is obtained, the curve of −1/G e (A 0 ) along with A 0 increasing from the value of deadband width can be depicted. Similarly, the value of ω is determined according to ω = 2π f for each curve of −1/G e (A 0 ). Combined with the Nyquist Curve of generator G gen (s), the system stability can be analyzed. As the oscillation frequency value is different, the negative reciprocal curve of the extended describing function G e and the Nyquist curve of generator intersect differently. In Fig. 8, the arrowhead indicates the change direction of the −1/ G e curve with the increase of A 0 . As the −1/ G e curve and the generator Nyquist curve do not intersect, the nonlinear system is stable. As they do, the point of intersection corresponds to a critical amplitude A 0 . When a disturbance of over A 0 amplitude occurs, the system becomes instable, otherwise stable.
Similarly given A 0 = 0.01pu and f = 0.1Hz, the influence of deadband width change on the system stability is analyzed. The deadband widths of hydro and thermal governors are set to be 0.0020 + b and 0.0015 + b respectively. Then the extended describing function G e is a function of b. The curve of −1/ G e (b) is depicted in Fig. 9 along with the Nyquist curve of generator. The arrowhead indicates the change direction of −1/ G e (b) with b increasing from 0. Shown in Fig. 9, −1/ G e (b) is initially encircled by the generator Nyquist curve and the system is instable. While b increases from 0, −1/ G e (b) passes through the Nyquist curve and the system is brought into the stable area. Therefore, it can be concluded that in regard to frequency oscillation, the system stability becomes higher with the increase of deadband width.
Note that in Fig. 8 and Fig. 9, the intersection point corresponds to different oscillation frequency ω for G e and G gen . While in a connected system, they are of the same frequency ω. The key point of Fig. 8 and Fig. 9 is to indicate that there exists an unknown point where G e and G gen can intersect with the same frequency. Then it leads to the purpose of proposed method to locate the point, that is to solve the critical amplitude A 0 and oscillation frequency ω 0 .
The value of critical amplitude A 0 is the key to the analysis of ultra-low frequency oscillation. The intersection point of both curves corresponds to the limit cycle, the creation condition of which is |G e | · G gen = 1 G e + G gen = −π 2 (15) To solve the simultaneous equations (15) provides the critical amplitude A 0 and its corresponding oscillation frequency ω 0 . The high value of A 0 indicates that the amplitude of disturbance to cause the ultra-low frequency oscillation is high, so is the system stability. Hence, the value of critical amplitude A 0 is a measure of the system stability.

A. SYSTEM DESCRIPTION
This section adopts the system in Fig. 2 to verify the proposed method. The configuration parameters are as follows.
Moreover, b 1 and b 2 are the deadbands of hydro and thermal governors respectively. By adjusting the value of b 1 and b 2 , the critical amplitudes to cause system oscillation are calculated through the proposed method. The results are presented in Table 1.

B. ULTRA-LOW FREQUENCY OSCILLATION ANALYSIS OF MULTI-DEADBAND SYSTEM
Focus on Case 1, 3 and 5 in this section. The deadband widths are different but within a common range in the three cases. It aims to test the effectiveness of the proposed method on different deadband widths. Establish the system of Fig. 2 in MATLAB/Simulink, set the deadband widths according to Case 1, 3 and 5, test the system with different amplitudes of disturbance, and the system stability can be observed.
Every setting of deadband in different cases corresponds to a specific critical amplitude. According to the analysis in Section IV.C, when the amplitude of disturbance is larger than the critical amplitude, the system is instable with a frequency oscillation. Otherwise, the oscillation does not last and the system is stable. The critical amplitudes are larger than the deadband widths in each case, then the two disturbances of less-than and greater-than critical amplitude (D1 and D2) are input into the system. The frequency response curves are presented in Fig. 10. In Case 1, the critical amplitude is 0.0116 pu. The disturbance of 0.0114 pu magnitude triggers a oscillation to decay gradually. The disturbance of 0.0117 pu is over the critical amplitude, so the oscillation persists to destabilize the system. The scenarios are similar in Case 3 and Case 5 and accord with the analysis in Section IV.C.

C. INFLUENCE OF DEADBAND WIDTH
The system established in Fig. 2 consists of a thermal and hydro generator, which is conducive to study the dead-band influence involving the frequency regulation of different energy sources units. Case 1 is deemed as the benchmark in this section and compared with the other cases.

1) INCREASE OR DECREASE BOTH DEADBAND WIDTHS
The two deadband widths are increased (Case 2) or decreased (Case 3) together. The critical amplitude is in positive correlation with deadband width. It is related to the deadband characteristic. With a wider deadband, the range for the disturbance to go through to activate the governor regulation is larger. The critical amplitude of disturbance to trigger the ultra-low frequency oscillation is thus higher. Hence, the system stability is higher. This is consistent with the analysis in Section IV.C.

2) DECREASE THE HYDRO OR THERMAL DEADBAND WIDTH SEPARATELY
Compare the critical amplitude from Case 4 (decrease the hydro deadband) and Case 5 (decrease the thermal deadband) with that from Case 1. It can be seen from the comparison that to decrease whether the hydro or thermal deadband decreases the threshold value of disturbance amplitude to trigger frequency oscillation. The risk of system disability increases. Meanwhile, it shows that to decrease the same width (0.0003 pu), the critical amplitude of Case 5 is higher than that of Case 4. It indicates that to decrease the hydro deadband is more conducive to deteriorate the system stability than to decrease the thermal deadband.

3) INCREASE THE HYDRO DEADBAND AND DECREASE THE THERMAL DEADBAND SIMULTANEOUSLY
In order to study the influence of different participation level of hydro and thermal governor due to the deadband limit in the frequency regulation, meanwhile to exclude the influence of deadband itself on the critical amplitude explained in section (1), Case 6 exchanges the deadband setting between the hydro and thermal governor. After the exchange, there are a same deadband setting of 0.0015 pu and 0.0020 pu in both Case 1 and Case 6. The difference in Case 1 is that the thermal governor is of lower deadband width and it responses to the disturbance firstly. After a period of the thermal governor regulation, the hydro governor participates in the regulation. It is just the opposite in Case 6.
With a disturbance of 0.01 pu magnitude as input, the system frequency oscillations of Case 1 and Case 6 are opposite. Because the disturbance amplitude is less than the critical amplitude of Case 1 (0.0116 pu), the solid curve of oscillation decays. While the disturbance curve exceeds the critical amplitude of Case 6 (0.0097 pu), the dashed curve of oscillation persists.
The critical amplitude of Case 6 is less than that of Case1. It indicates that when the whole participation level of governors remains the same, to increase the participation of thermal governor and decrease that of hydro governor is conducive to suppress the ultra-low frequency oscillation. For  the primary frequency regulation, a larger deadband width reduces the ability of system frequency regulation. Therefore, to avoid the system frequency decline significantly and also suppress the ultra-low frequency oscillation, it is practical to optimize the deadband setting of hydro and thermal governor to control the system frequency response.

D. HYDRO-THERMAL GENERATION RATIO ANALYSIS
The aim of the proposed method is to analyze a multi-machine power system with hydro and thermal generation. The previous analysis is based on a system with two units of one hydro and one thermal generation. This section focuses on the multi-machine system and the ratio of hydro and thermal generation. For this purpose, a 4-governor single-machine system is established shown in Fig. 11. Based on this model, three cases of different hydro-thermal configurations are studied.
In Case I, the system consists of two hydro units and two thermal units. In Case II, the system consists of three hydro units and one thermal unit. It is for the test of high proportion of hydro generation. In Case III, the system consists of one hydro unit and three thermal units. This case is to test the measure of increasing the thermal generation while decreasing the hydro generation to suppress the ultra-low frequency oscillation. In all cases, the hydro units share the same width of 0.0010 pu deadband and the thermal units share the other same width of 0.00080 pu deadband. Table 2 shows the critical amplitude of three cases. It can be seen from the table that the adjustment of hydro-thermal ratio affects the critical amplitude. By inputting a disturbance of 0.002 pu, the frequency curves from Case II and Case III are presented in Fig. 12. The disturbance causes a diverging oscillation in Case II while it causes a converging oscillation in Case III. The comparison proves that the critical amplitude gives a distinct boundary of the frequency stability. The appropriate configuration of hydro-thermal generation helps enhance the stability of power system.

VI. CONCLUSION
The variety of deadband widths in a multi-machine system makes it infeasible to apply the traditional describing function method in analysis of ultra-low frequency oscillation. This paper proposes a novel extended describing function method, in which the nonlinear composite system of multi-deadband is equivalent to a structure of an extended describing function and a linear transfer function. Then the Nyquist curve is adopted to analyze the system stability. If the negative reciprocal curve of the extended describing function intersects with the Nyquist curve of the generator, the critical amplitude corresponding to the intersection point is the minimum value of disturbance magnitude to cause frequency instability. The simulation result proves that the critical amplitude reflects the stability level of power system. By changing the deadband setting of different generation unit, the critical amplitude is obtained. The value of critical amplitude shows that the thermal governor is more conducive than the hydro governor to suppress the ultra-low frequency oscillation. By setting the appropriate configuration of hydro-thermal generation, the frequency stability of power system can be improved meanwhile the ability of frequency regulation is maintained. The method proposed in this paper is an extension of the describing function method. It plays an important role in expanding the stability analysis range from the single-machine to multi-machine system, from the single-nonlinear segment to multi-nonlinear segment system.