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IET Generation, Transmission & Distribution Research Article

Reduced-order model for computing frequency oscillation mode of power systems

ISSN 1751-8687 Received on 22nd November 2017 Revised 22nd February 2018 Accepted on 17th March 2018 E-First on 18th April 2018 doi: 10.1049/iet-gtd.2017.1800 www.ietdl.org

Lei Chen1, Ming Sun1, Xiaomin Lu1,2, Yong Min1 , Kaiyuan Hou3, Deming Xia3 1Department

of Electrical Engineering, Tsinghua University, Beijing, People's Republic of China Grid Nanjing Power Supply Company, Nanjing, People's Republic of China 3Northeast China Branch of State Grid Corporation of China, Shenyang, People's Republic of China E-mail: [email protected] 2State

Abstract: Frequency oscillations with very low oscillation frequencies were observed in several power systems. In frequency oscillations, the speeds of all generators change coherently and the frequencies in the system oscillate together. An equivalent model, which only preserves the governors, turbines, and an equivalent generator, is subsequently built. In the model, the intermachine rotor-angle oscillation modes are eliminated and only the frequency oscillation mode is preserved. First, a simplified relationship between the total generator output power and frequency is adopted to obtain an original reduced-order model. With the model, an initial estimation of the eigenvalue corresponding to the frequency oscillation mode is obtained. The initial estimation is used to obtain a more accurate relationship between the total output power and frequency, and then an improved reduced-order model is built. With the model, a more accurate eigenvalue estimation can be obtained. Owing to the low order of the proposed model, the computational burden is greatly reduced compared to the direct analysis with fully modelled systems. The test results verify the validity of the proposed method.

1 Introduction Frequency oscillations with very low oscillation frequencies, typically in the range of 0.01–0.1 Hz, occur in several power systems [1–5]. Periodic oscillations with frequency deviations of 0.05 Hz, with the 20–30 s period, have been observed in the Turkish power system [1]. Frequency oscillations of considerable amplitude and very low oscillation frequency were observed in the Colombian power grid [2, 3]. Some oscillations had deviations up to 1 Hz from the nominal 60 Hz with 12–20 s period and lasted for tens of minutes. One recent incident occurred on 28 March 2016 in the Yunnan power grid [5], a provincial grid in southern China. The Yunnan grid was asynchronously interconnected to other grids with only high-voltage direct current transmissions when the incident occurred. An oscillation arose and the frequency oscillated between 49.9 and 50.1 Hz with a period of 20 s. The oscillation lasted for 25 min and greatly threatened the security of the system. From the theory of small-signal stability, undamped oscillations are related with complex eigenvalues with zero or even positive real parts, and eigenvalue analysis is an effective tool to study the small-signal stability of power systems. Conventional studies focus on low-frequency oscillations, which are oscillations between groups of generators and belong to the small-signal rotor-angle stability [6]. Many methods are proposed for computing the rotorangle oscillation modes [7]. However, after careful investigation, the above-mentioned oscillations are not traditional rotor-angle oscillations [8, 9]. They are strongly related with the governors and turbines of generators, and are deemed as a result of the smallsignal instability of the system's primary frequency regulation process. The corresponding oscillation mode is called the frequency mode [8] or the common swing mode [9]. Conventional eigenvalue computation methods fully model the system, including governors and turbines, form the Jacobian matrix, and then directly compute the eigenvalues, in which the eigenvalues corresponding to the frequency oscillation mode are supposed to be included. However, these methods have some deficiencies. The order of the system is high, which greatly increases the burden of the eigenvalue computation. Moreover, the eigenvalues corresponding to the frequency oscillation mode mix with the rotor-angle

IET Gener. Transm. Distrib., 2018, Vol. 12 Iss. 11, pp. 2799-2803 © The Institution of Engineering and Technology 2018

oscillation modes and may be omitted when using some algorithms that compute partial eigenvalues. It is found that in frequency oscillations, the speeds or frequencies of all generators vary with the same phase and amplitude, which is a major difference from rotor-angle oscillations. Based on this feature, a reduced-order model that represents all the generators with an equivalent generator is proposed. The model only preserves the dynamics of the governors and turbines. The system order is greatly reduced. The intermachine oscillations are neglected and only the frequency oscillation mode is preserved. The efficiency of the frequency oscillation mode calculation can be greatly improved. The paper is organised as follows: Section 2 presents the original reduced-order model and its disadvantages in computing the eigenvalue of the frequency oscillation mode. An improved model is proposed in Section 3, and Section 4 presents the test results. Section 5 is the conclusion.

2 Original reduced-order model In frequency oscillations, the speeds or frequencies of all generators vary with the same phase and amplitude, and the frequencies in the synchronous grid oscillate together. The mode shapes in [2], the actual phasor measurement unit (PMU) records, and the simulation results in [3] all illustrate this phenomenon. Fig. 1 shows the frequencies recorded with PMUsat different locations including substations, hydropower plants, and thermal power plants in the Yunnan oscillation incident in China. The frequency curves at different locations are nearly the same. Fig. 2 is the mode shape of the frequency oscillation mode in the New England 10-machine 39-bus system (case 9 in Section 4.1, the corresponding eigenvalue is 0.0056 ± 0.1408i). We observed that the elements corresponding to the generator speeds in the right eigenvector are nearly the same. The amplitudes and phases of the generator speeds are the same. This is a major difference from rotor-angle oscillations. The linearised swing equation for generator i is T Ji

dΔωi = ΔPmi − ΔPei dt

(1) 2799

preservation of these dynamics guarantees the accuracy of results but increases the order of the system. If the transmission losses are neglected, i.e. ∑i ΔPei = ∑ j ΔPL j, and the loads PL j depend only on Δ f , i.e. PL j = PL0 j(1 + KLΔ f ) or ΔPL j = PL0 jKLΔ f , then

∑ ΔPei = ∑ ΔPL j = ∑ PL jKLΔ f i

j

j

0

= KL ∑ PL0 j Δω j

(3)

= KLsΔω Fig. 1 Frequencies at different locations in a frequency oscillation incident

Fig. 2 Mode shape of the frequency oscillation mode in the New England 10-machine 39-bus system

Under these assumptions, the model in Fig. 3 can be further simplified. We denote the transfer functions of governors and turbines as Ggov, i(s), Gt, i(s), and Gmi(s) = (ΔPmi(s)/Δω(s)) = Ggov, i(s)Gt, i(s). Further, we denote T Jeq = ∑i T Ji. We can obtain a reduced-order model shown in Fig. 4. The model is actually the average system frequency model in [10]. The system order is greatly reduced and the frequency oscillation mode can be computed with the model. When the load uses the static model and only considers the frequency sensitivity, and the network is lossless, the assumptions in the original reduced-order model are valid and the results will be accurate. However, in practical power systems, the load also depends on voltage. More precisely, dynamic load models should be considered. The network is lossy. The output power of the generator is affected by excitation control including AVR and PSS. Therefore, the relationship ∑i ΔPei = KLsΔω is no longer valid, and the results may have non-negligible errors. In the following section, an approach to mitigate this problem is proposed.

3 Improved reduced-order model The key to reduce the errors in eigenvalue computation is a more accurate relationship between ∑i ΔPei and Δω in Fig. 3. According to the theory of damping and synchronous torque of generators, the output power can be represented as ΔPei = KeDiΔωi + KeSiΔδi = KeDiΔωi + Fig. 3 Model of the system with an equivalent generator

ω0KeSi Δωi s

(4)

where KeDi and KeSi are the damping torque coefficient and the synchronous torque coefficient, respectively. Therefore

∑ ΔPei = ∑ KeDi + i

i

ω0 ∑i KeSi β Δω = α + Δω s s

(5)

This is a more accurate relationship between ∑i ΔPei and Δω. The task is to obtain an estimation of α and β. Assume that the linearisation of the fully modelled system is Fig. 4 Original reduced-order model

where T Ji, Δωi, ΔPmi, and ΔPei are the inertia constant, the speed deviation, the deviation of mechanical power, and the deviation of electromagnetic power, respectively. As the speeds are the same, we represent all Δωi with Δω and sum the equations together: = ∑ ΔPmi − ∑ ΔPei ∑ TJi dΔω dt i

i

i

(2)

All generators in the system are aggregated into a single entity with an equivalent inertia constant under a single frequency assumption. The model of the system is shown in Fig. 3. Inter-machine rotorangle oscillation modes are neglected in the model, and the model is used to study long-term frequency dynamics in [8]. Excitation control including automatic voltage regulator (AVR) and power system stabiliser (PSS), load dynamics, as well as network and equilibrium equations are included and affect ΔPei. The model can also be used to compute the frequency oscillation mode. The 2800

Δx˙ = AΔx ΔPe = BΔx

(6)

where Δx is the vector of state variables, and ΔPe is the vector of the generators’ output power. By denoting Δx = ΔΩ Δy T, where ΔΩ is the vector of the generator speeds, and Δy the vector of the other state variables, we obtain ˙ A11 A12 ΔΩ ΔΩ ΔΩ =A = ˙ Δy A21 A22 Δy Δy

(7)

Using the Laplace transformation, we obtain sΔy = A21ΔΩ + A22Δy

(8)

Δy = (sI − A22)−1 A21ΔΩ

(9)

Further


Fig. 5 Improved reduced-order model

Fig. 6 Diagram of excitation system and PSS

Fig. 7 Diagram of PID governor

Considering that all the elements in ΔΩ are equal to Δω, we obtain ΔPe = BΔx = B1

B2

ΔΩ Δy

= (B1 + B2(sI − A22)−1A21)ΔΩ

1 = (B1 + B2(sI − A22)−1A21) ⋮ Δω 1 K1(s) = ⋮ Δω Kn(s)

4 Test cases (10)

We can first obtain an estimation of the eigenvalue using the original reduced-order model in Fig. 4, λ0 = σ ± jωd, and then s = jωd is substituted into (10) to yield Ki(jωd) = ai + jbi. We can then obtain

∑ ΔPei = ∑ ai + j ∑ bi Δω i

i

i

(11)

Substituting s = jωd into (5) yields

∑ ΔPei = i

α− j

β Δω ωd

and loads, are considered. A more accurate estimation of the eigenvalue can be obtained with the improved model. The method can be iteratively used. However, iterations cannot guarantee the convergence to the actual eigenvalue, because (4) is mainly a physical understanding of the composition of ΔPei, but not a mathematically strict relationship between ΔPei and Δωi. Appropriate iterations can improve the accuracy, but the best number of iterations still need further investigation. Compared to the fully modelled system or the model in [8], the computational burden is reduced in the proposed model. The fully modelled system includes the dynamics of the governors, turbines, generators, exciters, AVRs, PSSs, and loads. The order of A in (6) is high and the computation of eigenvalues is time-consuming. Moreover, the eigenvalue corresponding to the frequency oscillation mode mixes with the rotor-angle oscillation modes. With the proposed method, the eigenvalue is to be computed twice, but the system order is greatly reduced. The original model includes the dynamics of the governors, turbines, and a first-order generator equation. The improved model includes the dynamics of the governors, turbines, and a second-order generator equation. The added computation is the matrix computation in (10). However, compared with the eigenvalue computation of the full matrix A, the computational burden is still reduced. Furthermore, the rotor-angle oscillation modes do not exist in the models and the analysis is focused on the frequency oscillation mode. Furthermore, compared to the original model, the accuracy of the eigenvalue computation is improved. The computational burden of the original model is the least, but its accuracy depends on the assumption that the transmission losses are neglected and the loads depend only on frequency, which hardly stands in practical power systems. The original model is applicable only when the requirement on accuracy is low, and the proposed improvement is necessary when better accuracy is demanded. The influences of network losses, load voltage sensitivity, and dynamics, as well as excitation controls including AVR and PSS, are all considered in the proposed model and a better eigenvalue estimation can be obtained.

4.1 New England 10-machine 39-bus system The method is tested in the New England 10-machine 39-bus system [11]. The network diagram and parameters of the system can be found in [11]. The generators use the fourth-order model with parameters in [11], and contain the excitation control, PSS, turbines, and governors. The excitation system and the PSS are shown in Fig. 6. The parameters are KA = 50, TA = 0.05 s, KSTAB = 9.5, TWA = 1.4 s, T1 = 0.154 s, and T2 = 0.033 s. G1–G5 are the hydro units with the transfer functions [12] described in (14). G6– G10 are the steam units with the transfer functions [12] described in (15). All governors are the proportional–integral–derivative (PID) type shown in Fig. 7 and (16). The parameters of the governors and turbines are shown in Table 1. It is noteworthy that although most actual frequency oscillations occurred in hydrodominant systems, we consider both hydro units and steam units in the test to show the general applicability of the proposed method

By comparing (11) with (12), we obtain the estimations of α and β: α=

∑ ai, i

β = − ωd ∑ bi i


(14)

1 + FHPT RHs 1 + T CHs 1 + T RHs

(15)

KDs2 + KPs + KI Δμ(s) 1 = − Δω(s) BPKI + s 1 + T Gs

(16)

Gst(s) =

(13)

The improved reduced-order model is shown in Fig. 5, where α and β are computed using the process above. Using the improved model, a new eigenvalue estimation can be obtained. In the model, the relationship between ∑i ΔPei and Δω is improved. From (10), we observed that none of the influence factors on ΔPei are omitted. All the influence factors on ΔPei, including exciters, AVRs, PSSs,

1 − TW s 1 + 0.5T W s

Ght(s) =

(12)

Ggov(s) =

Two types of load models are considered: i.

Static load model: The reactive load uses the constant impedance model. The active load uses the static model composed of constant impedance and constant power 2801

Table 1 Governor and turbine parameters of the New England 10-machine 39-bus system Unit Governor BP KP KI KD TG TW

FHP

TCH

TRH

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10

— — — — — 0.35 0.30 0.40 0.35 0.30

— — — — — 0.30 0.30 0.40 0.30 0.25

— — — — — 5.50 8.00 6.50 7.00 6.00

0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

0.90 0.80 1.00 1.10 0.95 0.83 1.05 1.23 0.70 0.99

0.95 0.70 1.00 1.10 0.93 0.83 1.02 1.23 0.90 0.90

Table 2 Results of eigenvalue computation Case Load model KL KZ Model 1

static

0

0.4

2

static

0

0.6

3

static

1

0.4

4

static

1

0.6

5

static

2

0.4

6

static

2

0.6

7

dynamic

— 0.3

8

dynamic

— 0.5

9

dynamic

— 0.7

detailed original improved detailed original improved detailed original improved detailed original improved detailed original improved detailed original improved detailed original improved detailed original improved detailed original improved

0.45 0.40 0.50 0.55 0.47 0.41 0.52 0.44 0.35 0.50

Eigenvalue

Damping ratio, %

Oscillation frequency, Hz

−6.87 −11.66 −5.39 −5.37 −11.66 −3.48 1.51 −1.45 2.25 2.33 −1.45 3.34 9.63 8.53 9.73 9.78 8.53 10.02 −6.74 −11.66 −4.90 −5.43 −11.66 −3.28 −4.40 −11.66 −2.01

0.024 0.028 0.023 0.023 0.028 0.021 0.024 0.028 0.023 0.023 0.028 0.021 0.024 0.028 0.023 0.023 0.028 0.021 0.024 0.028 0.022 0.023 0.028 0.021 0.022 0.028 0.020

U 2 + (1 − KZ ) (1 + KLΔ f ) (17) U0 ii. Dynamic load model: The load is composed of a constant impedance, with the proportion of KZ , and a first-order induction motor [13] considering mechanical transients shown in the below equation: PL = PL0 KZ

ds = Tm − Te dt

(18)

where s is the slip, T JM the inertia constant, T m the mechanical torque, defined as T m = k(λ + (1 − λ)(1 − s) ρ), and T e the electromagnetic torque. The typical parameters in [13] are used. 2802

2.00 1.50 1.50 2.00 2.50 — — — — —

0.0104 ± 0.1503i 0.0203 ± 0.1732i 0.0077 ± 0.1431i 0.0077 ± 0.1424i 0.0203 ± 0.1732i 0.0047 ± 0.1341i −0.0023 ± 0.1505i 0.0025 ± 0.1745i −0.0032 ± 0.1432i −0.0033 ± 0.1424i 0.0025 ± 0.1745i −0.0045 ± 0.1342i −0.0145 ± 0.1495i −0.0149 ± 0.1737i −0.0139 ± 0.1423i −0.0139 ± 0.1415i −0.0149 ± 0.1737i −0.0134 ± 0.1333i 0.0100 ± 0.1487i 0.0203 ± 0.1732i 0.0069 ± 0.1399i 0.0077 ± 0.1420i 0.0203 ± 0.1732i 0.0044 ± 0.1328i 0.0060 ± 0.1367i 0.0203 ± 0.1732i 0.0026 ± 0.1276i

considering frequency sensitivity, as shown in (17), where KZ , 1 − KZ are the proportions of the constant impedance load and constant power load, respectively, and KL is the load frequency sensitivity

T JM

0.20 0.15 0.18 0.23 0.17 0.12 0.20 0.20 0.13 0.24

Turbine

By setting the network resistances as 0, KZ = 0.0, and KL = 1.0, the actual eigenvalue of the frequency mode computed with the fully modelled system is −0.0003 ± 0.1782i. The eigenvalue computed with the original reduced-order model is −0.0003 ± 0.1784i. The results are very similar. In this case, the relationship ∑i ΔPei = KLsΔω and the original reduced-order model are valid. However, when KZ = 0.4, the actual eigenvalue becomes −0.0033 ± 0.1546i, whereas the result of the original model is still −0.0003 ± 0.1784i. The errors are not negligible and improvement is required. In the unmodified system with non-zero network resistances, the results of the proposed method are shown in Table 2, where ‘detailed’, ‘original’, and ‘improved’ represent the eigenvalues computed with the fully modelled system, the original reducedorder model in Fig. 4, and the improved reduced-order model in Fig. 5, respectively. For the original model, the average absolute values of the errors of damping ratios and oscillation frequencies are 4.28% and 0.041 Hz, respectively, while the values for the improved model are 1.31% and 0.012 Hz, respectively. The results of the improved model have small errors and are satisfactory for practical applications. IET Gener. Transm. Distrib., 2018, Vol. 12 Iss. 11, pp. 2799-2803 © The Institution of Engineering and Technology 2018

and the oscillation frequency is 0.037 Hz. The absolute values of the errors are 1.19% and 0.001 Hz, respectively. Satisfactory result is obtained with 0.059 s running time. The computational burden is greatly reduced.

5 Conclusion

Fig. 8 Mode shape of the frequency oscillation mode in the IEEE 50machine 145-bus system

In this paper, a reduced-order model for computing the frequency oscillation mode is proposed. The model only preserves the governors, turbines, and an equivalent generator. The original reduced-order model uses a simplified relationship between the total generator output power and frequency to obtain an initial estimation of the eigenvalue. Subsequently, the initial estimation is used to obtain a more accurate relationship between the total output power and frequency, and an improved reduced-order model is obtained. In the model, inter-machine rotor-angle oscillation modes are eliminated and only the frequency oscillation mode is preserved. A satisfactory eigenvalue estimation of the frequency oscillation mode can be obtained and the computational burden is greatly reduced compared to the direct analysis in the fully modelled system. The test results show the validity of the proposed method.

6 Acknowledgments The work was supported by the National Key R&D Program of China (2017YFB0902000), the National Natural Science Foundation of China (51377002), and the State Grid Corporation of China. Fig. 9 Speeds of all generators after a disturbance

7 References

Compared with the eigenvalue analysis in the fully modelled system, the computational burden of the proposed method is greatly reduced. For example, when using the dynamic load model, the order of the fully modelled system is 124, while the order of the original and the improved reduced-order models are 36 and 37, respectively. The system order and the burden of the eigenvalue computation are greatly reduced. Even with the matrix computation in (10), the total running time of the proposed method in MATLAB R2015b on a PC with 1.8 GHz CPU and 8 GB RAM is ∼7.7 ms, in which the running time with the original model takes 2.9 ms, while the running time with the fully modelled system is ∼34.5 ms. The computation time is greatly reduced.

[1]

4.2 IEEE 50-machine 145-bus system The method is also tested in the IEEE 50-machine 145-bus system [14]. The detailed models described in the previous section are used. The loads use the dynamic model consisting of 25 hydro units and 25 steam units. The order of the fully modelled system is 576. The eigenvalue corresponding to the frequency oscillation mode is −0.0078 ± 0.2238i, and the running time for the eigenvalue computation is 0.366 s. The damping ratio is 3.48% and the oscillation frequency is 0.036 Hz. The mode shape is shown in Fig. 8, in which we observed the coherence of the generator speeds. The speeds of all generators after a disturbance are shown in Fig. 9. The results are consistent with the eigenvalue. The orders of the original and the improved reduced-order models are 176 and 177, respectively. The eigenvalue with the original model is 0.0372 ± 0.3547i. The damping ratio is −10.43% and the oscillation frequency is 0.057 Hz. The absolute values of the errors reach 13.91% and 0.021 Hz, respectively, and are not negligible, even the running time is only 0.018 s. With the improved model, we obtain an eigenvalue of −0.0108 ± 0.2310i. The damping ratio is 4.67%


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