, and federico milano, fellow, ieee ieee proof
TRANSCRIPT
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IEEE TRANSACTIONS ON POWER SYSTEMS 1
Robust Frequency Divider for Power System OnlineMonitoring and Control
1
2
Junbo Zhao, Student Member, IEEE, Lamine Mili, Life Fellow, IEEE, and Federico Milano, Fellow, IEEE3
Abstract—Accurate local bus frequency is essential for power4system frequency regulation provided by distributed energy5sources, flexible loads, and among others. This paper proposes a ro-6bust frequency divider (RFD) for online bus frequency estimation.7Our RFD is independent of the load models, and the knowledge8of swing equation parameters, transmission line parameters, and9local Phasor Measurement Unit (PMU) measurements at each gen-10erator terminal bus is sufficient. In addition, it is able to handle11several types of data quality issues, such as measurement noise,12gross measurement errors, cyberattacks, and measurement losses.13Furthermore, the proposed RFD contains the decentralized esti-14mation of local generator rotor speeds and the centralized bus15frequency estimation, which resembles the structure of the decen-16tralized/hierarchical control scheme. This enables RFD for very17large-scale system applications. Specifically, we decouple each gen-18erator from the rest of the system by treating metered real power19injection as inputs and the frequency measurements provided by20PMU as outputs; then a robust unscented Kalman filter based dy-21namic state estimator is proposed for local generator rotor speed22estimation; finally, these rotor speeds are transmitted to control23center for bus frequency estimation. Numerical results carried out24on the IEEE 39-bus and 145-bus systems demonstrate the effec-25tiveness and robustness of the proposed method.26
Index Terms—Frequency estimation, frequency control, robust27statistics, decentralized estimation, dynamic state estimation, un-28scented Kalman filter, power system dynamics and stability.29
I. INTRODUCTION30
W ITH the increasing penetration of renewable energy-31
based generations, the total inertia of the synchronous32
power system is reduced significantly. As a result, the tradi-33
tional capacity-based requirements for primary reserve defini-34
tions may not be able to satisfy the frequency RoCoF and nadir35
limits [1], [2]. To tackle this potential frequency instability issue,36
it is expected by the transmission system operators (TSOs) that37
Manuscript received June 1, 2017; revised October 1, 2017; accepted Decem-ber 15, 2017. The work of J. Zhao and L. Mili was supported in part by the U.S.National Science Foundation under Grant ECCS-1711191. The work of F. Mi-lano was supported in part by the European Commission under the RESERVEConsortium under Grant 727481, in part by the EC Marie Skłodowska-CurieCIG under Grant PCIG14-GA-2013-630811, and in part by the Science Founda-tion Ireland under Investigator Programme under Grant SFI/15/IA/3074. Paperno. TPWRS-00834-2017. (Corresponding author: Junbo Zhao.)
J. Zhao and L. Mili are with the Bradley Department of Electrical and Com-puter Engineering, Virginia Polytechnic Institute and State University, FallsChurch, VA 22043 USA (e-mail: [email protected]; [email protected]).
F. Milano is with the School of Electrical and Electronic Engineering, Uni-versity College Dublin, Dublin 4, Ireland (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRS.2017.2785348
the wind farms [3], flexible loads [5], and energy storage devices 38
[6] would provide frequency regulations. However, to enable ef- 39
fective frequency regulations, reliable and accurate knowledge 40
of the local bus frequency is a prerequisite. 41
In the literature, the numerical derivative of the voltage phase 42
angle provided by Phasor Measurement Unit (PMU) through 43
a washout filter is used to define the local bus frequency [7], 44
[8]. However, as shown by Radman et al. [9], it may produce 45
physically implausible spikes in frequency due to the numerical 46
derivatives and consequently may exhibit instabilities if controls 47
are taken based on them. An alternative way to obtain local bus 48
frequency is through the phase-locked loop (PLL) technique 49
[10]. But it may be unreliable in presence of large step-input 50
speed changes, and has problems in presence of harmonics, 51
unbalance, etc. Furthermore, only a few load buses/substations 52
have PMUs or PLLs installed, which may prevent the system 53
from taking full advantages of local frequency controls. Finally, 54
measurements provided by PMUs and PLLs are always subject 55
to noise or even gross errors, communication losses, etc. [4]. For 56
example, it is shown in [5] that the measurement noise affects 57
the performance of the frequency regulator significantly, not to 58
mention the gross errors, measurement losses, etc. To address 59
these issues, a model dependent analytical expression of bus 60
frequency is proposed in [11] assuming comprehensive and ac- 61
curate models of the system. Milano and Ortega [12] improved 62
that approach and proposed a transient stability simulation- 63
based frequency divider. The main idea underlying this method 64
is to solve a steady-state boundary value problem, where 65
the boundary conditions are given by synchronous generator 66
rotor speeds. However, both methods assume accurate power 67
system dynamic models for time-domain simulations, which 68
is difficult to achieve in practice. In addition, the time-domain 69
simulations of large-scale power systems are computational 70
demanding, which may prevent them for the online control 71
applications. 72
This paper proposes a robust frequency divider (RFD) for 73
online bus frequency estimation. Our RFD is not dependent on 74
load models, and the knowledge of swing equation parameters, 75
transmission system line parameters and local PMU measure- 76
ments is sufficient. In addition, it is able to filter out measure- 77
ment noise, suppress gross measurement errors, handle cyber 78
attacks as well as measurement losses. To develop the proposed 79
RFD, it is observed that to obtain an accurate estimation of 80
bus frequency, accurate generator rotor speeds are required. In 81
the meantime, complicated generator and load models should 82
be avoided. To this end, we propose to divide RFD into two 83
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2 IEEE TRANSACTIONS ON POWER SYSTEMS
subproblems, namely the decentralized estimation of local gen-84
erator rotor speed using local measurements and the centralized85
bus frequency estimation. To address the first problem, we de-86
couple each generator from the rest of the power system by87
treating metered real power injection as model inputs and the88
frequency measurements provided by PMU, local meter devices89
or Frequency monitoring Network devices [13] as outputs. As90
a result, only the swing equation is required for rotor speed91
estimation and no detailed generator model is assumed. Since92
the local measurements can be subject to data quality issue93
when implementing an estimator, a robust unscented Kalman94
filter-based dynamic state estimator is proposed. Next, the local95
estimates are transmitted to the control center for bus frequency96
estimations, yielding two benefits: 1) the requirement of com-97
munication bandwidth is decreased notably as only estimated98
rotor speeds are communicated instead of voltage and current99
phasors; 2) the wide-area generator rotor angles are available for100
operator to achieve better situational awareness and carry out101
other applications, such as oscillatory modes monitoring, rotor102
angle stability analysis, etc. Last but not the least, thanks to the103
decentralized and centralized estimation scheme, the proposed104
method is suitable for designing controllers of very large-scale105
power systems.106
The remainder of the paper is organized as follows.107
Section II presents the problem formulation. Section III de-108
scribes the proposed RFD in detail and Section IV shows and109
analyzes the simulation results. Finally, Section V concludes the110
paper.111
II. PROBLEM FORMULATION112
In this section, the analytical relationship between bus fre-113
quency and generator rotor speeds will be presented first; then114
the limitations of this approach will be discussed thoroughly,115
and finally the problem statement will be declared.116
A. Analytical Relationship Between Bus Frequency and117
Generator Rotor Speeds118
When a disturbance occurs, such as transmission line faults,119
load shedding or generator tripping, power mismatch appears120
between the mechanical torque and electrical power at the gen-121
erator terminal buses. As a consequence, the generator rotor122
speeds will deviate from their nominal values. To re-synchronize123
generator with the rest of the power system, an increase or a de-124
crease in the rotor speed is actuated, which causes rotor angle125
oscillations as well. Due to such oscillations, the voltage phase126
angles of the buses that are adjacent to generators will encounter127
changes, which in turn causes a power mismatch. In this way, the128
electro-mechanical oscillations will be propagated throughout129
the entire power system with limited speed. Since we are in-130
terested in electro-mechanical oscillations and the propagation131
speed of such oscillations is much lower than that of the wave,132
the transient effects of wave propagation are therefore neglected.133
Based on the analysis above, it is clear that the spatial variations134
of the system frequency are characterized by synchronous gen-135
erator rotor speeds. Those frequency variations at each bus of the136
system are of vital importance for designing local controllers to137
enhance the frequency regulation capability of a power system 138
with high penetration renewable energy integrations. 139
Note that electro-mechanical oscillations can be characterized 140
by the magnitude and phase angle modulations of voltages and 141
currents as well since they are corresponding to the movement 142
of rotors of electric machines around the synchronous speed 143
[14], [15]. Thus, to estimate bus frequency of a transmission 144
system, we first need to analyze the relationship of the voltage 145
or current phasors between generators and system buses. This 146
relationship can be expressed by the balanced current injection 147
formula shown as follows: 148[IGIB
]=
[Y GG Y GB
Y BG Y BB + Y B 0
] [V G
V B
], (1)
where IG are generator current injections; V G are generator 149
internal electromotive forces (emfs); IB and V B are current 150
and voltage injections of the network buses, respectively; Y BB 151
is the power network admittance matrix; Y GG , Y GB and Y BG 152
are admittance matrices calculated by including the internal 153
impedances of the synchronous generators; Y B 0 is a diagonal 154
matrix, which takes into account the internal impedances of 155
synchronous generators at the generator buses. Since the load 156
current injections are negligible compared with that of the syn- 157
chronous generators [12], [18], (1) can be rewritten as: 158[IG0
]=
[Y GG Y GB
Y BG Y BB + Y B 0
] [V G
V B
]. (2)
By taking simple algebraic operations on the second row of 159
(2), we can derive the relationship between bus voltage vector 160
V B and the emfs of generators as follows: 161
V B = −(Y BB + Y B 0)−1Y BGV G = DV G. (3)
Taking time derivatives on both sides of (3) in rotating refer- 162
ence frame, i.e., dq frame, we get 163
dV B
dt+ jω0V B = D · dV G
dt+ jω0DV G, (4)
where ω0 is the nominal rotor speed. For more details of de- 164
riving (4), please see [12]. Define ΔωB = ωB − ω0 , ΔωG = 165
ωG − ω0 , the analytical relationship between bus frequency 166
and generator rotor speeds can be derived from (4), which is 167
expressed as follows [12]: 168
ωB = ω0 + D (ωG − ω0) . (5)
It can be observed from (5) that the bus frequency is correlated 169
with each synchronous generator in the system, but the degree of 170
participation of each generator on bus frequency is determined 171
by the transmission parameters. To speed up the calculation of 172
bus frequency from (5) without a relevant loss of accuracy, the 173
conductances of transmission lines utilized to calculate D can 174
be neglected [12]. 175
B. Limitations and Problem Statement 176
When implementing (5) to estimate bus frequency, there are 177
two possible ways: i) the dynamic simulation based-approach 178
and ii) the measurement-based approach. In the former ap- 179
proach, the transient stability program is used to obtain the 180
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ZHAO et al.: ROBUST FREQUENCY DIVIDER FOR POWER SYSTEM ONLINE MONITORING AND CONTROL 3
rotor speed of each generator, followed by the calculations of181
the bus frequencies through (5). However, to obtain good time-182
domain simulation results, accurate and detailed generator and183
load models are required, which may be difficult to achieve184
in practice. In addition, the time-domain simulations of large-185
scale power systems are computational demanding, which may186
not be suitable for the online control applications. By contrast,187
the measurement-based approach does not have such issues, but188
the metered generator rotor speeds and frequencies by PMUs189
or Frequency monitoring Network devices are assumed to be of190
high quality. However, this assumption may not hold true for191
practical power systems as the PMU measurements are usually192
subject to noise or even impulsive noise, gross errors, cyber at-193
tacks, communication losses, etc. Under those conditions, this194
approach will produce significantly biased results and subse-195
quently the control actions based on them may make the system196
even worse. Furthermore, it should be noted that the frequency197
of each bus is correlated with most generators, thus if just one198
rotor speed measurement is corrupted, its error may propagate199
to many other bus frequency estimations. It is thus indispens-200
able to make sure that every generator rotor speed is of good201
accuracy.202
Problem statement: given a limited number of PMUs installed203
at the terminal bus of each generator, a robust frequency divider204
is developed to address the data quality issues of PMU mea-205
surements; in the meantime, it should be model independent so206
as to mitigate the strong assumptions on the generator and load207
models, and finally, it should be fast to calculate and suitable208
for large-scale system online control applications. Note that in209
this paper, the rotor speed and frequency of each generator at210
its terminal bus are assumed to be monitored by PMUs. This211
is a reasonable assumption due to several reasons: 1) online212
monitoring of generators plays a major role in power system213
operation and control, thus it is given a high priority for PMU214
placement according to the NERC PMU placement standard215
[16]; 2) it is required by NERC standard [17] to have PMUs216
installed at the point of interconnection for power plant model217
validation and verification.218
III. PROPOSED ROBUST FREQUENCY DIVIDER219
By looking at (5), one may easily come up with the idea that220
this equation can be formulated as a regression problem, where221
the generator rotor speeds are measurements provided by PMUs222
while the frequency of each bus is the unknown state vector to223
be estimated. However, since the number of rows of the matrix224
D is larger than that of the columns, (5) cannot be treated as an225
estimation problem. Therefore, alternative approaches should226
be developed.227
Note that to obtain an accurate estimation of bus frequency,228
accurate generator rotor speeds are required. Thus, if measure-229
ment quality issues can be addressed locally at the generator230
bus through a local robust estimator, we are able to obtain231
accurate bus frequency estimates. To this end, we propose a232
decentralized-centralized bus frequency estimation framework233
shown in Fig. 1. Particularly, we first perform the robust un-234
scented Kalman filter-based dynamic state estimator at the local235
Fig. 1. Proposed decentralized-centralized bus frequency estimation frame-work.
generator substation or phasor data concentrator (PDC) level by 236
using the proposed model decoupling approach; then these local 237
estimates (generator rotor speeds and angles) are transmitted to 238
control center for bus frequency estimation through (5). Note 239
that if the decentralized dynamic state estimation at each gener- 240
ator substation is costly, the data of those generators associated 241
with PMU measurements will be transmitted to the PDC or 242
regional system operating center for decentralized estimation. 243
After that, local controls can be initiated if required, otherwise, 244
they will be further communicated to the control center for bus 245
frequency estimation and coordinated control. In the following 246
subsections, we will elaborate on each of the block. 247
A. Generator Model Decoupling Approach 248
Due to power unbalance and other control actions, the rotor 249
of the generator will accelerate or deaccelerate. Those electro- 250
mechanical dynamics can be captured by the swing equations 251
shown as follows [18]: 252
dδ
dt= ω − ω0 , (6)
2Hω0
dω
dt= TM − Pe −Dp(ω − ω0), (7)
where δ is the rotor angle; H is the generator inertia constant; 253
TM and Pe are the generator mechanical power and electrical 254
power outputs, respectively; it is assumed that TM remains the 255
same value as that in steady-state condition during transient 256
process, which is reasonable as the time constant of the governor 257
is very large;Dp is the generator damping constant. Note that the 258
generator first swing dynamics are affected by Pe significantly. 259
In other words, the generator swing equation is coupled with 260
its dq windings and the rest of the system through Pe . For 261
example, in the two axis generator model,Pe = E ′dId + E ′
q Iq + 262
(X ′q −X ′
d)IdIq , where E′d and E ′
q are the d-axis and q-axis 263
transient voltages, respectively;X ′d andX ′
q are generator d-axis 264
and q-axis transient reactance, respectively; Id and Iq are the d 265
and q axis currents, respectively. For more detailed model, the 266
expression is different. 267
Motivated by the aforementioned analysis, if Pe is measured 268
and taken as the model input while the measured frequency by 269
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4 IEEE TRANSACTIONS ON POWER SYSTEMS
PMUs is treated as outputs, the swing equation can be decoupled270
from the rest of the system. Furthermore, no information of271
the d and q damper windings is required. By doing that, no272
complex generator model is assumed. The physical mean of this273
decoupling approach can be explained as follows: when there274
is a disturbance at one point of the power system, synchronous275
generators will response to it through actions on the rotors; these276
responses reveal themselves in their real power injections and277
frequency. In other words, the generator swing dynamics are278
coupled with the rest of the system at the point of connection,279
and its interactions with the rest of the system are through the280
real power injections and frequency. If real power injections281
and frequency are measured by a PMU, its swing responses to282
the disturbance are captured completely and no other system283
information is required.284
B. Proposed Robust Decentralized Dynamic State Estimator285
The model decoupling approach enables a generator swing286
equation to be decoupled from the rest of the system model,287
which in turn allows us to rely only on local measurements to288
estimate the rotor speed and angle of a generator. The discrete-289
time state representation of the ith synchronous generator is290
xik = f i
(xik−1 ,u
ik
)+ wi
k , (8)
zik = hi
(xik ,u
ik
)+ vik , (9)
where xik is the state vector, including the generator rotor291
speed ωi and rotor angle δi ; zik is the measurement vec-292
tor that contains rotor speed zk1 provided by PMUs and fre-293
quency zk2 by local metering devices or frequency monitor-294
ing network devices [13]; f i(·) represents the discrete-time295
form of (6) and (7) while zik = [zk1 zk2 ]T and zk1 = ωi + vk1 ,296
zk2 = (1 + Δωi)f0 + vk2 ; f0 is the nominal system frequency;297
hi(·) = H ikx
ik and H i
k is a constant matrix that can be derived298
directly from the measurement equations of zk1 and zk2 ; wik299
and vik = [vk1 vk2 ]T are the process and observation noise, re-300
spectively; they are assumed to be Gaussian with zero mean and301
covariance matrices Qik and Ri
k , respectively; uik is the input302
that contains the real power injection of the ith generator.303
Based on the derived discrete-time state space equations (8)304
and (9), the dynamic state estimator (DSE) can be used to esti-305
mate the generator rotor speed and angle using local measure-306
ments. In this paper, the UKF is chosen as the basic DSE as it307
achieves a more balanced performance between computational308
efficiency and ability to cope with strong system nonlinearities309
than the extended Kalman filter, or the particle filter [19]. How-310
ever, UKF has been proved to be sensitive to gross errors, cyber311
attacks and loss of measurements, etc. [20]. To handle these312
issues, a robust Generalized Maximum-likelihood-type UKF313
(GM-UKF) is proposed. It consists of four major steps, namely314
a batch-mode regression form step, a robust pre-whitening step,315
a robust regression state estimation step, and a robust error co-316
variance matrix updating step. In the following subsections, we317
will discuss them in detail. Note that the index i is neglected for318
simplicity but without the loss of generality.319
1) Derive Batch-Mode Regression Model: Given a state320
estimate at time step k − 1, xk−1|k−1 ∈ Rn×1 , having a321
covariance matrix given by P xxk−1|k−1 , its statistics are captured 322
by 2n weighted sigma points defined as [19] 323
χik −1 |k −1
= xk−1|k−1 ±(√
nP xxk−1|k−1
)i, (10)
with weights wi = 12n , i = 1, ..., 2n, where n is the number 324
state variables for each generator. Then, each sigma point is 325
propagated through the nonlinear system process model (8), 326
yielding a set of transformed samples expressed as 327
χik |k −1
= f(χi
k −1 |k −1
). (11)
Next, the predicted sample mean and sample covariance ma- 328
trix of the state vector are calculated by 329
xk |k−1 =2n∑i=1
wiχik |k −1
, (12)
P xxk |k−1 =
2n∑i=1
wi(χik |k −1
− xk |k−1)(χik |k −1
− xk |k−1)T + Qk .
(13)
We define xk |k−1 = xk − Δk , where xk is the true state 330
vector; Δk is the prediction error and E[ΔkΔT
k
]= P xx
k |k−1 . 331
By processing the predictions and observations simultaneously, 332
we have the following batch-mode regression form: 333[zk
xk |k−1
]=
[Hk
I
]xk +
[vk
−Δk
](14)
which can be rewritten in a compact form 334
zk = Hkxk + ek , (15)
and the error covariance matrix is 335
W k = E[ek e
Tk
]=
[Rk 00 P xx
k |k−1
]= SkS
Tk , (16)
where I is an identity matrix; Sk is calculated by the Cholesky 336
decomposition technique. 337
2) Perform Robust Pre-Whitening: Before carrying out a ro- 338
bust regression, the state prediction errors of the batch-mode 339
regression form need to be uncorrelated. This can be done by 340
pre-multiplying S−1k on both sides of (15), yielding 341
S−1k zk = S−1
k Hkxk + S−1k ek , (17)
which can be further organized to the compact form 342
yk = Akxk + ξk , (18)
where E[ξkξk T ] = I . However, if outliers occur, the use of S−1k 343
for prewhitening will cause negative smearing effect. To handle 344
this issue, we first detect and downweight the outliers by means 345
of weights calculated using the projection statistics (PS) [21] 346
and a statical test applied to them. Those weights contribute 347
to the robust prewhitening and their functionals will be shown 348
later in the objective function. Specifically, we apply the PS to a 349
2-dimensional matrix Z that contains serially correlated sam- 350
ples of the innovations and of the predicted state variables. 351
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ZHAO et al.: ROBUST FREQUENCY DIVIDER FOR POWER SYSTEM ONLINE MONITORING AND CONTROL 5
Formally, we have352
Z =[
zk−1 − Hk xk−1|k−2 zk − Hk xk |k−1xk−1|k−2 xk |k−1
], (19)
where zk−1 − Hk xk−1|k−2 and zk − Hk xk |k−1 are the inno-353
vation vectors while xk−1|k−2 and xk |k−1 are the predicted state354
vectors at time instants k − 1 and k, respectively. The PS values355
of the predictions and of the innovations are separately calcu-356
lated because the values taken by the former and the latter are357
centered around different points. The implementation of PS can358
be found in [21].359
Once the PS values are calculated, they are compared to a360
threshold to identify outliers. According to our previous work361
[20], [21], Z can be shown to follow a bivariate Gaussian proba-362
bility distribution and the calculated PS values using Z follow a363
chi-square distribution with degree of freedom 2. As a result, the364
outliers can be flagged if their PS values satisfy PSi > χ22,0.975365
at a significance level 97.5% in the statistical test, and are further366
downweighted via [22]367
�i = min(1, d2/PS2
i
), (20)
where the parameter d is set as 1.5 to yield good statistical368
efficiency at Gaussian distribution.369
Remark: Except for the occurrence of outliers in rotor speed370
measurements, outliers may occur in the measured real power in-371
jections as well, and consequently, yielding incorrect predicted372
states, called innovation outliers [23]. In such condition, the pre-373
dicted state corresponding to incorrect real power injection will374
be flagged as outliers. Since the local measurement redundancy375
is not high, we will not downweight it directly, instead we pro-376
pose to replace the current real power injection by its previous377
value and obtain the new state predictions. By doing that, we378
can achieve a better statistical efficiency.379
3) Carry Out Robust Regression: To address the data quality380
issues, we develop a robust GM-estimator that minimizes the381
following objective function:382
J (xk ) =l∑
i=1
�2i ρ (rSi ) , (21)
where l = m+ n and m is the number of measurements; �i is383
calculated by (20); rSi = ri/s�i is the standardized residual;384
ri = yi − aTi x is the residual, where aTi is the ith row vector of385
the matrix Ak ; s = 1.4826 · bm · mediani |ri | is the robust scale386
estimate; bm is a correction factor; ρ(·) is the convex Huber-ρ387
function, that is388
ρ (rSi ) ={ 1
2 r2S i, for |rSi | < λ
λ |rSi | − λ2/2, elsewhere, (22)
where the parameter λ is typically chosen to be between 1.5 and389
3 to achieve high statistical efficiency in the literature [24].390
To minimize (21), the following necessary condition must be391
satisfied392
∂J (xk )∂xk
=l∑
i=1
−�iais
ψ (rSi ) = 0, (23)
where ψ (rSi ) = ∂ρ (rSi )/∂rSi is the so-called ψ-function. By 393
dividing and multiplying the standardized residual rSi to both 394
sides of (23) and putting it in a matrix form, we get 395
ATk Λ (yk − Akxk ) = 0, (24)
where Λ = diag(q (rSi )) and q (rSi ) = ψ (rSi )/rSi . By using 396
the IRLS algorithm [21], the state estimates at the j iteration 397
can be updated using 398
Δx(j+1)k |k =
(ATk Λ(j )Ak
)−1ATk Λ(j )yk , (25)
where Δx(j+1)k |k = x
(j+1)k |k. − x
(j )k |k. . The algorithm converges 399
when ‖Δx(j+1)k |k. ‖∞ ≤ 10−2 . 400
4) Update Error Covariance Matrix: After the convergence 401
of the algorithm, the estimation error covariance matrix P xxk |k 402
of the GM-UKF needs to be updated so that the state prediction 403
at the next time sample can be performed. Following the work 404
from [20], we derive the estimation error covariance matrix of 405
our GM-UKF as 406
P xxk |k =
EF [ψ 2 (rS i )]{EF [ψ ′(rS i )]}2
(ATk Ak
)−1(ATk Q�Ak
)(ATk Ak
)−1
(26)where Q� = diag
(�2i
). 407
Remark: the proposed robust DSE is supposed to be per- 408
formed for each generator substation. It can be implemented 409
at the control center as well if all the generator data and PMU 410
measurements are transmitted from local substations to it. How- 411
ever, there exist several concerns by doing so, such as increased 412
communication burden that is discussed in the next subsection 413
and the delayed local control, etc. Indeed, if all the calculations 414
are performed at the control center while some local controls 415
are required at this period, the estimated bus frequency for those 416
local controls can be delayed; by contrast, our robust DSE is first 417
conducted locally using local PMU measurements and its esti- 418
mated rotor speeds and angles can be used for local controls; in 419
the meantime, they can be transmitted to control center for bus 420
frequency estimation and coordinated control. In practice, if it 421
is costly to implement the decentralized DSE for each generator 422
substation, we can do it at the local phasor data concentrator 423
(PDC) level or regional system level. This can still save a lot 424
of communication burden and enable the timely local control 425
actions compared with the fully centralized strategy. 426
C. Bus Frequency Estimation 427
When the rotor speed and rotor angle of each generator are 428
obtained, they need to be communicated to the control center 429
for bus frequency estimation. Depending on the applications, 430
there are two ways to communicate the estimation results. 431
1) If the control center is only interested in monitoring and 432
regulating the system frequency, the rotor speed of each 433
generator is transmitted and the bus frequency is estimated 434
using (5). Compared with the conventional strategy, that 435
is, all the measured voltage magnitudes and angles, current 436
magnitudes and angles, and frequency by PMUs are com- 437
municated, the communication burden of our proposed 438
approach is only 20% of it; 439
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2) If the control center are interested in both frequency and440
rotor angle stability monitoring and control, rotor speed441
and angle estimates are communicated. In this case, it only442
requires 40% communication burden of the conventional443
strategy.444
Note that the total computing time of the proposed approach445
consists of two parts: the decentralized DSE and the projection446
of the rotor speed to bus frequency through (5). Since each447
robust DSE is performed locally and its estimates are communi-448
cated to the control center through its communication link, the449
proposed DSE is independent of the size of the power system.450
The only concern of the proposed RFD for very large-scale451
power system online applications is that D becomes rather452
dense, which requires a lot of computer memory. However, this453
is not a problem if we use the sparse matrices BBB , BG0 and454
BBG instead of D for the projection of rotor speeds to bus455
frequencies (see [12, eq. (20)]). As a result, the computational456
burden of this step is negligible.457
Remark: The reason that the complicated generator model is458
not required has been discussed in the model decoupling section.459
We discuss here how the proposed RFD is not dependent on load460
model. It is well-known that the power system dynamics are dif-461
ferent if different load models are assumed. However, although462
system dynamic behaviors are different, they are reflected on463
the variations of the rotor speeds of synchronous generators.464
Since the proposed RFD is based on such variations, load mod-465
els have been implicitly taken into account. This conclusion has466
also been verified by extensive simulation results in [12].467
IV. NUMERICAL RESULTS468
In this section, extensive simulations on the IEEE 39-bus469
test system will be carried out to demonstrate the effective-470
ness and robustness of the proposed RFD. Specifically, at t =471
0.5 s, the Generator 4 connected to bus 33 is tripped to simulate472
system disturbance. The transient stability simulations are per-473
formed to generate measurements and true state variables using474
the Matlab-based software PST with some revisions [25]. The475
fourth order Ruger-Kutta approach is adopted with integration476
step t = 1/120 s to solve differential and algebraic equations.477
The measured real power injection of each generator is taken478
as model input, while the measured generator rotor speed and479
frequency by PMU are treated as outputs/measurements. A ran-480
dom Gaussian variable with zero mean and variance equal to481
10−6 is assumed for system process noise. The generator model482
assumed for transient simulation is the detailed two-axis gen-483
erator model, whose parameter values are taken from [26]. The484
root-mean-squared error (RMSE) of all bus frequencies is used485
as the performance index while the estimated frequency at bus486
34 is taken for illustration. Note that, Generator 5 is connected487
to bus 34. The proposed non-robust UKF based method will be488
called DUKF, and the proposed robust UKF based method is489
called RDUKF while the original proposal [12] that works on490
D matrix directly will be called the D-method.491
A. Estimation Results With Noisy Measurements492
All the methods are tested with noisy measurements. Normal493
and large noise are considered; their noise covariance matrices494
Fig. 2. Comparing the estimated frequency at bus 34 by DUKF, RDUKF andD-method with normal measurement noise in the IEEE 39-bus system.
Fig. 3. RMSE of DUKF, RDUKF and D-method with normal measurementnoise in the IEEE 39-bus system.
are assumed to be 10−6I and 10−4I with appropriate dimen- 495
sions, respectively. The results are shown in Figs. 2–4, where in 496
Fig. 2 the rotor speed of Generator 5 is shown. From Fig. 2, we 497
observe that the rotor speed of Generator 5 is different from its 498
terminal bus frequency. This difference is caused by two factors: 499
(i) generator internal impedance and (ii) severity of the transient 500
(e.g., how much rotor speeds differ from each other). On the 501
other hand, by observing both Figs. 2 and 3, it is found that the 502
D-method is one of the most sensitive method to measurement 503
noise while our DUKF and RDUKF approaches are able to fil- 504
ter them out. This is because in D-method, the noise is directly 505
propagated through the rotor speed measurements to the bus 506
frequency. By contrast, our methods adopt the UKF to filter out 507
the noise, yielding better performance. When we increase the 508
measurement noise level, i.e., the covariance matrix is changed 509
from 10−6I to 10−4I , the results of D-method become even 510
worse while our methods can achieve comparable performance 511
as those in the former case (see Fig. 4). 512
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Fig. 4. RMSE of DUKF, RDUKF and D-method with large measurementnoise in the IEEE 39-bus system, where the noise covariance matrix is changedfrom 10−6 I to 10−4 I.
B. Impact of Observation and Innovation Outliers513
Due to cyber attacks, imperfect phasor synchronization, the514
saturation of metering current transformers or by metering Cou-515
ple Capacitor Voltage Transformers (CCVTs), to name a few,516
gross errors can occur in the PMU measurements [20]. As for517
our decentralized DSE-based bus frequency estimation prob-518
lem, there are two ways to induce outliers: i) the measured rotor519
speed by PMUs is contaminated with gross error, which is called520
observation outlier; ii) since the real power injection measured521
by PMUs can be contaminated with gross error, treating it as522
model input can yield incorrect rotor speed predictions, which523
is called innovation outlier. Note that, as D-method is working524
directly with rotor speed measurements, it is affected by obser-525
vation outliers while being independent of the innovation outlier526
caused by incorrect real power injection measurements. To this527
end, two cases are considered:528
Case 1: the measured rotor speed of Generator 5 is contam-529
inated with 20% error from t = 4 s to t = 6 s to simulate530
observation outlier.531
Case 2: the measured real power injection of Generator 5 is532
contaminated with 30% error from t= 3 s to t= 6 s to simulate533
innovation outlier.534
The test results for Case 1 are shown in Figs. 5 and 6. From535
these two figures, we find that the estimation results of the DUKF536
and the D-method are significantly biased in the presence of537
observation outliers. DUKF is less sensitive to the observation538
outlier compared with the D-method. By contrast, our RDUKF is539
able to suppress the observation outliers thanks to the robustness540
provided by PS and the GM-estimator, yielding negligible bias541
of the estimation. It should be noted that due to the smearing542
effect of applying (5) for bus frequency estimation, the estimated543
frequencies at many buses are affected by the incorrect rotor544
speed of the generator 5. This however does not happen in our545
RDUKF.546
The test results for Case 2 are shown in Figs. 7 and 8. As547
expected, the results of DUKF are biased in the presence of548
Fig. 5. Estimated frequency at bus 34 by DUKF, RDUKF and D-method withobservation outliers in the IEEE 39-bus system, where the measured rotor speedof Generator 5 is contaminated with 20% error from t = 4 s to t = 6 s.
Fig. 6. RMSE of DUKF, RDUKF and D-method with observation outliersin the IEEE 39-bus system, where the measured rotor speed of Generator 5 iscontaminated with 20% error from t = 4 s to t = 6 s.
Fig. 7. Estimated frequency at bus 34 by DUKF, RDUKF and D-method withinnovation outliers in the IEEE 39-bus system, where the measured real powerinjection of Generator 5 is contaminated with 30% error from t = 3 s to t = 6 s.
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Fig. 8. RMSE of DUKF, RDUKF and D-method with innovation outliers inthe IEEE 39-bus system, where the measured real power injection of Generator5 is contaminated with 30% error from t = 3 s to t = 6 s.
innovation outliers while the D-method is not affected. Due549
to the robustness of the proposed DSE, this innovation outlier550
has been suppressed. By comparison, RDUKF still outperforms551
D-method, yielding the best results.552
C. Loss of PMU Measurements553
Due to the failures of communication links between PMU554
and phasor data concentrator or cyber attacks, the PMU placed555
at Bus 34, where Generator 5 is connected, is assumed to loss556
its measurements from t = 3 s to t = 6 s. Therefore, the mea-557
surement set becomes unavailable during this time interval and558
their values are set equal to zero for simulation purpose. Please559
note that in this extreme case, both predicted and measured rotor560
speeds will be flagged as outliers. To enable the estimation of561
bus frequency by the proposed method, we advocate to either562
recover the missing data using [27] or perform short-term fore-563
casting of the PMUs using their spatial and temporal correlations564
[28]. The test results are presented in Figs. 9 and 10. It can be565
seen from these two figures that the estimated bus frequencies of566
both the DUKF and the D-method are biased significantly. But567
the DUKF approach is less sensitive to the measurement losses568
than the D-method. As a result, the proposed RDUKF can al-569
ways track the bus frequency reliably and accurately. It should570
be noted that the two mitigation approaches [27], [28] can be571
used in the situation that the measurements are temporally lost572
(a few seconds). For longer period, they may not be valid. Oth-573
erwise, the operator will be warned and the decentralized DSE574
stops before further careful investigations are done.575
D. Results on Large-Scale Systems576
To demonstrate the applicability of the proposed robust fre-577
quency divider for large-scale system, the 50-machine IEEE578
145-bus system is used. The dynamic data can be found through579
[29]. The generator located at bus 106 is tripped at t = 0.5 s to580
simulate the system disturbance. The following three scenarios581
are considered and tested:582
Fig. 9. Estimated frequency at bus 34 by DUKF, RDUKF and D-method withmeasurement losses from t = 3 s to t = 6 s in the IEEE 39-bus system.
Fig. 10. RMSE of DUKF, RDUKF and D-method with measurement lossesfrom t = 3 s to t = 6 s in the IEEE 39-bus system.
1) Scenario 1: only normal Gaussian noise is added to the 583
simulated data like the test done in Section IV-A; 584
2) Scenario 2: 10% of the measured generator rotor speeds 585
is contaminated with 20% error from t = 2 s to t = 5 s; 586
3) Scenario 3: 50% of the measured generator rotor speeds 587
are lost from t = 3 s to t = 6 s. 588
The test results of all three scenarios are displayed in Figs. 11– 589
13, where the RMSE of scenario 1 and the estimated frequencies 590
of bus 73 under scenarios 2 and 3 are represented accordingly. 591
Note that bus 73 is close to the tripped generator. Based on 592
these figures, we conclude that our proposed robust frequency 593
estimator still outperforms the other two alternatives in a larger- 594
scale system. These results are consistent with those for the 595
IEEE 39-bus system. It is interesting to find that even without 596
using missing data recovering approach [27], [28], our proposed 597
method is able to rely on good PMU measurements and the 598
predicted dynamic state variables for filtering, yielding good 599
estimation results. 600
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Fig. 11. RMSE of DUKF, RDUKF and D-method with normal measurementnoise in the IEEE 145-bus system.
Fig. 12. Estimated frequency at bus 73 by DUKF, RDUKF and D-method withoutliers in the IEEE 145-bus system, where 10% of the measured generator rotorspeeds is contaminated with 20% error from t = 2 s to t = 5 s.
Fig. 13. Estimated frequency at bus 73 by DUKF, RDUKF and D-method withmeasurement losses in the IEEE 145-bus system, where 50% of the measuredgenerator rotor speeds are lost from t = 3 s to t = 6 s.
TABLE IAVERAGE COMPUTING TIMES OF THE THREE METHODS AT EACH PMU SAMPLE
Scenarios D-method DUKF RDUKF
Section IV-A 0.02 ms 0.15 ms 0.37 msSection IV-B Case 1 0.022 ms 0.14 ms 0.45 msSection IV-B Case 2 0.021 ms 0.16 ms 0.46 msSection IV-C 0.021 ms 0.15 ms 0.46 msScenario 1 0.08 ms 0.66 ms 1.6 msScenario 2 0.09 ms 0.78 ms 2.1 msScenario 3 0.085 ms 0.79 ms 2.14 ms
E. Computational Efficiency 601
To validate the capability of the proposed method for online 602
estimation, that is, to be compatible with PMU sampling rate, 603
its computational efficiency is analyzed. All cases and scenarios 604
simulated in the previous sections are considered. All the tests 605
are performed on a PC with Intel Core i5, 2.50 GHz, 8 GB 606
of RAM. The average computing time of each method for ev- 607
ery PMU sample is displayed in the Table I. We observe from 608
this table that all methods have comparative computational effi- 609
ciency and their computing times are much lower than the PMU 610
sampling period, which are 16.7 ms and 8.3 ms for 60 samples/s 611
and 120 samples/s, respectively. On the other hand, although 612
RDUKF is the most time consuming method compared with 613
RDUKF and D-method, its computing time is negligible for 614
practical applications considering the PMU sampling speed. 615
Note that the decentralized DSE for each generator can be car- 616
ried out independently and in a parallel manner, which are very 617
fast to calculate and independent of the scale of the power sys- 618
tem. For very large-scale power systems, D becomes rather 619
dense and a numerical stable and computational efficient ap- 620
proach proposed in [12] is used to project the rotor speeds to 621
bus frequencies. The computational burden of this step has been 622
shown to be negligible [12]. In conclusion, the proposed method 623
is suitable for large-scale power system online applications. 624
V. CONCLUSION 625
A robust frequency divider (RFD) is proposed to estimate the 626
frequency of each bus in a power system. Our RFD consists 627
of two steps, namely the estimation of generator rotor speeds 628
through a robust decentralized UKF using local measurements, 629
and the projection of all generator rotor speeds to bus frequency. 630
The proposed RFD is model independent, and the knowledge 631
of local PMU measurements at each generator terminal bus and 632
transmission system line parameters is sufficient. Furthermore, 633
the proposed RFD is able to filter out measurement noise, sup- 634
press gross measurement errors, handle cyber attacks as well as 635
measurement losses. Extensive results carried out on the IEEE 636
39-bus and 145-bus systems demonstrate the effectiveness and 637
the robustness of the proposed method. A possible issue of the 638
proposed RFD is that the decentralized and centralized scheme 639
may produce some delays for the estimated bus frequencies 640
used for controllers. However, thanks to the advancement of 641
control techniques, the time delays can be effectively mitigated 642
[30], [31]. In the future work, we will design this type of robust 643
frequency regulator based on our RFD. 644
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Junbo Zhao (S’13) received the Bachelor’s degree in 733electrical engineering from Southwest Jiaotong Uni- 734versity, Chengdu, China, in 2012. He is currently 735working toward the Ph.D. degree at Bradley De- 736partment of Electrical and Computer Engineering, 737Virginia Polytechnic Institute and State University 738(Virginia Tech), Blacksburg, VA, USA. From May 739to August 2017, he did a summer internship at Pa- 740cific Northwest National Laboratory. He has written 2 741book chapters, published more than 30 peer-reviewed 742journal and conference papers, and 9 Chinese patents. 743
His research interests include power system real-time monitoring, operations 744and security that include power system state estimation, dynamics and stability, 745static and dynamic load modeling, power system cyberattacks and countermea- 746sures, big data analytics and robust statistics with applications in the smart grid. 747
Mr. Zhao is currently the Chair of the IEEE Task Force on Power System 748Dynamic State and Parameter Estimation, the Secretary of the IEEE Working 749Group on State Estimation Algorithms, and the IEEE Task Force on Synchropha- 750sor Applications in Power System Operation and Control. 751
752
Lamine Mili (LF’17) received the Electrical Engi- 753neering Diploma from the Swiss Federal Institute of 754Technology, Lausanne, Switzerland, and the Ph.D. 755degree from the University of Liege, Liege, Belgium, 756in 1976 and 1987, respectively. 757
He is currently a Professor of Electrical and Com- 758puter Engineering, Virginia Tech, Blacksburg, VA, 759USA. He has five years of industrial experience with 760the Tunisian electric utility, STEG. At STEG, he 761worked in the Planning Department from 1976 to 7621979 and then at the Test and Meter Laboratory from 763
1979 until 1981. He was a Visiting Professor with the Swiss Federal Insti- 764tute of Technology, the Grenoble Institute of Technology, the Ecole Superieure 765D’electricite in France, and the Ecole Polytechnique de Tunisie in Tunisia, and 766did consulting work for the French Power Transmission company, RTE. His 767research has focused on power system planning for enhanced resiliency and 768sustainability, risk management of complex systems to catastrophic failures, 769robust estimation and control, nonlinear dynamics, and bifurcation theory. He 770is the co-founder and co-editor of the International Journal of Critical Infras- 771tructure. He is the chairman of the IEEE Working Group on State Estimation 772Algorithms. He is the recipient of several awards including the U.S. National 773Science Foundation (NSF) Research Initiation Award and the NSF Young In- 774vestigation Award. 775
776
Federico Milano (S’02–M’04–SM’09–F’16) re- 777ceived the M.E. and Ph.D. degrees in electrical engi- 778neering from the University of Genova, Italy, in 1999 779and 2003, respectively. 780
From 2001 to 2002, he was a Visiting Scholar with 781the University of Waterloo, Waterloo, ON, Canada. 782From 2003 to 2013, he was with the University of 783Castilla-La Mancha, Ciudad Real, Spain. In 2013, he 784joined the University College Dublin, Ireland, where 785he is currently a Professor of Power System Control 786and Protections. His research interests include power 787
system modeling, stability analysis, and control. He is an IET Fellow. He is or 788has been an Editor of the IEEE TRANSACTIONS ON POWER SYSTEMS and IET 789Generation, Transmission & Distribution. 790
791