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A Generic Architecture for Robust Automatic Detection and
Suppression of Sub-Harmonics
Jens Hulsmann, Andreas Buschermohle, Christian Lintze, Werner Brockmann
Smart Embedded Systems Group
University of Osnabruck
Albrechtstraße 28
49074 Osnabruck
{jehuelsm,andbusch,clintze,wbrockma}@uos.de
Abstract: Sub-harmonic phenomena become a critical issue as properties of the powergrid change. In this paper we propose a generic architecture to detect and suppress thesesub-harmonics automatically in a robust way. Therefore different algorithms to analysethe signals from the power grid, like wavelet- and prony analysis, are used and extendedto rate and incorporate signal quality. The results of these algorithms are dynamicallyfused using according to their trustworthiness in order to achieve a detection as robustas possible. Afterwards the intervention needed to suppress a detected sub-harmonic isdetermined and applied by remote load management in our case. First experimentalresults show the validity of this trustworthiness-based architecture.
1 Motivation
Today’s power grid is subject to impressive changes. The characteristics of generation,
distribution and consumption of electrical power have changed a lot and will change a lot
more in the future. The installation of many renewable energy sources in favour of singular
big power plants and the resulting distributed power generation is the major issue. Along
with it come increased, uncontrollable dynamics in the amount of power that is generated
and a multi directional energy flow. The uncontrollability is caused by the fact that it
is not possible to influence the primary energy sources of renewable energies, like wind
and light, effectively. Also the power consumption changes: For example the upcoming
e-mobility adds a non-trivial amount of rechargeable batteries to the grid. The amount of
other non-linear loads like switch-mode power supplies and LED-light rises to an extend,
which is no longer negligible. On the other hand, more and more components of the grid can
be controlled directly by remote load management (RLM) and by techniques like demand
side management (DSM). Thus the (inner) dynamics of the power grid get excessively
complicated. All this leads to a high uncertainty about the system under control.
Beyond the energy flow level, severe problems concerning power quality arise: The in-
creased amount of harmonic, inter-harmonic distortions and sub-harmonic stability issues
cause a significant danger to components of the grid, endangers the overall stability, and
1455
bears a risk for blackouts [dAE00]. Severe sub-harmonic distortions are caused by the
interaction of different, individually controlled, grid-components [AAVN99]. Hence the
interrelation of the components, and not the single component itself, is the cause of a sub-
harmonic. Furthermore sub-harmonics can built up, if a component induces an oscillation
below the main frequency (e.g. 50Hz). For example shadowing effects in wind farms are
known to be a source of such characteristics. These problems are known for a long time, but
only received little attention, as they rarely occurred in the past, where only few components
tended to cause sub-harmonic behaviour. But now the situation changes significantly due to
the reasons mentioned above.
The classical way of tackling these undesired phenomena in the power grid is modelling
the affected part of the grid formally to identify their causes and tackle them directly. With
the highly dynamic distributed power generation and the huge number of generators and
other non-linear components this approach gets increasingly complicated and expensive,
if not even intractable. To be able to effectively cope with sub-harmonics in the power
grid, a flexible, generic architecture for the detection and suppression of sub-harmonics
with a robust and model free approach is needed. The respective system needs to work
autonomously under dynamic conditions in presence of a lot of uncertainties that range from
noisy data to unknown and changing grid participants and topology. One key issue for such
model free systems, is to prevent a ”garbage in - garbage out” behaviour: If the algorithms
are supplied with uncertain data, it should not generate a random output or at least mark
this output as uncertain. It also has to be easily deployed (at any place) within the grid
structure. As the power grid is a safety critical infrastructure, a reliable, real time detection
of sub-harmonics and a safe strategy for intervention is needed. In this paper we thus deal
with a generic systems architecture for robust detection and suppression of sub-harmonics
without the need of formal modelling of the grid, that can handle the dynamically changing
uncertainties. The performance of such a system can be measured by the speed and the
robustness of the detection and the amount of intervention and system excitation it causes
in the power grid.
2 State of the Art
2.1 Identification of Sub-Harmonics
Sub-harmonics in the grid result in drifts and flicker on the frequency ν and the effective
voltage VRMS . They typically occur on the time scale of seconds, but periodic excitations
can also cause them on time scales up to some minutes. Besides formal modelling ap-
proaches, different techniques to identify sub-harmonics are known. They tackle different
parts of the problem: In [BdAD07] the exact measurement of VRMS is discussed. A
high accuracy is needed, as the amplitude of the sub-harmonics is generally low (in the
beginning). Due to the different time scales, a high sampling rate and long measurement
periods are needed, which leads to huge amounts of data. Simply analysing these data with
Fourier analysis is difficult, as it misses some information from the signal and requires
well adapted sampling intervals [Leo10]. Instead, wavelet methods yield good results in
1456
identifying sub-harmonics [Tse06]. Additionally the frequency of the sub-harmonic can
simply be determined via autocorrelation.
To describe a sub-harmonic properly, its frequency, its amplitude and its damping is needed.
One approach to identify these parameters is prony analysis [HDS90, LRS03]. Here the
signal is written as a sum of modes M , the modes amplitude Am, the modes damping σm,
its frequency fm and its phase φm(Eq. 1).
VRMS(t) =
M∑
m=1
Am
2eσmt cos (2πfmt+ φm) (1)
It has been shown that one can approximate this sum for a point in time N by reformulation
and discretizing it into a linear combination of past measurements:
VRMS [N ] = a1VRMS [N − 1] + · · ·+ aNVRMS [0] (2)
Now a measured signal can be substituted in Eq. 2 to approximate the signal parameter in
Eq. 1 stepwise. A detailed description of the method is given e.g. in [LRS03].
Each of these methods has some drawbacks under certain conditions. Wavelet analysis
cannot easily identify the frequency of a sub-harmonic, as the wavelet bands do not
correspond to a certain frequency. A pseudo-frequency for each band has to be determined
and is inaccurate by construction. Dynamic changes in the amplitude of the voltage lead
to high energy in other bands of the wavelet analysis and hinders the identification of
the band with the sub-harmonic. Autocorrelation can only identify the frequency of the
sub-harmonic, but can neither calculate the amplitude nor the damping. Additionally, it is
relatively sensitive to noise. Prony analysis seems to be the solution to all these problems,
but due to the needed approximation algorithms hidden inside, it is very slow and has no
determined runtime; it is not even guaranteed that it converges any time at all. This is a huge
problem for systems that are reliant on actual real time behaviour like the system discussed
in this paper. Additionally prony analysis is problematic for non-sinusoidal waveforms,
which do not meet the formulation in Eq. 1. As a sub-harmonic is not always a plain
sinusoidal waveform [AAVN99], prony analysis fails in accurately analysing this part of
sub-harmonics. Furthermore, any dynamic change in the characteristics of the electrical
signal has a negative effect on the reliability of all these algorithms.
2.2 Countermeasures to Fight Sub-Harmonics
The avoidance and suppression of sub-harmonics in the power grid is done on two different
levels: In the control of a synchronous generator a power system stabilizer adjusts the exci-
tation by means of the polar angle. Besides the adjustments in the generators, the reactive
power mainly is compensated passively. The most important parameter that influences the
sub-harmonic characteristics besides the oscillations from the components is the reactive
power [AAVN99, AF08, FEH05]. In some cases special devices that actively compensate
reactive power are installed to stabilize the grid, so called FACTS-devices [SS98, JAG06].
These dedicated countermeasures are expensive and need a lot of maintenance. Additionally,
1457
they perform no specific intervention on the occurrence of a sub-harmonic, as they are
always part of the grid. Nowadays the specific intervention due to the occurrence of a
sub-harmonic by adjusting (reactive) power consumption automatically by means of RLM
or DSM is possible.
It is important to note, that due to the highly non-linear behaviour of sub-harmonic phenom-
ena, a small change in a parameter can cause a strong ”(de)tuning” of the effects that cause
a sub-harmonic. Thus, under certain circumstances, even small actions can already stabilize
or destabilize the grid. In order to allow robust and reliable detection and suppression
of sub-harmonics without the burden of broad modelling of the network at the point of
installation, we describe a systems architecture in the following which is designed for
handling the dynamically varying uncertainties.
3 AMIGO-Approach
3.1 Architectural Overview
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Figure 1: Overview of the AMIGO system architecture
The hierarchical architecture of the Autonomous Multi-Objective Intelligent Grid Opti-
mizer (AMIGO) is organized in the three subgroups of upsteam, decision making and
downstream which are organized in the three architectural layers (Fig. 1), handling different
levels of abstraction. The measurements from the grid are aggregated (upstream) such that
the decision layer can assess the grid state along with the actual uncertainties and take a
1458
decision about an intervention. After a decision is made, it is processed in the downstream
in two stages to interact with the grid.
In the whole process, beginning with the measurement, several outer and inner uncertainties
accumulate: The measurement is subject to noise and other disturbances, the aggregation
algorithms use discretizations and approximations and depend on different underlying
assumptions which render their results useless, if these assumptions are not met. Therefore,
we extended the upstream and the decision making with a framework which explicitly
processes these dynamically changing uncertainties, called Trust Management [BBH10,
BBHR11] (see Sec. 3.2), in order to improve the robustness of the system. Hence the
particular upstream modules process the data as follows: At first, the voltage Vreal is
measured in the grid. The data pre-processing extracts characteristic numbers, like the root
mean square (RMS) voltage VRMS and the frequency ν, from Vreal and determines the
trustworthiness of these results for the actual situation. Afterwards the feature extraction
calculates the amplitude a, the damping d and the frequency f of a potential sub-harmonic
with different algorithms, calculates the trustworthiness of these results again, and fuses the
results accordingly.
At the decision layer, a classification unit detects the type(s) of sub-harmonic from the
features, and the criticality evaluator calculates the potential of the sub-harmonic to result
in a crash of the grid. Both modules evaluate the trustworthiness of their inputs to be able
to rate the trustworthiness of their outputs too. Based on these informations, the decider
module decides if and which (abstract) action, like decreasing reactive load or increasing
real power, is needed to ensure the stability of the grid. Thus it operates in a non-continuous
way and takes an event-based action in order to prevent unnecessary excitation of the system
under control and to prevent closed loop effects like positive feedback with the resulting
action build-up. It is important to note, that the AMIGO system is in no way intended to
perform actions in or above the frequency of the sub-harmonic oscillation. The performed
action is designed to change the state of the local a few times, so that it ”detunes” in the
above mentioned way.
The implementation of this action is carried out by the downstream modules: If a new action
is indicated, the dispatcher module, which knows about the concrete actors to influence the
grid, determines the concrete desired actions for each actor. These actors are devices like
RLM/DSM components, compensation plants, device controller settings or FACTS-devices.
The specific selection of the parameters that are tuned by the AMIGO is subject to a trade
off: Directly adjusting controller parameters of certain grid devices is a powerful measure,
but bears a high risk of worsening the power grid dynamics. In contrast to that, one could
for example slightly adjust the working point of the controllers by changing the load via
RLM/DSM, and thus avoid to bring the grid into unknown dynamics: The new working
point could also be caused randomly by a real load and thus is not a problem for the grid
per se. However, only the dispatcher module needs to be adapted to the current place of
installation of the AMIGO system and the present actors, as they are specific to the grid
and the local situation.
1459
3.2 Trust Management
The information about the trustworthiness, or uncertainty respectively, of a signal in a
technical system can be determined in different ways [BBH10]. For sensors often a sensor
model or additional information from other sensors is used. For example, a sensor reading
close to the limit of the measuring range often is not very trustworthy due to non-linear
sensor effects or the accuracy of a sensor is dependent on the working condition determined
by another sensor. IN either case, the trustworthiness of results can hence not be better in
the further processing steps, then that of the data they rely on, except there was redundancy
in the data source or additional information is provided dynamically by a system model.
In the framework of Trust Management, the trustworthiness is reflected by a Trust Signal
attribute which is a meta-information to a normal signal, e.g. a sensor-reading, or to
an internally generated signal, which depends on uncertain information. Also system
components or functional modules can be attributed with a Trust Signal ϑ. In our case,
potentially all sources of uncertainty are reflected gradually by Trust Signals. A Trust
Signal ϑ has a scalar value from the interval [0, 1], called the Trust Level, and indicates the
trustworthiness of the signal/component it is associated with. Two rules generally apply
here: If it is 1.0, the related data can be fully trusted, hence they can be handled as normal.
If the Trust Level is 0.0, the related data must not influence the output. It is important
to note that the Trust Signal reflects the trustworthiness of information. Thus it is not a
probabilistic representation, because it does not depend on or declare anything about the
statistical properties of the data it is assigned to.
The module, which receives the Trust Signal enhanced data, has to decide in which way
it incorporates the regular and the Trust Level data into the processing of its output. If
the input data are not trustworthy enough, it can switch to a fallback strategy or gradually
fade out the influence of the affected input(s). As the modules are normally part of a data
processing chain, every module should again make a statement about the trustworthiness
of its output, according to its specific data processing and the trustworthiness of its inputs.
This is a generic concept which is applicable throughout the whole system architecture.
Nevertheless, it is only applied to the upstream processing and the decision making part here
because it turned out, that they are the most critical part in the detection and suppression of
sub-harmonics. The implementation of these modules and the incorporation of the Trust
Management approach is described in the following.
3.3 Upstream Processing
In the data preprocessing module, the root mean square voltage VRMS and the main fre-
quency ν are derived from the measured voltage Vreal. As Vreal is subject to noise and
harmonics, extraction of VRMS is potentially not trustworthy. To measure this trustwor-
thiness, the Trust Level ϑVRMS is calculated by comparison with a sliding mean: VRMS is
filtered with this mean, and the squared difference of the original and the filtered signal
is measured. To get the Trust Level in [0, 1] it is scaled with an empirically determined
1460
factor. The main frequency ν of the signal is determined via zero-crossing detection. As
the zero-crossing detection gets worse with the amount of noise on the signal (see Fig. 4
for an example), its Trust Level is derived from the variance of Vreal by scaling to [0, 1].
The feature extraction calculates different, more expressive features based on VRMS . To
identify a sub-harmonic, one needs its frequency f , its amplitude a and, to estimate its
further progress, the damping factor d of the amplitude (d < 1.0: amplitude decreases, d >1.0: amplitude increases). These features are calculated redundantly with autocorrelation,
wavelet- and prony analysis. Autocorrelation only yields a frequency estimate, wavelets
only yield amplitude and damping and prony analysis yields all three. But the quality of all
results dynamically depends on the actual situation (see Sec. 2.1).
The sub-harmonics frequency f can be estimated by autocorrelation of VRMS .
RVRMS(τ) =
∫
∞
−∞
VRMS(t)VRMS(t− τ) dt
The wanted frequency is found by searching for τ1st max = argmaxτ (RVRMS) with f =
1/τ, f ∈ [0, 50Hz] . The Trust Level for this result is linked to the clearness of the
found maxima. To judge it, the second largest maximum is also calculated and the relative
difference of R(τ1st max) and R(τ2nd max) gives ϑfac. Hence the Trust Level is low, if similarly
strong maxima occur.
The wavelet-analysis transforms the signal into a time-frequency-domain. For the AMIGO
we choose a 14 bands Haar-wavelet decomposition with a one second window. The signal is
resampled to contain 8192 measurements in this window. The Trust Levels of the 14 bands
over time are calculated by transforming ϑVRMS along with the wavelet transformation: As
each segment of each wavelet band is based on a sharp time window of the analysed signal,
the average Trust Level from these intervals can be used.
The frequency of the sub-harmonic f now should correspond to the pseudo-frequency1
of the band with the highest energy. As this can easily fail due to noise and amplitude
changes, it is double checked with the frequencies calculated by the two other algorithms.
If the band is not consistent with them, the Trust Level of all information derived from the
wavelet-analysis is 0.0, which means that they are not considered further. If it is consistent,
the energy of this band gives the estimate awl. The Trust Level ϑawl is given by a comparison
between the (scaled) band amplitude and the amplitude of VRMS in the window, which
should be similar. Additionally it is limited by minimum with the mean Trust Level of the
whole signal in the band. This ensures that only trustworthy input data result in trustworthy
outputs.
The wavelets damping factor estimate dwl for the signal is calculated according to the
energy in the identified band B. Therefore the energy at the beginning and at the end of the
considered window of one second is set into relation. As all bands b contain at least small
parts of the sub-harmonics energy, they should have consistent damping factors dbwl, where
b ∈ [1..14] \B is the number of the band. This is because VRMS is a superposition of the
1Every wavelet band contains a spectrum of frequencies from the original signal, which is dependent on
the used wavelet. But for every band, one can calculate the frequency with the highest contribution, called
pseudo-frequency.
1461
Wavelet Analysis Prony Analysis Autocorrelation
Trusted Fusion Trusted Fusion Trusted Fusion
請 awl
請 dwl
請 dpr
請 fpr
請 fac
請 apr
awl
dwl
apr
dpr
fpr
fac
a 請 â d 請 d f 請 f^^^ ^
^
Figure 2: Fusion of signals with Trust Management
sub-harmonic and noise and where a constant amount of noise has a damping factor of 1.0,
thus does not contribute to the damping factors. Consequentially, consistency means that if
dBwl is above 1.0 all other dbwl should be above 1.0 and vice versa. Hence its Trust Level
ϑdwl is calculated by comparison of the damping factor in the other wavelet bands: ϑd
wl is
the relative number of bands with a consistent damping. Afterwards ϑdwl is also limited by
a minimum-t-norm with the mean Trust Level of the whole signal in the band.
The prony analysis yields all three wanted parameters (f , a and d). As mentioned above,
the problem is that it may not converge in a given time. If this happens, all its results get a
Trust Level of 0.0. If it converges, the Trust Levels are calculated based on the fact that
prony analysis is designed on for sinusoidal signals. As the signal is noisy and not all
sub-harmonics have a clean sinusoidal characteristic, the Trust Levels ϑapr, ϑ
dpr and ϑf
pr are
calculated by comparison via back-transformation: After the prony analysis is done, the
estimated parameters are substituted in Eq. 1 and the resulting values V ′
RMS are compared
with the values of VRMS for each measurement N by calculating the mean squared error e(Eq. 3).
e =N∑
i=0
(VRMS(ti)− V ′
RMS(ti))2/N (3)
All three Trust Levels2 are calculated based upon the comparison with an empirically
determined allowed error emax by a linear equation (Eq. 4).
ϑ∗
pr =
{
0 for e >= emax
emax−eemax
for e < emax
(4)
Additionally the Trust Levels ϑ∗
pr are again limited with the mean Trust signal of the
VRMS-data they are calculated from.
2noted by the superscript *
1462
All different estimates for the parameters f , a and d are now fused dynamically according
to their Trust Levels (Fig. 2). The details of this fusion are found in [BBH10]. The outline
is the following: First a Trust Level-weighted average of the inputs is calculated. The
Trust Levels of each input are fused by an s-norm to an intermediate Trust Level. Than the
degree of contradiction between both inputs is calculated as their relative difference to the
mean. Afterwards the degree of contradiction is fused with the intermediate Trust Level by
a t-norm. This leads to the following overall behaviour of the fusion:
• Inputs with a high Trust Level have a high influence on the result.
• If two algorithms yield similar results, the Trust Level of the fusion in higher than
the highest Trust Level of a single input.
• If two algorithm contradict each other and both have a high Trust Level, despite this,
the result has a low Trust Level.
All in all the Trust Management in the upstream processing considers the individual
weaknesses of each algorithm dynamically for the given data and their inherent uncertainties.
It thus yields robust features for the detection of a sub-harmonic. As these features are
also equipped with Trust Levels, further processing modules can judge them accordingly in
order to archive results with maximal reliability.
3.4 Decision Making
On the decision layer, the results from the feature extraction are evaluated for two different
criteria: The classification determines, if it is a sub-harmonic with a simple threshold on
the amplitude a. On a positive detection, criticality of the sub-harmonic γ is evaluated by
additionally looking at the damping factor d according to Eq. 5.
γ(a, d) = (1− eFaa)(1− eFdd) . (5)
The parameters Fa and Fd have to be chosen by an expert or have to be determined
empirically, as they represent the characteristics of sub-harmonics the grid can handle. The
Trust Level of γ is the minimum of the Trust Levels ϑa and ϑd.
Afterwards the decider module determines if an action is needed: The kind of sub-harmonic,
its criticality and the corresponding Trust Levels are evaluated according to certain rules
derived from knowledge about the principle system behaviour, but without knowledge about
the concrete location in the grid or about available actors according to the following rules:
• If the situation is very critical, and we trust this result, then choose efficient and
potent actions.
• If the situation is very critical, and we do not trust the result, then carefully apply
save but expensive worst-case actions.
• If the situation is somewhat critical, and we somewhat trust this result, then choose
basic actions and wait for lower criticality or better information.
1463
• If the situation is not critical, and we trust this result, then do nothing, respectively
retract former actions.
• If the situation is not critical, and we do not trust this result, then do nothing.
As the sub-harmonics are mainly caused by an inappropriate amount of reactive power in
the grid, it must be adjusted according to the decision. The action can be parametrized with
a strength and a desired time characteristic.
As the AMIGO should not influence the grid too much, the action can be retracted slowly, if
the grid is stable again. If the sub-harmonic returns, the retraction of the action is cancelled.
This ensures a stable grid by only some goal-directed steps while it helps to keep the
headroom of the actor high, up to now without a model or concrete knowledge about the
grid. After such an abstract action is determined, it is applied to the grid by the downstream
processing.
3.5 Downstream Processing
To apply an action, it has to be scheduled to the available actors. Therefore the dispatcher
module knows about all manipulatable actors at hand, their actual state, their limitations
and characteristics. It is hence the only module requiring concrete knowledge about the
local situation in the power grid. The actors, which the AMIGO can handle, range from
lightweight, indirect actions with a high delay like (price responsive) DSM over the direct
adjustment of the real and reactive power by RLM up to the activation of FACTS-devices
or by changing their set points. Modifying parameters of the controllers is also possible but
not intended in order not to change the grid dynamics.
The dispatcher selects one or more actors matching kind and amount of intervention and
the desired time characteristics. Strategies to prioritize the actors can be implemented
to satisfy commercial or safety interests. Different cost metrics or limits for actors can
be implemented. The same holds for the retraction of the actions. A simple scheme like
”first applied is first removed” can be used, but more complex ones are possible. The
current implementation we investigate exemplarily in the next section only contains a
simple mechanism to adjust the real and reactive power directly.
4 Exemplary Investigations
4.1 Experimental Setup
To investigate the proposed architecture, we used a grid model that produces sub-harmonics
[DC89]. The model (Eq. 6) is relatively simple: It has four state variables (V, ω, δ, δm)
and 19 parameters. V is the amplitude of the voltage, ω the current frequency. δ and δmcharacterize the phase of the generators to each other.
1464
˙δm = ω
Mω = −dmω + Pm + E2
mYm sin(θm) + EmYmV sin(δ − δm − θm)
Kqw δ = Qd −Q0 −Q1 −KqvV −Kqv2V2
KpvT V = Pd + P0 + P1 −KpvV −Kpw
Kqw
·Kqw δ
(6)
It models the formation of components visualized in Fig. 3. A load is connected to two
generators. The left one represents an infinite bus E0 with its admittance Y0, the right is
an equivalent circuit for the rest of the grid given. It models the network’s non-linearity
by a smaller generator with a non-linear model for a synchronous generator Em and the
admittance Ym. In between the load consumes the energy. Although the model has not
this many components, it shows characteristic non-linear effects like instability caused by
bifurcations and limit cycles that generate drifts and flicker in voltage V and frequency ω[FEH05]. As the AMIGO-architecture is only partly dependent on the grid topology, the
complexity of this model is high enough to evaluate the performance.
~ ~
Y0 Ym
LoadE0 Em
AMIGO
Figure 3: Model to generate sub-harmonics
The AMIGO just measures the local voltage at the load and influences the load parameters.
The important parameters to influence the development of sub-harmonics are the real
and the reactive loads P∗ and Q∗. To set up a realistic scenario, we perform an artificial
measurement process on the grid, to get the voltage Vreal. It is derived from the state
variables and the main frequency ω0 = 50Hz:
Vreal(t) = V (t) · sin(ω0 · t + δ(t)) + N(t) (7)
Additionally a measurement noise N(t) ∈ [−1, 1] (scaled, band limited white noise) is
added. A short period of Vreal(t) is shown exemplarily in Fig. 4. The 50 Hz sine wave is
overlain by noise with an amplitude higher than that of the sub-harmonic.
To investigate the grid under dynamic conditions, the normally fixed parameters P1 and Q1
are replaced with datasets taken from real parts of the power grid. For Q1 the ”Individual
household electric power consumption Data Set ” from [BL13] is used. The real power load
1465
3.97 3.98 3.99 4 4.01−1
−0.5
0
0.5
1
time [s]
Vre
al
Figure 4: Voltage Vreal(t) with noise derived from the state of the model according to Eq. 7
of the city Kiel, Germany is used for P1 [Kie13]. To fit to the model and still to alter the
parameters in a realistic way, the data is rescaled and accelerated. The scaling is performed
with the constraint that 5% of the values lay in a critical parameter range. To increase
volatility the minute-based data is speeded-up by a factor of 6.
The actors of the AMIGO in this setup are direct manipulators of P and Q or the ratio
P/Q with individual constraints and time-behaviour (the speed of parameter change is
limited). The implementation of the model is done in MATLAB/Simulink. The AMIGO is
connected to the model via TCP. This interface allows to simply replace the model with
real measurement devices and actors.
4.2 Results
We investigate an integral scenario with the active AMIGO and compare it to the one
without. The Parameters P1 and Q1 are dynamically changing like they would in a power
grid near its load limit. The rest of the parameters is set like in [FEH05]. In Fig. 5 the state
variable V from the model is visualized3. In Fig. 6 the derived criticality is shown.
To examine the effect the AMIGO has on the grid, the grey lines in Fig. 5 and Fig. 6
show the development without any intervention: The amplitude of the sub-harmonic rises
and at t=75s the voltage starts to leave the specified range of 1 ± 0.2pu. The criticality
reflects this correctly, but as the sub-harmonic gets stronger and stronger, and non-sinusoidal
characteristics grow, the Trust Level of the criticality determination falls.
The black lines in Fig. 5 and Fig. 6 show the development with an active AMIGO for the
same initial situation: First, the criticality rises similar to the scenario without the AMIGO
until t=32s. Here the AMIGO decides to lower the reactive power carefully by a small
amount and the actors are set accordingly. The criticality does not rise afterwards, but it
also does not fall significantly. Hence the AMIGO lowers reactive power again and raises
3Note that this is not the noisy signal Vreal that the AMIGO works on (Eq. 7, see Fig. 4 for a short time period),
but the amplitude which is derived from the complete model state and is overlain by noise. The visualisation of
Vreal over a time long enough to see the sub-harmonics is impossible on paper.
1466
Time [s]
V
0 10 20 30 40 50 60 70 80 90 1000.85
0.9
0.95
1
1.05
1.1
1.15
UnstabilizedStabilized
Figure 5: State variable voltage V in pu over time in seconds
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Figure 6: Criticality of sub-harmonic and its Trust Level over time
real power simultaneously at t=42s. This yields the desired result: The criticality falls and
finally the network is stabilized.
Further details on the performance of the algorithms are visualised in Fig. 7: We see the
progress of the amplitude and the damping factor estimated by the different algorithms, the
fused result, and the according Trust Levels. The wavelet analysis is not very accurate in
calculating the amplitude and thus has a low Trust Level; the damping is calculated more
trustworthy and the results from the prony analysis show a high overall trustworthiness.
After the first action is applied by the AMIGO, the Trust Levels of results from the wavelet
analysis drop to 0.0, as the frequency band cannot be determined correctly due to the
resulting dynamic change in amplitude. After the intervention, these results are trustworthy
again, as the amplitude stabilizes.
During the intervention, the Trust Levels of the prony analysis results fall, as different signal
characteristics hinder a completely trustworthy result. In the examined situation, the prony
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Figure 7: Measured amplitudes of the sub-harmonic and their damping factors over time
analysis shows to be more trustworthy than the wavelets. Thus, the fused features tend to
the result of the prony analysis. Nevertheless, if both give the same value, the Trust Level
of the fused result is still higher than the particular Trust Levels of each input. Throughout
the complete scenario, the fused results still maintain a high quality as the algorithms back
up each other dynamically. The Trust Level of the result of the fusion indicates this, even
during the intervention of the AMIGO.
5 Conclusion
In this paper we proposed a generic architecture, named AMIGO, to automatically detect
and suppress sub-harmonics with actors like remote load management in presence of
uncertain. The complex and dynamic dissemination of uncertainties in the different signals
and in the results of the detection-algorithms are handled with Trust Management. This
approach requires no formal modelling of the grid and needs only basic knowledge about
the local situation. For a first investigation of the architecture we used data with real world
characteristics and a non-linear model to provoke sub-harmonics. It shows that the AMIGO
approach stabilizes the network by adjusting load parameters automatically. The key for
robust and reliable operation is the Trust Level based sensor fusion, which dynamically
combines strengths and weaknesses of the signal analysis algorithms and allows robust
detection and suppression of sub-harmonics
Acknowledgement
We would like to thank the group of students who worked with us on this application,
namely: Dominik Abraham, Sven Galenski, Thorsten Gedicke, Rolf Thomas Hanel, Felix
Igelbrink, Julian Imwalle and Jonas Schneider. We also would like to thank the reviewers
for the valuable comments.
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