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

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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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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.

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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.

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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 *

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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.

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• 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

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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.

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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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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.

1468

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