Three-dimensional maps of power quality loss based in the power tensor theory

This paper presents the results of a project which was aimed at developing a flexible visualisation tool to be used mainly by power system operators. The project concentrated on a technique for assessing the deterioration of power transfer quality in electrical networks based on the power tensor theory. One of the envisioned tool's essential goals was to aid system operators by providing images of the system with easily identifiable characteristics related to loss or overall deterioration of power quality. The result was a user-friendly, three-dimensional graphic interface. The tool's effectiveness has been illustrated via different scenarios created using the IEEE 13-bus test feeder as an example.


INTRODUCTION
Effective power system operation means that power system engineers and operators must analyse vast amounts of information.A key challenge in systems containing thousands of buses is to present this data in a form allowing a user to intuitively and quickly assess the state of the system (Santoso et al, 2000;Gu and Bollen, 2000).This is particularly true when trying to analyse electrical network power quality.Therefore, constructing maps regarding power quality loss or deterioration due to steadystate and non-stationary events is of great importance for controlling, protecting and analysing power systems.Typically, efforts at drafting maps are focused on analysing the voltage signal.This is done by simulation programmes and electrical events using sophisticated measuring and monitoring the voltage (instantaneous values or RMS) at specific nodes of the particular system being studied.This information is analysed by systems' experts for detecting and classifying disturbances present in the voltage waveform (Gauoda et al., 2002;Wang et al., 2004;Duque et al., 2006;Cerqueira et al., 2006;Mosoum et al., 2010).The results are subsequently displayed by using in two-and three-dimensional visualisation techniques (Weber, and Overbye, 2000;Overbye et al., 2003;Overbye, and Weber, 2001;Klump and Weber, 2002;Sun and Overbye, 2004;Xu et al., 2006;Overbye et al., 2007;Evrenosoglu et al., 2007;Milano, 2009).
This form of presenting the state of an electrical system's variables to the network operator can be helpful when making decisions, but it can become a very tedious task which is difficult to interpret if many indicators must be analysed simultaneously.It should also be borne in mind that the bars may represent many power systems.
This paper describes a technique for three-dimensional visualisation and animation of the deterioration of power quality in electrical networks based on power tensor theory.The technique's objective is to provide electrical network images having easily identifiable characteristics by constructing graphical schemes or maps of power quality loss.The tool's effectiveness is illustrated via different scenarios which were created using the 13 bus distribution system reported by the Institute of Electrical and Electronics Engineers' (IEEE) subcommittee on distribution system Three-dimensional maps of power quality loss based in the power tensor theory Mapas tridimensionales de pérdida de la calidad de potencia basados en la teoría tensorial de la potencia A.J. Ustariz-Farfan 1 , E.A. Cano-Plata 2 , and H.E. Tacca 3 analysis (Distribution Planning Working Group Report, 2001).

THREE-DIMENSIONAL VISUALIZATION AND ANIMATION a) Background Review
The importance of an intuitive and detailed visualisation of results produced by a disturbed power system has been discussed since late in the past century.Two-dimensional visualisations based on one-line diagrams have been relatively popular due to power system operators' familiarity with oneline diagrams and the importance of geographical information in power system operation and control.
Examples of this can be found in work where twodimensional contour plot are proposed for visualising voltage levels in each power system bus (Weber, and Overbye, 2000;Overbye et al., 2003;Overbye, and Weber, 2001).Similarly, the contour plot technique has been further developed for visualising a variety of data such as power flows, marginal prices, available transfer capability, contingency analysis, etc (Klump and Weber, 2002;Sun and Overbye, 2004;Xu et al., 2006;Overbye et al., 2007).All these references focus on static data visualisation and are basically two-dimensional plots.By contrast with the two-dimensional visualisation techniques listed, three-dimensional visualisation and animation has also been proposed (Evrenosoglu et al., 2007;Milano, 2009).

b) Proposed Visualization Technique
The basic functioning of the proposed visualisation and animation tool is shows in Figure 1.A three-dimensional visualisation and animation tool needs two source files, one for topological data (Simulink) and another for numerical data (EMTP-ATP).The three-dimensional plots in this work have been obtained by computing the convex hull enveloping the values obtained from simulations onto a high-resolution three-dimensional surface.For example, if a power network's power quality deviator factor is considered than the number of available deviator factor values is equal to the number of connected meters, typically not being sufficiently high enough to adequately fill up the surface.To overcome this issue, a grid was created having a high number of points through polynomial interpolation.Next, the height of each point of the grid was determined by using Delaunay triangulation (Skiena, 1998).The resulting surface was finally coloured using a contour map and the one-line diagram of the network was superposed.
MATLAB was used to compute the Delaunay triangulation and plot results.The main functions used for displaying graphics schemes are polynomial interpolation (griddata.m),convex hull (convhull.m),Delaunay triangulation (delaunay.m)and contour maps (colormap.m).

POWER TENSOR THEORY a) Instantaneous Power Tensor Definition
The proposed technique was based on the definition of an evolutionary expression of instantaneous power called "instantaneous power tensor" for geometrically interpreting electric phenomena behaviour, analogous to deformation studies in the mechanics of solids.Ustariz et al., using the tensor product, defined the instantaneous power tensor as follows (Ustariz et al., 2010A): where, ℘ ĳ was the instantaneous power tensor, u i was the phase voltage tensor and i j was the line current tensor.The instantaneous power tensor was expressed in an n-phase and m-wire system as: Here, the main diagonal terms (u i i i ) referred to the instantaneous energy flow.Moreover, the differences between the transposed terms (u i i j -u j i i with i ≠ j) referred to the instantaneous energy exchange between phases i and j whose average was zero (Ustariz et al., 2010B).Therefore, ℘ĳ was a single expression which involved the two conventional components of instantaneous power (active and reactive).

b) Power Quality Deviation Factor
The measurement and evaluation of nonconformities in a disturbed power system could be quantified by defining a new overall power quality index called instantaneous power quality deviation indicator (IDI).It would be evaluated using the following expression: ( ) where the term ideal ℘ ij would be defined as the ideal power tensor.Calculating an ideal power tensor involves defining an ideal electrical system as being a circuit consisting of a sinusoidal, balanced voltage source (ideal voltage tensor) feeding a resistive, balanced linear load (ideal current tensor).Moreover, the root mean square of IDI pq has been called the power quality deviation factor (DF) DF pq , so that: Here, the interval defined between t 1 and t 2 matched the temporal observation window's width.A 12-cycle window is recommended when working frequency is 60 Hz Std.61000-4-30, 2003).In (4), zero indicated a null deviation regarding an ideal power system.
The notable features of definition (4) are the following: • It provides quick assessment of power transfer quality deterioration from the ideal in terms of maximum power transfer; • It is regardless of whether the disturbance arises from the load or the source; and • It does not use controversial or misleading quantities such as traditional indexes' average, maximum or weighting values.Additionally, the power quality deviator factor can have several useful applications: • Sites having poor power quality could be easily ranked to determine priority for power quality improvements; • It can be easy to represent by constructing graphical schemes or maps of power quality loss in large electrical networks; and • System operators will have a single index which they can correlate with planning, protection or maintenance practice.

c) Comparison Criteria
Power quality deterioration assessment is traditionally based on individual indicators, such as total harmonic distortion (THD X ), unbalance factor (UF X ) and reactive power factor (PF R ) (Milanez and Emanuel, 2003;Emanuel, 2004;Sharon et al., 2008).The conventional definitions are: • Total harmonic distortion of the voltage and current 1 1 ; • Unbalance factor of the voltage and current Expressions ( 5) to (7) were based on (IEEE Std-1459, 2000), where sub indexes 1 and H indicated fundamental and harmonic frequency components.Super indexes + referred to the direct sequence component; S and P referred to apparent and active power.

SIMULATIONS
Different scenarios have been created in this section, using IEEE 13-bus test feeder (see Figure 2) to illustrate the benefits of using proper visualisation methods in monitoring power quality in electrical networks based on power tensor theory.The IEEE-13-bus test feeder presents some very interesting characteristics, such as being short and relatively highly loaded for a 4.16 kV feeder, having one substation voltage regulator consisting of three single-phase units connected in wye, overhead and underground lines having a variety of phasing, shunt capacitor banks, in-line transformer, unbalanced spot and distributed loads.The IEEE 13-bus test feeder was implemented in EMTP-ATP (see Figure 3).
On the other hand, the IEEE 13-bus test feeder presented three types of steady-state disturbance: voltage imbalances, load unbalance and reactive power flows.These disturbances were measured and the results are summarised in Table I.As expected, nodes 652 and 611 (single-phase sections) had the highest levels of quality loss (143.44% and 122.69% respectively) followed by nodes 646 and 645 (two-phasing sections: 114.42% and 110.67%, respectively).These nodes also had the highest UF V , UF I and PF R indexes, except for the PF R index in node 611, which was low by the shunt capacitor banks.The linear load connected at node 634 was changed for a six-pulse adjustable-speed drive (ASD-6P) in this scenario.The main ASD-6P parameters were 480 V nominal line voltage, nominal 600 kVA rated three-phase power and 0.8 power factor.

DFpq-factor [%]
The base case was modified to characterise a non-linear load connection in the IEEE 13-bus test feeder and construct a power quality loss map in steady state.Figure 6 shows voltage and current waveforms at the ASD-6P entrance (node 634).The remaining nodes in the modified test system had a noticeable influence on harmonic distortion propagation (Table 2).
From the results summarised in Table 2, as expected, nodes 634 and 633 became the next in importance in terms of quality deterioration.These nodes had a 45% increased loss of quality compared to the base case.A single-phase fault (in phase-b) was caused in the line connecting nodes 632 and 671.The fault occurred at t=1.1s, lasting 70ms and fault impedance was 0.5 Ω.    Figure 10 shows the DF pq -factor detected as a nonstationary event in the system's different nodes.Figure 10 clearly shows that nodes 650 and 632 had the highest levels of quality loss during the single-phase fault.Quality loss compared to the values before the fault changed very little in the rest of the nodes; no changes were seen in nodes 634 and 652.On the other hand, using three-dimensional animation, one can intuitively show how the whole system was affected by quality loss during the single-phase fault.Figure 11 gives six snapshots showing the propagation of a transient disturbance for the IEEE 13-bus test feeder.Unfortunately, the snapshots did not fully illustrate the propagation of a transient disturbance as well as the three-dimensional animation did; animation clearly showed how quality loss was propagated from one zone to another of the particular system.

CONCLUSIONS
This paper has described a software package developed as a system operator aid.The tool was based on a technique for assessing power quality deterioration in electrical networks, based on power tensor theory.It has shown a user-friendly interface allowing a user to construct a system diagram for a new power system or edit an existing one to make desired modifications.
Three-dimensional animation and colour contouring techniques have been used for visualising the integral evaluation of power transfer quality deterioration in power systems.
The examples of different cases led to verifying the proposed indicator's applicability when measuring power transfer quality deterioration.
Future work will concentrate on improving the threedimensional visualisation technique.
A promising development would be the inclusion of new technology, such as involving a geographical information system when constructing graphical schemes or maps.

ACKNOWLEDGEMENT
This work has been partly supported by the Universidad Nacional de Colombia, Manizales

Figure 4 .
Figure 4. Map of quality loss and one-line diagram of the IEEE 13-bus test feeder for scenario 1

Figure 5 .Figure 6 .
Figure 5. IEEE 13-bus test feeder modified for scenario 2Figure5shows the modified IEEE 13-bus test feeder circuit which has been simulated in EMTP-ATP.This circuit shows the adjustable-speed drive model used.

Figure 7 Figure 7 .
Figure7compares power quality loss maps and twodimensional visualisation for the base case, as in the case of harmonic distortion.

Figure 9 .
Figure 9. Waveforms upstream of the fault: (a) voltage and (b) current As can be seen in Figure 9(b), phase-b current became increased upstream of the fault because of with little resistance being offered by the fault impedance.Moreover, the singlephase fault also occurred in system sag and voltage swell nodes, as shown in Figure 9(a).It can be seen that phase-b voltage became decreased by almost 50% while phase-a and phase-c voltage became slightly increased.

Figure 1 .
Figure 1.Frames from the three-dimensional animation of the quality loss map in scenario 3

Table 1 .
Power Quality Index Measurement in Scenario1

TABLE 2 POWER
QUALITY INDEXES MEASUREMENT IN SCENARIO 1 Individual indexes (%)Overall index (%