1
A Method for Optimally Localizing Power Quality Monitoring Devices in Power Systems C. Dzienis, Student Member IEEE, P. Komarnicki, Student Member IEEE and Z. A. Styczynski, Senior Member IEEE
Abstract — This paper presents a method for optimally localizing power quality monitoring devices in power systems. The method is based on calculating the load flow in the frequency domain. The computation algorithm also obtains the Jacobi matrix that reflects the working point of a power system. Singular value decomposition enables obtaining information about nodes sensitive to harmonic. These nodes are where the power quality monitoring systems should be installed. Index Terms — Harmonic Load Flow, Singular Value Decomposition, Modeling, Power Quality Monitoring System
T
I. INTRODUCTION
he continuous growth of harmonics in power systems has mainly is the result of the increasing number of non-linear loads. Today, nearly every electrical device is equipped with an integrated energy conversion unit (power converter), which transforms the energy into any form, thereby allowing for more efficient operation of the equipment. These properties are especially used in technical processes. Not only end-loads have a strong non-linear character [1]. Appliances that enable improved energy transmission are also equipped with nonlinear power converters. For instance, the main function of equipment such as STATCOM or TCSC is to exchange the reactive-active energy flow and thus to increase a power system’s capacity. On the one hand, such equipment compensates for the reactive power in the system but, on the other hand, distortions like higher harmonic are injected into the network [2]. Moreover, a significant increase of renewable energy sources can be expected in the next few years, which, because of their operation principle, generate a certain portion of the harmonic into a power system. Therefore, the occurrence of harmonics and their propagation pathways deserves careful investigation and incorporation into analysis of network operation. An excellent method to analyze disturbance in an electrical network is to install a power quality monitoring system. This equipment helps completely monitor a network, identify the source of a disturbance and carefully control a system. The high costs of such systems means obviate installation in an This work is supported by the Deutsche Forschungsgemeinschaft (DFG). C. Dzienis and Z. A. Styczynski work at Otto von Guericke University Magdeburg, P. Komarnicki works at the Fraunhofer IFF, Magdeburg, Germany (email:
[email protected] magdeburg.de,
[email protected],
[email protected])
entire network. Therefore, an effective method ought to be developed to localize the optimal place to install a power quality monitoring system in a network. The points most jeopardized by disturbances would have to be found and a monitoring system installed at these points. Generally, the following three aspects have to be taken into account when searching for sensitive nodes: The reference point where complete information about the harmonic voltage is measured continuously (in most cases, this point is the PCC node), the network structure and its parameters and, finally, the injected harmonic power in individual nodes which should be carefully measured so that complete information about the harmonic profile is available. Once the information about an analyzed system is obtained, the harmonic load flow can be calculated. This is done with the iterative procedure of the Newton-Raphson algorithm. This algorithm calculates the so-called Jacobi matrix, which symbolizes a working point of the power system [3]. Using tools from linear algebra, this matrix can be divided into three orthogonal matrices. These matrices transport information about sensitive points in the network. Moreover, most of the harmonic sources can be recognized with this procedure [4]. This paper presents this procedure. The full modeling algorithm is presented and used to determine the nodes for optimal placement of a power quality measurement system. II. POWER SYSTEM MODELING In most cases, an electrical system is asymmetrical. The onephase nonlinear load has influenced this so greatly that it is impossible to consider a system in a symmetrical coordinate. In such cases, a complete network system should be implemented with the help of full three phase models. This requires full models of equipment such as transformers, distribution lines and loads [5]. Three methods can be employed to obtain the power system parameters. The first method is to measure parameters, which is very safe approach that delivers very accurate results. In particular, if the parameters are used not for common computation, special development procedures can provide additional information on searched parameters. The second method is to compute the parameters based on the geometric and physical properties of the elements. The third and very popular method is to accept the parameters provided by the manufacturers of the elements. This method is used to create network element models.
2 n
A i
c
IA
Ic
IC
Ib
IB
Ia
A
c
j b
C
B
i
b
C
a
j
B a
Figure 1. Transformer model – Dy5 sequence group.
In Figure 1 presents the power transformer model created for the Dy5 sequence group that is the most popular group in power systems. Equation 1 specifies the physical properties of this element. Yij symbolizes the admittance of the transformer based on the secondary side of the transformer and n expresses the ratio between the secondary and primary sides of the transformer. Since the magnetizing impedance is many magnitudes larger than the leakage impedance, what is seen as open circuits from the network perspective has been disregarded in this model. IA, IB, IC and VA, VB, VC denote the quantities of the primary side of the transformer (i-node). However, Ia, Ib, Ic, Va, Vb, Vc correspond to the quantities of the secondary side (j-node).
⎡I A ⎤ ⎢I ⎥ ⎢ B⎥ ⎢I C ⎥ ⎢ ⎥ ⎢I a ⎥ ⎢I b ⎥ ⎢ ⎥ ⎢⎣I c ⎥⎦
⎡ 2 ⎢ n2 ⎢ 1 ⎢− 2 ⎢ n ⎢ 1 ⎢− 2 = Y ij ⎢ n 1 ⎢− ⎢ n ⎢ 0 ⎢ ⎢ 1 ⎢ ⎣ n
1 n2 2 n2 1 − 2 n 1 n 1 − n −
0
1 n2 1 − 2 n 2 n2
1 n 1 − n
0
−1
0
0
−1
0
0
−
1 n 1 − n
0
0 1 n 1 − n
1⎤ n⎥ ⎥ 0 ⎥ ⎡V A ⎤ ⎥ ⎢V ⎥ 1 ⎥⎢ B ⎥ n ⎥ ⎢⎢V C ⎥⎥ ⎥ V 0 ⎥⎢ a ⎥ ⎥ ⎢V b ⎥ ⎢ ⎥ 0 ⎥ ⎢⎣V c ⎥⎦ ⎥ ⎥ − 1⎥ ⎦
−
−1
1 h(RijT + jX ijT )
i
(1)
RijT =
uijkN UijNT 100% SijNT uijRN U ijNT
IA
C
B
a Ia
Z Cc
IC
2 2 = RijT + X ijT
(3)
100% SijNT
where uijkN is the rated short circuit voltage, uijRN reflects the
j
b Ib
YB
(2)
c
Ic
ZBb
IB
where h is harmonic order and RijTp and XijTp are transformer primary side resistance and reactance for a fundamental frequency of 50Hz respectively. These quantities are calculated from the rated short circuit voltage as follows (3): Z ijT =
Z Aa
A
The parameters in the matrix allow expressing the rotation of the voltage in the transformer windings so that the real phase shift and magnitude for harmonics can be calculated. Moreover, measurements are taken on the lower voltage level of the electrical network. The full transformer model can be used to compare the computed currents in the higher voltage level with the current registered at the reference point. It can be applied as a plausibility rule to verify the calculations. The method in [5] was applied to compute the admittance of the transformers for each frequency studied (2): Y ij (h) = Z ijTp =
rated resistance voltage drop for fundamental frequency. In order to apply the calculated parameter in the given transformer model, the calculated impedance must be converted from primary to secondary side. This involves applying an n transformer ratio. Generally, the calculations were performed based on the datasheet available for each transformer. Since the system being investigated has not been operating a long time, the obsolescence factor of the transformers, which influences the parameters, can be disregarded. The other sequence groups of the transformers can be mathematically represented with Kirchhoff’s laws [5]. The next important model that must be applied for a detailed simulation of an electrical system is a model of the distribution wire or cable. This requires the type of representations pictured in Figure 2. Basically, the model consists of two main branches, the serial branch reflecting the conductor’s inductive-resistive character and the parallel branch describing the wires’ capacitive behavior. It is assumed that any electrical interaction between ground and phase wires appears. Moreover, the distances between the nodes do not exceed 300m so that the effects associated with current and voltage delays between the beginning and end of a wire terminal do not have to be taken into account. Therefore, the model can be computed from primitive parameters and can be applied to system equations as single six-pole.
YC
Yb
YA
Yc
Ya
Figure 2. Full model of the distribution wire or cable.
The equation (4) describes the connection between the nodes of a power system. The model makes no assumptions for mutual inductance between the different wires because the system investigated was constructed from cable with a tape shield so that the cross currents generally result from the significant capacity between conductor and shield. ⎡ 1 ⎢Z +Y A ⎢ Aa ⎢ 0 ⎡I A ⎤ ⎢ ⎢I ⎥ ⎢ B ⎢ ⎥ ⎢ 0 ⎢I C ⎥ ⎢ ⎢ ⎥=⎢ 1 I ⎢ a⎥ ⎢ − Z Aa ⎢I b ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎣⎢I c ⎦⎥ ⎢ ⎢ ⎢ 0 ⎣⎢
0
0
1 +Y B Z Bb
0
0
−
1 +Y C Z Cc
−
1 Z Aa 0
0 −
1 Z Bb
0
0 0
0
0
1 Z Bb
1 +Y a Z Aa
0
0
1 Z Cc
1 +Y b Z Bb
0
0
0
−
⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎡V A ⎤ ⎢ ⎥ 1 ⎥ ⎢V B ⎥ − ⎥ Z Cc ⎥ ⎢V C ⎥ ⎥ ⎢V a ⎥ 0 ⎥⎢ ⎥ ⎥ ⎢V b ⎥ ⎥ ⎢⎢V ⎥⎥ 0 ⎥⎣ c ⎦ ⎥ 1 +Y c ⎥ Z Cc ⎦⎥ 0
(4)
Just like the transformer parameters, the detailed cable parameters were taken from datasheets. The end loads are represented as injected harmonic reactive and active power as can be seen in Figure 3. Each load-node is modeled with three inputs containing the three reactive and active harmonic powers.
3
III. SINGULAR VALUE DECOMPOSITION METHOD
(h)
P a,b,c c
a
The harmonic voltages and currents appearing in the electrical network can be described with the following set of nonlinear algebraic equations (5):
b
i t
(h)
Q a,b,c
(h) (h) Pa Q a
(h) (h) P c Qc
Pi ,hp = V
Qih,p = V
(h) (h) Pb Q b
t Figure 3. The model of the (end) load as harmonic power profile for each harmonic order. 10
Current 5th [A] Phase A Phase B Phase C
7.5 5 2.5 0 11:00 15:00 Voltage 5th [V] 3
19:00
23:00
03:00
07:00
Phase A Phase B Phase C
2 1 0 11:00 15:00 cosφ 5th [-] 1
11:00
19:00
23:00
03:00
07:00
11:00 Phase A Phase B Phase C
0.5 0 -0.5 -1 11:00
15:00
19:00
23:00
03:00
i ,p
h i ,p
n
∑ ∑V { }
q∈ a,b,c j =1
n
∑V
∑
q∈{a,b,c } j =1
h j ,q
Y
h
Y j ,q
h ij , pq
cos(δ ih,p − δ ih,p − µ ijh,pq )
(5)
h
sin(δ ih,p − δ ih,p − µ ijh,pq ) ij ,pq
where Pi,p(h) and Qi,p(h) are the active and reactive power, respectively, for hth harmonic order at node i; |V|i,p(h) is a module of the voltage and δi,p(h) and µij,pq(h) denote angles for voltage and admittance, respectively. The n index symbolizes the number of nodes in the electrical system considered and p und q are indexes of the individual phases. The expressions combine the influences of network structure (Yij(h) factors obtained from the nodal admittance matrix configured from the given models in the previous section) and input power (Pi,p(h), Qi,p(h) indices acquired from measurements for example) on harmonic changes. In order to solve this set of equations, expression (5) should be linearized according to the Newton-Raphson procedure [7] and, after being clearly arranged in a matrix, can be written in the following formula (6):
07:00 11:00 Time [hh:mm]
⎡ ∆ P ( h ) ⎤ ⎡ J P( h( δ) ) = ⎢ (h ) ⎢ (h ) ⎥ ⎣ ∆ Q ⎦ ⎢⎣ J Q ( δ )
Figure 4. Fifth harmonic profile from one of the measured stations (Station 2, see Figure 5).
A mobile power quality measuring system measured the injected reactive and active harmonic powers. The measurements were taken on the low voltage side of the transformers because the medium voltage side was inaccessible. Moreover, the medium voltage measurement must be taken with two instrument transformers and this can contribute to significant measurement error. The injected powers were obtained from a simple compartment of current and voltage as well cosϕ, that determine the ratio between injected or absorbed active and reactive power. Figure 4 presents a simple example of one such harmonic profile. It was measured for the fifth harmonic at Station 2. At first glance, a simple correlation exists between the voltage at the node and the currents. If the current is high then the voltage drop is low and vice versa. This indicates that the fifth harmonic is absorbed at this node. The positive cosϕ factor confirms this. The value of cosϕ is positive for almost the whole registered interval. Moreover, the fifth harmonic is the same for two phases. Only phase C deviates from the two other phases. The low level of the fifth harmonic in this phase is a positive aspect. Strong cosϕ factor fluctuation was measured for this conductor and indicates moments when the loads connected to the system by phase C injected the harmonic into the network. There is no observable impact on the voltage profile because of the small values in current.
h
J P( h(V) ) ⎤ ⎡ ∆ δ ( h ) ⎤ (h) ⎥ ⎥⎢ J Q( h(V) ) ⎥⎦ ⎢⎣ ∆ V ⎥⎦
(6)
where J(h)P(δ) , J(h)P(V) , J(h)Q(δ) , J(h)Q(V) are the Jacobi matrices obtained from differentiating the function of active and reactive power. ∆δ(h), ∆|V|(h) denote the vector of the changes of angle and magnitude for each harmonic. ∆P(h), ∆Q(h) symbolize the vectors of active and reactive power changes, respectively. Figure 5 presents a graphic interpretation of the Jacobi matrix parameters. A simple example helps explain the method. 10
Harmonic voltage V(h) [%]
9 8 7 6 ∆V(h)
5
1
Working point 1
4
tanφ
3
1
1
∆V(h)
2 Working point 2 1 0 0
∆P(h)
tanφ 20
2
40
2
∆P(h) 2
60
80 100 120 Harmonic power P(h) [W]
Figure 5. Illustrative interpretation of Jacobi matrix parameters.
4
Figure 5 illustrates the two different working points in the network for given harmonic h. They are expressed by two different coefficients of the Jacobi matrix, which can be ∂P(h)/∂V(h)|V1=tanϕ1 and interpreted as tangents ∂P(h)/∂V(h)|V2=tanϕ2. If tanϕ1 is smaller than tanϕ2, then, for the same active power changes of the harmonic h (P1(h)= P2(h)), the changes of the harmonic voltage V2(h) are lower than the changes of the voltage V1(h). Consequently, point 1 is more sensitive than point 2. As can be seen in equation 6, the changes of the voltages are a multidimensional function of the reactive and active harmonic power injected at individual nodes. In order to find the system’s most sensitive points, singular value decomposition (SVD) should be used to factor the Jacobi matrix into the group of three matrices [3], [4]. The following can then be represented (7): ⎡ v (j1h ) ⎤ ⎡ ∆δ ( h ) ⎤ 2 n 1 ⎢ ⎥ ( h) ( h) ⎢ ( h ) ⎥ = ∑ ( h ) ⎢ M ⎥ w j1 K w j 2 n ⎣∆V ⎦ j =1 σ j ⎢v ( h ) ⎥ ⎣ j 2n ⎦
[
⎤ ⎥ ⎣∆Q ⎦
]⎡⎢ ∆P
( h)
( h)
(7)
Equation 7 makes clear that the smallest singular values (weight factors) σj(h) for j∈ {l,…,2n-1,2n} make the greatest contribution to the harmonic voltage changes and are indicators of the proximity to the harmonic steady state stability limit. These values generate two left wj(h) and right vj(h) vectors that consign information either about the weak nodes (sensitive due to harmonic voltages or angles changes), i.e. vector vj(h), or indicate the places that most disturb the network with reactive or active power injections, i.e. vector wj(h) [8]. Those points that are the sources of the most disturbance or exhibit significant sensitivity are precisely the point where a power quality measurement system should be installed. IV. EXPERIMENTAL RESULTS The method will be carefully explained by analyzing a real electrical network. The system considered consists of six medium voltage nodes interconnected by a ring or spurs. The nodes are interconnected on the medium voltage level of 10kV.
An external power system supplies the system at two points that are continuously monitored and have been selected as reference points for calculations. The other five nodes are connected with the during the step-down transformer’s low voltage stage. The nonlinear impact of the loads is reflected by the harmonic profile, the fundamental properties of which were explained in the preceding section. Figure 6 presents the investigated system. The loads inject or absorb a large portion of harmonics for which the THDi was chosen as the indicator. The current in Station 2 is especially characterized by distortions equaling approximately 40%. The harmonics flow for other stations is smaller and increases to a maximum of 25%. Since the network impedances are small, these distortions do not contribute significantly to voltage disturbances (THDu). This factor lies within the prescribed limits of 5%. As mentioned above, the harmonic load flow must be calculated in order to compute the points sensitive to low frequency disturbances or the places that contribute significantly to distortion circulations. Matlab software was used to implement the entire procedure for a three phase network as m-file scripts. The results are presented in Figure 7 and Figure 8 where the voltage harmonic profile were presented for the fifth and seventh, respectively, in a twentyfour hour interval. The calculations based on the primary side of the transformer. The frequencies fifth and seventh have the most significant impact on the system. In both cases, the calculations agree with the measurements. Only a slight deviation is apparent and stems from the measurement of the harmonic powers not being exactly synchronized with the measurement of the harmonic voltage at PCC. Moreover, the varying accuracy of the two measurement devices causes the differences. The calculated voltage profile demonstrates that the system studied is resistant to the disturbances. The voltage values are far below the limits, e.g. the calculated and measured magnitudes of the fifth harmonic do not exceed 3V, which is about 1% of the nominal voltage in 400V networks. th 3 5 Harmonic [V]; Phase A
Measurement Calculation
2 1 0 11:00
15:00
19:00
23:00
03:00
th 3 5 Harmonic [V]; Phase B
07:00
11:00
Measurement Calculation
2 1 0 11:00 15:00 19:00 5th Harmonic [V]; Phase C 3
23:00
03:00
07:00
11:00
Measurement Calculation
2 1 0 11:00
Figure 6. Investigated medium voltage network with labeled THDu and THDi.
15:00
19:00
23:00
03:00
07:00 11:00 Time [hh:mm]
Figure 7. Calculated and measured voltage profile for the fifth harmonic at station 4.
5 7th Harmonic [V]; Phase A Measurement Calculation 2
19.
Disturbance level [-]
3
0.4
1
Harmonic order [-] 17.
0.3
0 11:00 15:00 19:00 23:00 7th Harmonic [V]; Phase B 3 Measurement Calculation 2
03:00
07:00
11:00
0.2
13.
0.1 11.
1 0 11:00 15:00 19:00 23:00 7th Harmonic [V]; Phase C 3 Measurement Calculation 2
03:00
07:00
11:00
0 1 12 18
1 0 11:00
7.
6
5.
Node number [-] 24 15:00
19:00
23:00
03:00
07:00 11:00 Time [hh:mm]
Figure 8. Voltage profile calculated and measured for the seventh harmonic at station 4.
The same holds true for the seventh harmonic, i.e. the magnitude is lower and less than 1%. Interesting in this calculation (fifth and seventh harmonic) is the symmetrical voltage harmonic profile. This cannot be assumed to be the network’s normal working state. The voltages obtained for each harmonic can be used to determine the actual working point of the network, which is defined by the Jacobi matrix. To simplify the procedure, the Jacobi Matrix has only been calculated for one working point. To improve the accuracy of searches for weak points and disturbances sources, calculating a long time interval is expedient. A statistic approach can be used and information about characteristic points is more certain. Taking its three phase structure into account, the investigated network has thirty-six nodes, without the slack node (PCC). The first fifteen nodes are medium voltage and other are low voltage. Each Jacobi-matrix was decomposed by SVD implemented in Matlab. The calculation revealed that the one the last singular value is significant smaller then other one, so that the singular vector according only to this singular value can take into considerations. The W vector classifies disturbance sources, where the first elements (in this case up to thirty-six) deliver information on active power sources of distortion and the remaining elements deliver information on reactive power sources. Figures 9 and 10 present the results. In Figure 9, the most significant source (active power changes) for node 24 (fifth) appears at Station 1 on the low voltage side and for node 31 (seventh) at Station 3 on the low voltage side. The influence of the active power changes on other frequencies is absolutely uniform so that the system does not have any characteristic points where active disturbance sources are especially harmful. Only node 25 due to fifth harmonic and 31 due to seventh can be reason of the voltage changes for these harmonics respectively. We say here first about active parts of harmonics power.
30 3.
36
Figure 9. Disturbance sources qualified according to the SVD approach (active power changes) 19.
Disturbance level [-] 0.4
Harmonic order [-] 17.
0.3 0.2
13.
0.1 11. 0 1 7.
6 12 18
5.
Node number [-] 24 30
3.
36
Figure 10. Disturbance sources qualified according to the SVD approach (reactive power changes) 19.
Sensitivity level [-] Harmonic order [-]
0.4
17.
0.3 0.2
13.
0.1 11. 0 1 7.
6 12 18 Node number [-]
5. 24 30 36
3.
Figure 11. Sensitivity level qualified according SVD approach (harmonic voltage changes)
6
The findings of an analysis of the influence of reactive power disturbances were extremely interesting. The system is more sensitive to this type of power change because the network has impedances where the reactive (capacity and inductivity) predominates over the active (resistance). The disturbed currents generally cause reactive voltage drops. From a system perspective, this is seen as reactive power absorption or generation. Assuming that power generation cannot appear on the medium voltage nodes (see above), only nodes from 16 can be taken into consideration. In this case, most disturbance sources for each frequency can be found in nodes 24, 28, 30 and 31. There are stations 1, 2 and 3. The reactive power changes at these nodes can particularly disturb the system. It is best to reduce the reactive power injection or absorption at these nodes. This can be done by installing higher harmonic filters at these points. The sensitive nodes can be determined by analyzing the V singular vector. Only the nodes of the medium voltage network should be analyzed because the voltage changes on these nodes correspond to the voltage changes at corresponding nodes on the low voltage side. Node 13 (station 5) is the most sensitive point and is close to the ring so that potential disturbances can flow from both side of the network. What is more, that the other station which accompany with ring connection exhibit the increased sensitivity level. Therefore, the monitoring system should be installed at these nodes. Node 13 in particular should be regularly measured for low frequency disturbances. A better assessment of the sensitivity of individual nodes is obtained by applying the statistical procedure that analyzes many network states and produces quantitative information.
V. SUMMARY This paper presents an effective method for determining the optimal localization of power quality monitoring systems. They can be installed at the points where increased levels of sensitivity to low frequency disturbances are identified. The asymmetrical character of the loads, especially in the case of nonlinear devices, led to the development of the full three phase model of network elements to obtain very exact modeling results. The models developed were implemented in the harmonic load flow algorithm. The calculation of the harmonic flow also determines the Jacobi matrix, which corresponds to a network’s working point. By using SVD to factor the Jacobi matrix, the most sensitive nodes and/or the strongest disturbance sources can be found in the network. The monitoring systems should be installed there. VI. REFERENCES [1] [2] [3] [4]
[5] [6] [7] [8]
R. C. Dugan, M. F. McGranaghan, S. Santoso, H. W. Beaty, Electrical power systems quality, 2nd ed., New York: McGraw-Hill, 2003. E. Acha, M. Madrigal, Power System Harmonics: computer modelling and analysis, Chichester: Wiley, 2001, ISBN 0471521752. A. Z. Gamm, I. I. Golub, A. Bachry, Z. A. Styczynski, “Solving Several Problems of Power Systems Using Spectral and Singular Analyses,” IEEE Transaction on Power Systems, Vol. 20, NO. 1 February 2005. C. Dzienis, A. Bachry, Z. Styczynski, „Full Harmonic Load Flow Calculation in Power System for Sensitivity Investigation,” in Proc 1seventh International Zurich Symposium on Electromagnetic Compatibility Conference in Singapore, pp. 646-649. G. J. Wakileh, Power system harmonics: fundamentals, analysis and filter design, Springer, Berlin Heidelberg New York, 2001, ISBN 3540422382. R. Roeper, Short-circuit Currents in Three-phase Systems, John Wiley and Sons, 1985. J. Machowski, J. W. Bialek, J. R. Rumby, Power System Dynamics and Stability, John Wiley & Sons, 1997, ISBN 0-471-97174X. P-A. Löf, G. Andersson, D.J. Hill, “Voltage Stability Indices for Stressed Power Systems,” IEEE Transactions on Power Systems, Vol. 8, No.1 February 1993.
