Abstract: Present distribution network voltage design practice limits the distributed ... more active approach to network voltage control can significantly increase ...
Network voltage controller for distributed generation C.M. Hird, H. Leite, N. Jenkins and H. Li Abstract: Present distribution network voltage design practice limits the distributed generation capacity that can be connected to 11 kV networks. It has been shown previously that adoption of a more active approach to network voltage control can significantly increase distributed generation capacity. One way to do this is to control the target voltage of automatic voltage control relays at primary substations. A basic design for a controller to do this has been created, comprising three algorithms. A statistical state estimation algorithm estimates the voltage magnitude at each network node supplied by the primary substation, using real-time measurements, network data and load data. The estimate accuracy depends on the number and placement of real-time measurements. Studies using an 11 kV feeder model showed that an acceptable accuracy could be obtained with one or two measurements. The state estimator uses pseudo measurements for unmeasured loads. A load model was constructed using load profiles to calculate the pseudo measurements. The calculated pseudo measurements were inaccurate, but it was found that acceptable state estimate accuracy could be obtained with inaccurate pseudo measurements. A control algorithm alters the AVC relay target voltage, based on the maximum and minimum node voltage magnitude estimates. A simulation on a four-feeder network showed that the algorithm enabled the generator power export to be more than doubled.
List of symbols x Ns zi Nm fi s2i J Pinj Qinj n vi vk yi yk yik gik bik Dx x H R Cx sv2i Pu,w,h Pinj,h Nc
state variable number of state variables ith measurement number of measurements function relating ith measurement to state variables variance of ith measurement measurement residual active power injection at node i reactive power injection at node i number of nodes voltage magnitude at node i voltage magnitude at node k relative phase angle at node i relative phase angle at node k yiyk line conductance between node i and node k line susceptance between node i and node k vector of state variable increments vector of state variables Jacobian matrix of functions fi(x) diagonal matrix of measurement variances s2i matrix of state variable variances variance of node i voltage magnitude estimate active power for wth customer of uth load class for hth half-hour active power injection at node i at time h number of customers of load class u assigned to node i
r IEE, 2004 IEE Proceedings online no. 20040083 doi:10.1049/ip-gtd:20040083 Paper received 2nd May 2003. Online publishing date: 13 February 2004 M. Hird is with Econnect Ltd, Energy House, 19 Haugh Lane Industrial Estate, Hexham, Northumberland, NE46 3PU, UK H. Leite, N. Jenkins and H. Li are with the Department of Electrical Engineering and Electronics, UMIST, PO Box 88, Manchester, M60 1QD, UK
150
vref vmax vmin vupper vlower vd t rx JðxÞ g(x) gdes f 1
automatic voltage control relay target voltage maximum node voltage magnitude minimum node voltage magnitude upper network design voltage magnitude limit lower network design voltage magnitude limit automatic voltage control relay dead band width simulation time gradient of J(x) generalised function of x desired value of g(x) vector of function fi
Introduction
The connection of generators to distribution networks is attracting a great deal of attention in the electrical power industry internationally. In Europe this is being driven by ambitious national targets for renewable energy and combined heat and power generation, set by governments in response the 1997 Kyoto Protocol on climate change. In the USA, the main driver is that the transmission and distribution networks are operating close to full capacity in places, which creates an opportunity for new distributed generation. In the UK there are complex regulatory, commercial and technical factors that restrict the connection of such distributed generators. Three technical factors that can limit the capacity of a distributed generation project are fault level, thermal limits and voltage limits. Present network voltage design practice, suitable for passive networks with unidirectional power flow, restricts distributed generator capacity due to voltage limits, particularly on 11 kV networks. Adoption of a more active approach to 11 kV network voltage control could significantly increase the capacity of distributed generation. 2 Present 11 kV voltage control and distributed generation In the UK, 11 kV network voltage is controlled by tap-changers on the 33/11 kV transformers at primary IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004
substations. Tap-changers are generally controlled by automatic voltage control (AVC) relays to maintain the 11 kV bus-bar voltage within a dead band, as the substation load varies with time of day and season. AVC relays usually incorporate a time delay between detecting an out of range voltage and starting a tap-change operation, so that shortterm voltage fluctuations do not cause tap operations. In this way customers’ voltage is maintained within the statutory limits of 230 V+10%–6% and 11 kV76%. The winter peak load generally causes the maximum voltage drop along the 11 kV feeders supplied by the primary substation. The AVC target voltage, which is the centre of the dead band, is chosen to ensure that the minimum customer voltage at peak load is above the lower statutory limit. When a generator is connected to an 11 kV feeder, its active power export reduces the power flow from the primary substation and so reduces the voltage drop along the feeder. If the generator power export is larger than the feeder load, power flows from the generator to the primary substation and this causes a voltage rise between the primary substation and generator. Present network design practice is to limit the generator capacity to the level at which the upper voltage limit is not exceeded with maximum generation and minimum load. 3
For a distribution network state estimator, network topology and impedance data, information about customer loads and a few real-time measurements are required. The control block compares the voltage estimates with the voltage limits and alters the AVC relay target voltage to bring any out-of-range voltages within the limits. Figure 2 shows the voltage controller hardware configuration. The voltage controller is sited at an 11 kV primary substation. It uses local measurements, such as 11 kV busbar voltage and feeder current and measurements from remote terminal units, which are sited at key network nodes, such as a generator point of connection or a node at which large voltage variations are expected. The voltage controller and remote terminal units communicate for example by public data network or low power radio. The voltage controller output is connected to the AVC relay target voltage input. voltage controller
local measurements
The restrictions placed on generator capacity by present network design practice can be overcome if a more active approach to network voltage control is adopted. The following control actions could be taken to prevent a generator from causing an overvoltage: (i) decrease AVC relay target voltage (ii) increase generator reactive power import (iii) decrease generator active power export.
estimates
load data
Fig. 1
network data
control
AVC relay set point
limits
Voltage controller functional diagram
To ensure that all customer voltages are within statutory limits, the voltage controller requires an accurate knowledge of the voltage at each network node. Distribution networks have very few measurements, with 11 kV networks typically having measurements only at primary substations. State estimation provides a method for estimating the voltage at each node of the network from the available information. IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004
Fig. 2
4
4.1
Voltage controller hardware configuration
Statistical distribution state estimator
Method
4.1.1 Algorithm: For application in the 11 kV voltage
The combination of these control actions could significantly increase the capacity of generation that can be connected and hence the generator annual energy export [1]. The first control action enables the largest increase in generator energy export and is likely to have the lowest cost. As a simple approach to enabling an increase in generator energy export, a possible design for a controller for the AVC relay target voltage has been created. Figure 1 is a functional diagram of the voltage controller. There are two blocks: state estimation and control. The measurements, estimates and AVC relay target voltage, shown with black arrows, are real-time signals. The other inputs, shown with white arrows, are offline data.
state estimation
remote terminal unit
remote terminal unit
11 kV voltage controller
measurements
AVC relay
controller, a statistical distribution state estimator has been constructed, using the simplest applicable established techniques. These are drawn from transmission network state estimation and recent distribution network state estimation research [2–9]. A weighted least squares formulation is used, with the node voltage magnitudes and relative phase angles as state variables, as defined by (1) [2]. This formulation is used in transmission state estimation [3] and is applicable to radial and meshed networks, with bidirectional power flows. The solution of (1) is the set of state variables that minimises the difference between the measurements and the measurement values calculated from the state variables, weighted by the measurement accuracies: min
fx1 ;x2 ;xNs g
¼
J ðx1 ; x2 ; xNs Þ
Nm X ½zi fi ðx1 ; x2 ; xNs Þ 2 s2i i¼1
ð1Þ
Three types of function fi are used, as defined in (2)–(4). These relate the state variables, vi, vk, yi and yk to the measurements Pinj, Qinj and vi [2]: n X vi vk ðgik cos yik þ bik sin yik Þ ð2Þ Pinj ¼ k¼1
Qinj ¼
n X
vi vk ðgik sin yik bik cos yik Þ
ð3Þ
k¼1
vi ¼ vi
ð4Þ
Equation (1) is solved using the Newton–Raphson method, which is used in load flow and transmission state estimation 151
and is suitable for use with the low line inductive reactance to resistance ratios, e.g. X/R ¼ 0.5, found in distribution networks. Equation (5) is the resulting iterative vector equation, which is evaluated until Dx is close to zero [2]. Equation (5) is derived in the Appendix: 2 3 ½z1 f1 ðxÞ
6 ½z2 f2 ðxÞ 7 6 7 ð5Þ Dx ¼ ðH T R1 HÞ1 H T R1 6 ½z3 f3 ðxÞ 7 4 5 .. . For state estimation on a distribution network, generally Nm{Ns. A necessary condition for a weighted least squares state estimation algorithm to have a unique solution is NmZNs [2]. Pseudo measurements, derived from offline data, can be used to provide unmeasured quantities, so that NmZNs. This approach has been used in both transmission and distribution state estimators [2, 4–6]. In the case considered, the unmeasured quantities are Pinj and Qinj at load nodes and so pseudo measurements are used for these. The measurements are assumed to be independent normal random variables, with variance s2i being greater for pseudo measurements than for real-time measurements. The state variable estimates are normal, because the measurements are assumed to be normal. The algorithm estimates both the state variable expected values and variances. Equation (6) is used to calculate the state variable variances, as is done in transmission state estimation bad data detection [7]: C x ¼ ðH T R1 HÞ1
ð6Þ
Equation (6) is based on a linearisation of the functions fi(x) for small deviations of x from a particular value of x. The expected value of x is chosen as the point of linearisation and so (6) can be applied for small variances of x [8]. As (5) contains Cx, evaluating it yields both the expected value and the variance of x.
4.1.2 Voltage magnitude estimate variance: It is important to know the voltage magnitude estimate variance, because it affects the voltage range within which the voltage controller must keep all of the node voltage magnitude estimates. Figure 3 shows a normal node voltage magnitude estimate for two expected values, E1[vi] and E2[vi] and one variance s2vi . E1[vi] and E2[vi] are three standard deviations away from lower and upper voltage limits, vlower and vupper, respectively. This implies a 99.73% level of confidence that the actual voltage magnitude is between
3σvi
voltage vlower E1[vi ]
E2[vi ] vupper
Fig. 3 Normal node voltage magnitude estimates for one standard deviation and two expected values showing the voltage range within which the voltage controller must keep the node voltage magnitude estimates 152
4.1.3 Network model: A nine node 11 kV feeder model, as shown in Fig. 4, was constructed from real network data. The model consists of the feeder topology, line impedances and three types of measurements: real-time voltage magnitudes, real-time power injections and pseudo power injections. 1
primary substation
Fig. 4
5
loads
9
loads generator
11 kV feeder model
Winter maximum loads were used as the node power injection measurement expected values. Node voltage magnitudes were calculated by a load flow using the winter maximum loads and these were used as the node voltage magnitude measurement expected values. The standard deviation of each measurement was calculated as the product of one-third of the measurement accuracy and the expected value. Three accuracy classes were used: 0.5% of 11 kV for real-time voltage magnitudes, 3% of expected value for real-time power injections and 20300% of expected value for pseudo power injections. The load power factor was taken as 0.98 lagging and separate variances were calculated for active and reactive power measurements. The value of each measurement was calculated by adding to the expected value a normal random number, scaled by the measurement standard deviation.
4.2
probability 3σvi
vlower and vupper. If s2vi is decreased and the 99.73% level of confidence is maintained, E1[vi] and E2[vi] can be moved closer to the limits. This is desirable, as it allows a larger generator power export. Increasing the number of real-time measurements generally decreases s2vi , as real-time measurements are more accurate than pseudo measurements. However, it is desirable to use few real-time measurements, as they are expensive. An economic balance needs to be found between the increase in generator revenue enabled by increasing E1[vi]–E2[vi] and the cost of the real-time measurements needed for the increase. A study was therefore carried out to investigate the relationship between voltage magnitude estimate variance and measurement placement and accuracy.
Results
The statistical state estimation algorithm was used to calculate the node voltage magnitude expected values and variances for the nine-node feeder model. The results are shown in Figs. 5–7. The x-axis of each Figure is the distance along the feeder and each point is plotted at the position of a node. Nodes 1 and 2 are very close together. The y-axis of each Figure is the node voltage magnitude estimate range (abbreviated to ‘voltage range’), which corresponds to three standard deviations, given as a percentage of 11 kV. Figure 5 shows the voltage range for a single real-time voltage measurement at node 1 and various pseudo measurement accuracies. The voltage range is close to the real-time measurement accuracy of 0.5% at node 1 and increases with distance along the feeder, due to the less accurate pseudo measurements, which imply uncertainty in the power flow in the feeder. For pseudo measurement accuracies of 20–100%, the voltage range is less than 4%. IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004
12
300% 100% 50% 20%
voltage range, %
10
8
6
4
2
0 0
5
10
15
20
distance, km
Fig. 5 Node voltage magnitude estimate range against distance from the primary substation for different pseudo measurement accuracies
3.5 node 1 nodes 1 & 2 nodes 1 & 5 nodes 1 & 9 nodes 1,5 & 9
3.0
voltage range, %
2.5 2.0 1.5 1.0 0.5 0 0
5
10
15
20
distance, km
Fig. 6 Node voltage magnitude estimate range against distance from the primary substation for different voltage measurement placements 3.5 none node 1 nodes 1 & 2 nodes 1 & 5 nodes 1 & 9
3.0
voltage range, %
2.5 2.0 1.5 1.0 0.5 0
0
5
10
15
20
distance, km
Fig. 7 Node voltage magnitude estimate range against distance from the primary substation for different power measurement placements
For a pseudo measurement accuracy of 300% an estimate can still be made, but its range is wide. Figure 6 shows the voltage range for pseudo measurement accuracy of 100% and real-time voltage measurements at various nodes. Comparing the plots for a IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004
measurement at node 1 and for measurements at nodes 1 and 2, the addition of the second measurement reduces the voltage range at nodes 1 and 2 from 0.5–0.35%, which is smaller than the measurement accuracy. The second measurement increases the certainty in the voltage at these nodes. It does not significantly change the uncertainty in the power flow in the feeder and so does not significantly change the voltage range at the other nodes. The plot for measurements at nodes 1 and 5 shows the voltage range at these nodes to be less than the measurement accuracy and the voltage range between the two measurement nodes to be even smaller. This shows that the decrease in voltage range due to the two measurements is largest between the measurement nodes. The voltage range along the remainder of the feeder is determined by the less accurate pseudo measurements and so increases with distance along the feeder. The plot for measurements at nodes 1 and 9 shows the voltage range at these nodes to be slightly less than the measurement accuracy and to rise between the two measurement nodes. This is because the voltage drop between nodes 1 and 9 is known accurately, as it is determined by the real-time measurements, but the apportioning of this voltage drop between the branches that make up the feeder is not known accurately, as it is determined by the less accurate pseudo measurements. The voltage range is largest at the node where the effect of the pseudo measurements is largest. The reason for this not being halfway between the measurements, at node 5, is that approximately half the feeder load is at node 2. This measurement placement gives the smallest maximum estimate range for two voltage measurements. The plot for measurements at nodes 1, 5 and 9 shows an estimate range smaller than the measurement range at all nodes. Figure 7 shows the voltage range for pseudo measurement accuracy of 100%, a real-time voltage measurement at node 1 and real-time power measurements at various nodes. Comparing the plots for a power measurement at node 1 and no power measurement, the additional measurement has reduced the voltage range slightly. The measurement accurately determines the power flow between nodes 1 and 2, but because the large load at node 2 is determined by a pseudo measurement, the power flow in the remainder of the feeder still has a large uncertainty. Comparing the plots for a power measurement at node 1 and at nodes 1 and 2, the addition of the second power measurement reduces the voltage range along the whole feeder. This is because the two power measurements accurately determine the power flow in the remainder of the feeder. The plot for power measurements at nodes 1 and 5 shows a small decrease in voltage range, compared with that for a measurement at node 1. This is because the node 5 power injection measurement only slightly decreases the uncertainty in the power flow in the feeder. The plot for power measurements at nodes 1 and 9 shows a decrease in voltage range along the whole feeder, compared with that for a measurement at node 1. This is because the two measurements, being at the two ends of the feeder, accurately determine the power flow in the feeder. The node 9 power injection measurement accurately determines the power flow in the node 8 to 9 branch and the corresponding voltage drop. The voltage range at node 9 is therefore only slightly higher than at node 8. The following observations can be made from the study results. For 100% pseudo measurement accuracy and two real-time voltage measurements, the voltage range was less 153
than 1.5%. For 100% pseudo measurement accuracy, a real-time voltage measurement and two real-time power measurements, provided that the power measurements accurately determined the feeder power flow, the voltage range was less than 2%.
7
Load model
For each node power injection that is not measured, the statistical state estimation algorithm requires a pseudo measurement comprising an expected value and a variance. These can be calculated using a load model based on load profiles [4, 6, 9]. Load profiles describe patterns of electricity demand over time and are used to facilitate electricity trading [10]. A load-profile-based load model was constructed for the network model described above. The load model was constructed using UK Electricity Association generic load profiles and real network data. Each load profile consists of an active power value for each half-hour of a generic year for the national average customer of each of eight load classes. As the network data was from a rural network, all the customers belonged to one of two load classes: ‘Domestic Unrestricted’ or ‘Domestic with Economy 7’. The network data for each customer was its load class and the location of the distribution transformer to which it was connected. Each customer was assumed to be the national average. Node i active power injection for half hour h was calculated as follows: Pinj;h ¼
Nc 2 X X
Pu;w;h
ð7Þ
u¼1 w¼1
The sum of the load node active power injections calculated using the model was compared with four months’ actual half-hour metered primary substation and generator active power injection, neglecting line losses. The agreement between the modelled and actual power injections was not good. The assumption that each customer was the national average was the largest source of inaccuracy in the model. However, the state estimation study showed that a reasonable node voltage magnitude estimate range could be obtained with 100% pseudo measurement accuracy and so the load model was not further refined. 6
6.1
Control
6.1.1 Network model: The network shown in Fig. 8 was constructed using the same network data that was used for the state estimator and load model studies. The feeder to which the generator is connected is the same feeder as that shown in Fig. 4. The primary substation is located between nodes 1 and 2 and has one 33/11 kV 1 MVA transformer with a tap-changer. The model includes a representation of a modern AVC relay [11, 12]. The tap step was 1.43% and the AVC relay dead band vd was 3.0%. The network operating voltage limits are 11 kV+1.2%6%. The primary substation supplies four 11 kV feeders with the minimum summer load at 0.98 power factor. A synchronous generator, with power factor excitation control operating at unity power factor, is connected at node 7.
9
2
1
4
5
6
node node with measurement generator tap-changing transformer
3
~
Fig. 8
11 kV network model
Voltage magnitude measurements are taken at nodes 2, 6 and 7.
6.1.2 Algorithm: The inputs to the control algorithm are the maximum vmax and minimum vmin node voltage magnitudes. The control algorithm compares vmax and vmin with the upper vupper and lower vlower network voltage limits, respectively. If vmax 4 vupper or vmin o vlower, the control algorithm lowers or raises the AVC relay target voltage vref, respectively. vref is altered in steps of vd/2 within the range vupper–vd/24vref4vlower+vd/2. The AVC relay compares the 11 kV bus-bar voltage v2 and vref and if v24vref+vd/2 it instructs the tap-changer to tap down, so reducing v2, whereas if v2ovref–vd/2 it instructs the tap-changer to tap up, so increasing v2.
6.2
Results
The simulation results are shown in Figs. 9 and 10. Both xaxes show simulation time. The simulation used deterministic node voltages, with v7 being the maximum network voltage and v6 the minimum. vref was initially set at 10.967 kV.
Method
A control algorithm has been constructed for application in the 11 kV voltage controller. The algorithm was simulated on a network model to demonstrate the increase in generator capacity that it enables.
154
11
~
1000 900 active power, kW
5
10
800 700 600 500 400 300 200 100 0 1
2
3
4
T5
6
7
8
9
time t,s
Fig. 9
Simulated generator active power output against time
Figure 9 shows the generator active power output increasing linearly with time. Figure 10 shows v7 increasing until v74vupper at time t1 and the generator active power is 400 kW. The control algorithm reduces vref by vd/2, but the IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004
11.4
and the co-operation of Scottish Hydro-Electric Power Distribution Ltd.
network voltage limits V7
voltage, kV
11.2 11.0
9
V2 and V6 AVC relay dead band
10.6 10.4
t1 t2 T
10.2 0
1
2
3
4
5 6 time t, s
7
8
9
Fig. 10 Simulated voltage at nodes 2, 6 and 7 of 11 kV network model against time, showing operation of control algorithm
condition v24vref+vd/2 is not satisfied and so the AVC relay takes no action. The control algorithm waits 0.3 s for the tap-changer mechanism to operate and at time t2 the control algorithm again reduces vref by vd/2, as v74vupper. The condition v24vref+vd/2 is now satisfied and so the AVC relay bypasses its time delay and immediately instructs the tapchanger to tap down. After the 0.3 s time delay of the tap-changer mechanism, at time T the tap changes, so that v7ovupper. The voltage change causes a small transient in the generator active power output, shown in Fig. 9. The generator continues to increase its active power output until v7 ¼ vupper and the generator active power output is 860 kW. The condition v64vlower is satisfied throughout the simulation. The plots for v2 and v6 are close together as the network load and consequent voltage drop are small. The simulation demonstrates that the control algorithm enables a doubling of the distributed generator active power output on this network with minimum summer load. 7
Conclusions
Three algorithms have been created for use in a controller for the target voltage of AVC relays at 11 kV primary substations. The purpose of the voltage controller is to enable an increase in the distributed generator capacity that can be connected to the network. A statistical state estimation algorithm estimates the voltage magnitude at each network node supplied by the substation. The estimate accuracy is dependent on the number and placement of real-time measurements. An acceptable accuracy was obtained with one or two measurements on an 11 kV feeder model. The state estimator uses pseudo measurements for unmeasured loads. A load model was constructed using load profiles to calculate the pseudo measurements. The load model was inaccurate, but was not further refined, because it was found that acceptable state estimate accuracy could be obtained with inaccurate pseudo measurements. A control algorithm alters the AVC relay target voltage, based on the maximum and minimum node voltage magnitude estimates. The algorithm enabled the generator power export to be more than doubled on a network model. 8
References
10.8
Acknowledgments
The authors would like to acknowledge the support of EPSRC, Econnect Ltd. and Qualitrol Hathaway IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004
1 Liew, S.N., and Strbac, G.: ‘Maximising penetration of wind generation in existing distribution networks’, IEE Proc., Gener. Transm. Distrib., 2002, 149, (3), pp. 256–62 2 Wood, A.J., and Wollenberg, B.F.: ‘Power, generation, operation and control’ (Wiley, New York, 1996, 2nd Edn.), pp. 453–508 3 Wu, F.F.: ‘Power system state estimation: a survey’, Electr. Power Energy Syst., 1990, 12, (2), pp. 80–87 4 Lu, C.N., Teng, J.H., and Liu, W.H.E.: ‘Distribution system SE’, IEEE Trans. Power Syst., 1995, 10, (1), pp. 229–240 5 Baran, M.E., and Kelley, A.W.: ‘State estimation for real-time monitoring of distribution systems’, IEEE Trans. Power Syst., 1994, 9, (3), pp. 1601–1609 6 Li, K.: ‘State estimation for power distribution system and measurement impacts’, IEEE Trans. Power Syst., 1996, 11, (2), pp. 911–916 7 Monticelli, A.: ‘State estimation in electric power systems’ (Kluwer, Boston, 1999) 8 Handschin, E., Schweppe, F., Kohlas, C., and Fiechter, J.A.: ‘Bad data analysis for power system state estimation’, IEEE Trans. Power Appar. Syst., 1975, 94, (2), pp. 329–337 9 Ghosh, A.K., Lubkeman, D.L., and Jones, R.H.: ‘Load modelling for distribution circuit state estimation’, IEEE Trans. Power Deliv., 1997, 12, (2), pp. 999–1005 10 Allera, S.V., and Horsburgh, A.G.: ‘Load profiling for energy trading and settlements in the UK electricity markets’. Presented at DistribuTECH Europe DA/DSM Conference, London, UK, October, 1998 11 VA TECH REYROLLE ACP Ltd.: ‘Technical manual of Super/ MicroTAPP voltage control relays’ (UK, 2000) 12 Calovic, M.S.: ‘Modeling and analysis of under-load tap-changing transformer control systems’, IEEE Trans. Power Appar. Syst., 1994, 103, (7), pp. 1909–1915
10 Appendix: Derivation of weighted least squares iterative equation [2] An iterative method is required to solve (1), because (2) and (3) are nonlinear. The method used is to form the gradient of J(x) and then force this to zero using the Newton– Raphson method. The following is the general form of the equation that is to be solved: gdes gðxÞ ¼ 0
ð8Þ
Perturbing x by Dx, expanding g(x+Dx) about x in a Taylor’s series and ignoring all higher order terms gives: gdes gðxÞ ½g0 ðxÞ Dx ¼ 0
ð9Þ
, Dx ¼ ½g0 ðxÞ 1 ½gdes gðxÞ
ð10Þ
To solve (8), (10) is evaluated iteratively. An initial value for x is first guessed and Dx is evaluated. Then xnew ¼ x+Dx is evaluated and (10) is re-applied using xnew until either Dx becomes very small or g(x) comes close to gdes. To apply this method to (1), the gradient of J (x) is first formed: 3 @J ðxÞ 6 @x1 7 7 6 7 6 rx JðxÞ ¼ 6 @J ðxÞ 7 6 @x2 7 5 4 .. . 2
ð11Þ
155
2
@f1 @f2 @f3 6 @x1 @x1 @x1 6 6 @f1 @f2 @f3 6 6 @x @x @x 2 2 2 ¼ 26 6 6 @f1 @f2 @f3 6 6 @x3 @x3 @x3 4 .. .. .. . . . 2 3 ½z1 f1 ðxÞ
6 ½z2 f2 ðxÞ 7 6 7 6 7 6 ½z3 f3 ðxÞ 7 4 5 .. .
32 1 . . . 7 6 s2 1 76 76 6 7 76 76 76 76 76 76 56 4
3 1 s22 1 s23 ..
3 ½z1 f1 ðxÞ
6 ½z2 f2 ðxÞ 7 6 7 ¼ 2H T R1 6 ½z3 f3 ðxÞ 7 4 5 .. .
and R is the diagonal matrix of measurement covariances: 2 2 3 s1 6 7 s22 6 7 2 R¼6 ð15Þ 7 s3 4 5 .. .
7 7 7 7 7 7 7 7 7 7 7 5
To minimise J(x), rxJ(x) is set equal to zero and the Newton–Raphson method, as shown in (10), is applied: @rx JðxÞ 1 ½rx JðxÞ
ð16Þ Dx ¼ @x
.
ð12Þ
The Jacobian of rxJ(x) is calculated by treating H as a constant matrix: 8 2 39 ½z1 f1 ðxÞ > > > > > = 6 ½z2 f2 ðxÞ 7> @rx JðxÞ @ < 7 T 1 6 ð17Þ ¼ 2H R 6 ½z3 f3 ðxÞ 7 4 5> @x @x > > > > > . ; : ..
ð13Þ
¼ 2HT R1 H ð18Þ 8 2 39 ½z1 f1 ðxÞ > > > > > > = < 6 ½z2 f2 ðxÞ 7 1 T 1 1 6 7 T 1 ð19Þ Dx ¼ ðH R HÞ 2H R 6 ½z3 f3 ðxÞ 7 > 4 5> 2 > > > > .. ; : . 2 3 ½z1 f1 ðxÞ
6 ½z2 f2 ðxÞ 7 6 7 Dx ¼ ðH T R1 HÞ1 H T R1 6 ½z3 f3 ðxÞ 7 ð20Þ 4 5 .. .
2
where H is the Jacobian of f (x): 2 @f1 @f1 6 @x @x 2 6 1 6 @f2 @f2 6 6 H ¼ 6 @x1 @x2 6 @f3 @f3 6 6 @x1 @x2 4 .. .. . .
156
@f1 @x3 @f2 @x3 @f3 @x3 .. .
3 ...7 7 7 7 7 7 7 7 7 5
ð14Þ
IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004
