Monitoring, protection and fault location
in power distribution networks
using system-wide measurements
A thesis submitted for the degree of
PhD in Engineering Sciences
Pierre Janssen
Pierre Janssen
Thesis Director:
Prof. Jean-Claude Maun
President of the committee:
Prof. Pierre Mathys
Members of the committee:
Prof. Johan Gyselinck
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Abstract
This work takes place in the context of distribution grids with high level of distributed generation, for example in microgrids. With high level of distributed generation, it has been shown that selective, fast and sensitive network protection is expected to be more difficult. Furthermore, during system
restoration, the accurate fault location could be more challenging to assess, thereby increasing the average outage duration.
Thanks to cost reductions and improvement of information and communication technologies, future distribution networks will probably have advanced communication infrastructures and more
measurement devices installed in order to manage the increasing complexity of those networks, which is primarily caused by the introduction of distributed generation at the distribution level.
Therefore this thesis investigates how the monitoring, protection and fault location functions can be improved by using system-wide measurements, i.e. real-time measurements such as synchronized voltage and current measurements recorded at different network locations. Distributed synchronized measurements bring new perspectives for these three functions: protection and fault location are usually performed with local measurements only and synchronized measurements are not common in monitoring applications. For instance, by measuring distributed generators infeed together with some feeder measurements, the protection is expected to be more sensitive and selective and the fault location to be more accurate.
The main contribution of this work is the use of state estimation, which is normally only used for network monitoring, for the protection and the fault location.
The distribution system state estimation is first developed using the classical transmission system approach. The impact of the placement of the measurement devices and of a relatively low measurement redundancy on the accuracy, on the bad data detection and on the topology error identification capabilities of the estimator are discussed and illustrated. This results in
recommendations on the placement of the meters.
Then, a backup protection algorithm using system-wide measurements is presented. The coherence of the measurements and the healthy network model are checked thanks to a linear three-phase state estimation. If the model does not fit to the measurements and if the estimated load is too high or unbalanced, a fault is detected. The advantages of the method are that the voltage measurement redundancy is considered, improving the detection sensitivity, and that load models may be considered in the algorithm, avoiding the need to install measurement devices on every line of the network. Finally, two new impedance-based fault location algorithms using distributed voltage and current recordings are proposed. By defining statistical errors on the measurements and the network
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Acknowledgements
I would like to thank Prof. Jean-Claude Maun for giving me the opportunity to work on this research. I want to express my gratitude for the freedom he gave me and the trust he put on me during these three years.
I am also grateful to Dr. Tevfik Sezi for his support and advice, especially for reorienting the topic of the thesis at the right time. His high level view of power systems truly contributed to this document. I would like to thank all the colleagues of the department for the interesting discussions and great working atmosphere. A special thanks to Matthieu Loos for the technical discussions and review of this document.
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Contents
Chapter 1.
Introduction ... 1
1.1 Context ... 1
1.2 Approach of the thesis ... 4
1.3 Contributions and structure of the thesis ... 4
1.4 Publications ... 5
Chapter 2.
Distribution system state estimation ... 7
2.1 Introduction ... 7
2.1.1 State estimation in transmission systems ... 8
2.1.2 State estimation in distribution systems ... 9
2.2 State estimation ... 10 2.2.1 Problem statement ... 11 2.2.2 Solution ... 12 2.2.2.1 Linear systems ... 13 2.2.2.2 Nonlinear systems ... 13 2.2.3 Observability ... 14
2.2.4 Accuracy of the state estimate ... 14
2.3 Measurement model ... 15
2.3.2 Synchrophasors ... 17
2.3.3 Power measurements ... 17
2.3.4 Voltage magnitude measurements ... 18
2.3.5 Unsynchronized phasors ... 19 2.3.6 Pseudo-measurements ... 21 2.3.6.1 Load forecasting ... 22 2.3.7 Virtual measurements ... 24 2.4 Illustrations ... 24 2.4.1 Benchmark network ... 24 2.4.2 Simulation results ... 25 2.4.2.1 Measurement set ... 25 2.4.2.2 Convergence characteristic ... 27 2.4.2.3 State estimate ... 27
2.4.2.4 Visualization on a map of the network ... 29
2.4.2.5 Impact of uncertain flow direction ... 30
2.5 Measurement placement impact on the state estimation accuracy ... 31
2.5.1 Accuracy metrics ... 32
2.5.2 Simulation results ... 33
2.5.3 Metering placement to maximize the state estimate accuracy ... 36
2.6 Bad data detection and identification ... 37
2.6.1 Redundancy analysis ... 38
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2.6.3 Leverage of the measurements ... 39
2.6.3.1 Computation of hat matrix ... 40
2.6.4 Detection and identification method ... 40
2.6.4.1 Chi square distribution test ... 40
2.6.4.2 Normalized residual test ... 41
2.6.5 Illustrations ... 42
2.6.5.1 Redundancy of the measurement sets ... 42
2.6.5.2 Simulations of bad data ... 43
2.6.5.3 Display of the residuals ... 44
2.7 Topology error detection and identification ... 46
2.7.1 State of the art ... 47
2.7.2 Method ... 48
2.7.3 Illustration ... 49
2.7.3.1 Full example with the base measurement set... 49
2.7.3.2 Impact of the flow repartition ... 51
2.7.3.3 Metering set impact ... 52
2.8 Measurement system design ... 52
2.8.1 Design criteria and meter placement impact summary ... 53
2.8.1.1 Observability ... 53
2.8.1.2 State estimate accuracy ... 54
2.8.1.3 Bad data detection capabilities ... 54
2.8.1.4 Topology error ... 55
2.8.1.5 Robustness to the loss of measurements ... 55
2.8.2 Measurement placement rules ... 55
2.9 Conclusions ... 57
Chapter 3.
Protection using system-wide measurements ... 59
3.1 Introduction ... 59
3.2 Protection challenges in distribution networks with distributed generation ... 60
3.2.1 Conventional protection of distribution network ... 60
3.2.2 Impact of distributed generation on conventional distribution network protection ... 61
3.2.3 Protection issues in microgrids ... 63
3.2.4 Summary of the protection challenges ... 67
3.2.5 Solutions proposed in the literature ... 68
3.2.5.1 Protection based on local measurements only ... 68
3.2.5.2 Adaptive protection ... 68
3.2.5.3 Limited communication with neighboring relays ... 69
3.2.5.4 Measurements coming from other locations ... 70
3.3 Protection algorithm ... 70
3.3.1 Scope of the proposed protection ... 70
3.3.1.1 Hypotheses ... 71
3.3.2 Summary of the method ... 72
3.3.3 Measurements pre-processing ... 75
3.3.4 State estimation for fault detection ... 75
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3.3.4.2 Selection of the state variables and measurement equations ... 76
3.3.5 Fault detection ... 77
3.3.5.1 Load forecast error detection ... 78
3.4 Simulation results ... 80 3.4.1 Test system ... 80 3.4.2 Measurement set ... 82 3.4.3 Selectivity ... 83 3.4.3.1 Protection zone 2 ... 83 3.4.3.2 Protection zone 1 ... 85 3.4.4 Robustness... 87 3.4.5 Sensitivity ... 88
3.4.5.1 Impact of the fault current level... 88
3.4.5.2 Impact of the fault location ... 90
3.4.5.3 Impact of the measurements weights ... 91
3.5 Unobservable protection zones ... 92
3.5.1 Causes of unobservability ... 93
3.5.2 Dealing with a lack of voltage measurement... 93
3.5.3 Illustrations ... 95
3.6 Topology errors ... 98
3.6.1 Method ... 98
3.6.2 Illustrations ... 101
3.6.2.1 Test system and fault isolation rules ... 101
3.6.2.2 Test results ... 103
3.7 Practical implementation ... 105
3.7.1 Technological implementation ... 105
3.7.1.1 Structure of computation ... 106
3.7.1.2 Implementation of the intelligent electronic devices ... 107
3.7.1.3 Communication infrastructure ... 107
3.7.1.4 Synchronization of the measurements ... 108
3.7.2 Measurement set selection... 108
3.7.3 Parameters of the algorithm ... 109
3.7.4 Update of the pseudo-measurement parameters ... 110
3.8 Conclusions ... 113
Chapter 4.
Fault location using system-wide measurements ... 115
4.1 Introduction ... 115
4.1.1 Fault location in transmission systems ... 116
4.1.2 Fault location in distribution systems ... 118
4.2 Principle of the proposed fault location algorithms ... 121
4.2.1 Measurements and data needed ... 121
4.2.2 Stages of the fault location algorithms ... 122
4.3 Fault location by transfer and aggregation of the measurements ... 125
4.3.1 Estimation of faulted line terminal voltages and currents ... 125
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4.3.2.1 Single-ended fault location ... 128
4.3.2.2 Synchronized two-ended fault location ... 132
4.3.2.3 Unsynchronized two-ended fault location ... 134
4.3.2.4 Confidence interval of the fault distance estimate ... 135
4.3.3 Fault location candidates selection ... 136
4.4 Fault location using state estimation ... 136
4.4.1 Computation of faulted line terminal voltage and currents ... 136
4.4.2 Fault distance calculation ... 138
4.4.3 Fault location candidates selection ... 139
4.4.4 Comparison of the two fault location algorithms ... 139
4.5 Illustrations ... 139
4.5.1 Test system ... 140
4.5.2 Automated fault location ... 141
4.5.2.1 Faulted line identification ... 141
4.5.2.2 Fault type identification ... 145
4.5.3 Synchronization of the measurements during the fault location... 146
4.6 Fault location accuracy ... 147
4.6.1 Sources of fault location errors ... 148
4.6.1.1 Modeling errors ... 148
4.6.1.2 Measurement and parameter errors ... 149
4.6.1.3 Impact of the fault conditions ... 150
4.6.2 Assessment of a confidence interval of the fault distance estimate ... 150
4.6.3 Simulation results ... 151
4.6.3.1 Confidence interval of the fault location estimate ... 152
4.6.3.2 Sensitivity analysis ... 154
4.6.4 Discussion about the meter placement impact ... 161
4.7 Topology errors ... 162
4.7.1 Method ... 163
4.7.2 Illustrations ... 164
4.7.2.1 Examples with voltage measurement redundancy ... 165
4.7.2.2 Examples without voltage measurement redundancy ... 167
4.8 Conclusions ... 169
Chapter 5.
Conclusions ... 171
5.1 Summary, conclusions and contributions ... 171
5.2 Proposals for future work ... 173
References ... 177
Appendix A
Condition number of a matrix ... 187
Appendix B
Parameters of the CIGRE MV benchmark network ... 188
Appendix C
Additional simulation results of the protection algorithm ... 189
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Chapter 1.
Introduction
1.1 Context
Concerns about the environment, the need to become energy-independent and the liberalization of electricity markets are causing a massive introduction of distributed generation in electrical
distribution networks. Indeed, renewable generation such as wind turbines or photovoltaic panels and embedded generation such as co-generating plants are by nature distributed and are often being connected at the distribution level rather than at the transmission level.
Another change in distribution networks is the introduction of new types of loads whose behaviors are difficult to forecast: loads responsive to the electricity price or to the availability of renewable energy (demand side management), and electric vehicles.
The massive introduction of distributed generation and these new loads create stresses on distribution networks. For example, distributed generation may disturb the voltage level of the network, overload network components or disturb protective relays (Walling et al., 2008). The difficulty to forecast the load and the intermittency of renewable generation is also a difficulty for the daily network operation. In the meantime, customers still expect a very high reliability of power supply.
Figure 1-1 Sample microgrid (Chaudhuri)
An example of network with high distributed generation level is the microgrid. Microgrids are networks that have the possibility to island from the main grid, for example in case of a transmission system outage (Hatziargyriou et al., 2007), and are then only supplied by distributed generation. The CIGRE C6.22 definition of microgrids is: “Microgrids are electricity distribution systems containing loads and distributed energy resources, (such as distributed generators, storage devices, or
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Therefore several operation, control and protective functions could have to be adapted to allow the connection of more distributed generation. For microgrids, totally new concepts could be needed to provide the appropriate quality of supply.
In this thesis, three functions are studied: the monitoring, the protection and the fault location in power distribution networks with a high level of distributed generation.
Firstly, because loads and generation patterns are more uncertain, distribution networks would need to have a better monitoring of their actual state. Indeed, a better network monitoring is the first stage before any other network management functions, such as voltage control or network reconfiguration, can be performed.
Secondly, the protection of networks with high penetration of distributed generation is expected to be more complicated because distributed generation is changing the fault behavior of distribution networks. These sources are distributed, which makes the repartition of the current in the network more complex during faults than in a passive network. In some areas, for example in islanded microgrids, the total fault current will also be closer to the load current because the distributed generators have lower short-circuit power than large power stations; especially if the distributed generator is interfaced through an inverter that limits considerably its contribution to the fault current. If the protection practices are not adapted, it will result in a loss of sensitivity, selectivity or slower fault clearing time and hence to a reduction of the system reliably.
Thirdly, for the same reasons, the accurate fault location could be more complex to assess. In case of a power system disturbance, the main focus of the utility is to keep the duration of the interruption as short as possible. This requires fast identification of the fault location, isolation of the defective components and power supply restoration. Lower fault level and distributed sources make the fault location more difficult to assess; which would delay the system restoration and thus reduce the reliability of the network.
In a microgrid context, the three functions discussed above are especially difficult to tackle and new concepts need to be proposed.
Trends in power distribution systems
With the development of information and communication technologies (ICT), relatively powerful and inexpensive computers are now available. Modern telecommunication systems are reliable, have high bandwidth and their price is constantly decreasing. Therefore, intelligent and communicating devices could be used in distribution systems to help the utilities to keep or improve the reliability and the quality of the power supply.
This is often denoted as part of the smart grid concept. Basically, the smart grid denotes the use of modern technology in power networks in order to manage the networks more efficiently. In Europe, they are strongly related to the technology enabling a higher penetration of distributed generation in a more cost effective way than, for example, by building new lines.
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control intelligence, monitoring abilities and the capability of extensive communications directly to a SCADA system.” (Strauss, 2003). These IEDs could therefore be used for several functions: network monitoring, network protection, disturbance recording, power quality measurement, control switching devices (circuit-breakers, sectionalizers), etc.
In power systems, accurate time synchronization of measurements taken at remote locations is now also possible and will probably be widely used in future distribution networks, either with a GPS signal (Phasor Measurement Unit (Phadke, 1993)), or via wires by measuring the communication delays (e.g. with the IEEE 1588 standard (Fodero et al., 2010)). An accurate synchronization of the measurement reduces the difficulty of processing measurements taken at different operating points and allows the use of phasors in a common frame.
Another change is that the networks could be operated in more meshed topologies than today’s radial ones in order to incorporate more distributed generation.
The differences between transmission and medium voltage distribution network will therefore probably fade away: the topology will be more complex and the level of monitoring and automation will increase.
These evolutions have several implications for the three functions studied in this thesis.
First of all, with more measurements, a better network monitoring is possible. The state estimation function, which is common in transmission systems, would be introduced in distribution networks. State estimation is essential to provide a good evaluation of network state but also to detect bad input data or detect topology errors for example. Moreover, because most of the measurement devices will be accurately synchronized, there will be a revolution in the network monitoring approach as
synchronized measurements provide a direct state measurement instead of providing a means to estimate it.
Secondly, another possible evolution is to expect that the future protection systems will be more based on communications. Protection is traditionally based on local measurements only. But the introduction of distributed generation and reduced fault current level create a need for communicating protection systems. This evolution is also possible with the experience gained with line differential protection over the last two decades. Very selective and sensitive protection is indeed possible with
communications but the security and dependability of the protection system has to be analyzed. Moreover, if the fault current is getting closer to the load current, such as in an islanded microgrid, there is a change in the objectives of the protection system: the fault must be primarily eliminated for the safety of people and to provide high system reliability whereas avoiding damage to the
components of the network (e.g. cables or transformers) is less a concern. This could have implications for instance on the maximum fault clearance times.
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1.2 Approach of the thesis
In this context, the objective of this thesis is to develop algorithms using measurements distributed in the network, also called system-wide measurements, for the monitoring, protection and fault location in networks with high penetration of distributed generation. The measurements could be retrieved through a fast and reliable telecommunication network from intelligent electronic devices located at primary or secondary substations, distributed generator’s locations or reclosers.
The approach of this work is that any measurements taken at different locations or a priori information about the network can be used to provide a better network state monitoring, more sensitive and
selective protection and more accurate fault location. For example, the distributed measurements could be used to take into account the distributed generator’s infeed during normal situations or during faults. The algorithms proposed in this work are all based on a phasor analysis, i.e. based on the 50 Hz component of the signals. The final goal is to allow for more distributed generation connection, to provide solutions for microgrids with low fault current level and to improve the reliability of the power supply.
However, because the cost of the system should remain low, it is not realistic to assume that sensors are placed everywhere in the network. Hence there will be some uncertainties that must be considered in these algorithms, mainly the exact load repartition.
A primary protection based on local measurements would be still necessary but is not studied here. The protection using system-wide measurements would be used as a backup protection that would have to selectively clear complex fault scenarios that would be more likely to happen in networks with important distributed generation.
This thesis aims at developing the core of the algorithms. The problem studied in this thesis is to develop the core of the algorithms. The practical telecommunication, synchronization and calculation infrastructure are outside the scope of this work. Moreover, the signal processing, such as the phasor calculation, is not studied here since this is well documented in the literature, for example in protective relays (Schweitzer et al., 1993), in PMUs (Phadke, 1993) or for offline disturbance analysis (Bollen et al., 2006).
1.3 Contributions and structure of the thesis
The main contribution of the thesis is the use of a state estimation framework for the fault analysis. State estimation is the algorithm that processes the redundant network measurements (e.g. voltage or current measurements) to compute the most likely network state. This algorithm is generally only used for the monitoring of power systems in normal conditions.
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synchronized measurements resulting in a linear state estimation allowing fast calculations that are necessary for a protection function.
The thesis is structured in three main chapters.
The first chapter deals with the monitoring of distribution networks. The chapter presents a state estimation algorithm using the node voltages as state variables. Compared to transmission system state estimation, the developed estimator is three-phase which is necessary in order to manage the load unbalance present in distribution. The main contribution of this chapter is the discussion of the implication of working with a low measurement redundancy compared to transmission system state estimation. The limitations of the standard detection and identification of bad data of the topology error identification methods are explained and illustrated. The impact of the placement of
measurements on the state estimation performances are illustrated with simulations. These simulations result in some simple metering placement rules that could be used to design a measurement system for new networks or to reinforce existing measurement sets. A new algorithm dealing with three-phase unsynchronized phasor measurements is also proposed.
The second chapter presents a new backup protection algorithm using system-wide measurements. This backup protection would rely on the system-wide measurements to provide a selective and sensitive backup to a primary protection based on local measurements. The method computes state estimation with the measurements and the network model which checks automatically Kirchhoff’s laws on a given protection zone. The advantages are that measurement devices do not have to be placed on every line of the network; it considers the voltage measurements and its redundancy, and the load current angle and unbalance is taken into account. It is shown that, with additional processing and communications, it is sometimes possible to diagnose topology errors and load forecast errors as well. The third chapter presents two new impedance-based fault location algorithms dealing with system-wide measurements. Another contribution is the calculation of a confidence interval of the fault location estimate using a linearization of the fault distance equation or using Monte Carlo simulations. This calculation is then used with simulated faults in order to estimate the possible fault location accuracy and the factors influencing the accuracy of the proposed algorithms. A method to identify topology errors during the fault location is also proposed.
1.4 Publications
The following publications have been made during the research work. Papers presented at international conferences:
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Pierre Janssen, Tevfik Sezi and Jean-Claude Maun, Distribution System State Estimation Using Unsynchronized Phasor Measurements, Innovative Smart Grid Technologies (ISGT Europe), 3rd IEEE PES International Conference and Exhibition on, pp. 1-6, Berlin, 2012 Pierre Janssen, Tevfik Sezi and Jean-Claude Maun, Meter Placement Impact on Distribution
System State Estimation, 22nd International Conference on Electricity Distribution (CIRED), pp. 1-4, Stockholm, 2013
Patent:
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Chapter 2.
Distribution system state estimation
2.1 Introduction
Distribution system operators are facing technical challenges caused by the integration of distribution generation. Intermittent distributed generation may temporarily cause voltage problems or overload lines and transformers (Walling et al., 2008). Another new source of stress on distribution networks are new types of loads difficult to forecast such as electric vehicles, storage systems or dynamic loads responsive to the electricity price or to other criteria (‘demand side management’).
Traditionally, few monitoring devices were installed in distribution systems. This approach was acceptable because the load was relatively predictable and there was no significant penetration of distributed generation, hence few control actions were required during daily network operation. But with these new stresses on the distribution network, network operators need an enhanced monitoring system to enable the management of the network closer to its limits, i.e. to stay within acceptable operating conditions, to maintain the quality of supply and to improve the network efficiency (Grenard et al., 2011).
For example, overvoltage problems or component overloading caused by distributed generators must be first detected before the appropriate control actions, such as voltage and Var control or optimal network reconfiguration, can be decided.
Other examples of network operation tasks that benefit from an enhanced monitoring system are the losses minimization, outage detection and network reconfiguration after an outage (Hoffman, 2006). In a planning perspective, the network operator will have a better overview of the network utilization which will ease the planning of the reinforcements of the network (Vinter et al., 2009).
State estimation is therefore a necessary tool in modern distribution system control centers to manage unpredictable loads and distributed generation. State estimation is the algorithm that will process the measurements gathered on the network to obtain its most likely operating condition, to detect and eliminate erroneous measurements and to identify network configuration errors.
This chapter presents an algorithm for the state estimation of distribution systems. Another major contribution of this thesis is the use of state estimation for protection and fault analysis functions. This chapter introduces thus the state estimation concepts necessary for the understanding of the next two chapters as well.
The chapter is organized as follow. After a state of the art of state estimation in transmission and distribution systems, the distribution system state estimation algorithm is presented. Next, the bad data processing and topology error identification methods are explained and illustrated. If the measurement infrastructure is limited, the resulting potential limitations of these functions are highlighted.
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proposes recommendations for the meter placement to obtain the state estimation performances required to manage the network.
2.1.1 State estimation in transmission systems
A state estimator processes the real-time measurements to provide useful on-line information about the current state of the network for the energy management system (EMS) (Figure 2-1).
Examples of EMS functions are the monitoring operational constraints such as the voltage limits or lines loading, computation of losses, online security assessment, etc. The network state monitoring also helps to take decision in the long term, for instance to decide future grid reinforcements.
Figure 2-1 Relation between state estimation and energy management functions
State estimation is a mature technology in transmission systems (Monticelli, 1999), (Abur et al., 2004). State estimation involves several processing stages shown in Figure 2-1.
The state estimation is based on a bus/branch network model. Therefore, the topology processor establishes first a logical bus/branch model from the physical network connectivity using switch or circuit breaker status and analog measurements.
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When erroneous measurements are still present after the prefiltering stage, if the measurement redundancy is large enough, they are detected and eliminated during state estimation by bad data processing functions.
In transmission systems, the measurement redundancy index, defined as the ratio of the number of measurements to the number of state variables, is generally very high (between 2 and 3), thereby making the estimator accurate and robust.
Recent developments in transmission state estimation concern for example the use of phasor
measurement units (PMUs) which provide synchronized measurements at higher sampling frequencies than conventional measurements or the application on very large power systems (Gomez Exposito et al., 2011), (Huang et al., 2012).
PMUs allow a direct monitoring of the state variables. Therefore, network monitoring is moving from state estimation to direct state measurement. But PMUs are synchronized by GPS signals that are not 100% reliable because the GPS signal can be easily jammed (Fodero et al., 2010). Therefore, state estimation function is still needed. However, another means of measurement synchronization is via wires: the Precision Time Protocol (IEEE standard 1588) is a protocol used to synchronized clocks throughout a computer network which provides very accurate (accuracy of less than 1 µs) and reliable synchronization (Fodero et al., 2010). The future state estimation will thus change of paradigm. For example, line parameters and topology errors will be a more important source of error than in today’s state estimation.
2.1.2 State estimation in distribution systems
As explained in the introduction, state estimation is becoming necessary in distribution to manage unpredictable loads and generation. The characteristics of distribution networks are first explained below and a short review of the state estimation methods proposed for distribution is given. Compared to transmission networks, distribution networks differ in the following characteristics (Lehtonen, 2003):
Their topology is radial or weakly meshed.
The load is unbalanced. In the United States single phase lateral feeders and loads are
common practice (Haughton et al., 2012). Therefore, single phase equivalents do not contain enough details and three-phase estimators are needed.
The lines have high r/x ratios, making a decoupling of the active and reactive subproblems not possible. For example, the ratio may be more or less equal to one for medium voltage
overhead lines and will be generally greater than one for underground cables. Some lines segments may be very short (lower than hundreds of meters in an urban environment). Because of the lower number of customers, these networks have very limited measurement
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The loads are more difficult to forecast than in transmission networks, because the number of customers per MV/LV transformer is smaller, resulting in greater randomness. Further, with the introduction of electric vehicles or demand response, the load randomness could become even higher.
These characteristics explain that distribution system estimators may be very different from transmission system state estimators.
When field-measurements are not available at all, the state is approximated with a load flow solution. A ladder-type load flow algorithm can be used because of the weakly meshed topologies
(Zimmerman, 1995). Confidence bounds on the estimated variables can be obtained by using worst case load flow scenarios or by using probabilistic load flows (Valverde Mora, 2012).
In order to take advantage of the real-time measurements, load allocation algorithms are widely used, e.g. (Hoffman, 2006), (Roytelman, 2006). The principle of these algorithms is to first scale the load estimates so that the load estimate fits the feeder flow measurements. The node voltages and the losses are then computed from these load estimates. The load estimates are next updated until the losses plus the load fit the feeder measurements. The limitations of load allocation algorithms are that they cannot handle voltage measurements and have difficulties to handle the meter imprecision. Some authors proposed to use the branch currents as state variables instead of the node voltages (Baran et al., 1995), (Deng et al., 2002), (Wang et al., 2004). These state estimators are called branch-current-based state estimators and can be understood as an extension of load allocation algorithms. They are said to be numerically robust because they can handle easily both very short and long lines. But these estimators are primarily designed for radial networks and may be difficult to apply to weakly-meshed networks. Indeed, with meshed topologies, taking all the branch currents as state variables will give redundant information. To understand this, for a network containing N nodes and n
b
N branches, there would be N current state variables, but since the network is meshed we will haveb n
b N
N , thus the number of voltage state variables would be lower which means that the branch currents contain redundant information. This topology limitation goes against the current trend to operate distribution networks e.g. in closed-loop rather than radially because of the reduced losses and reduced voltage problems (Nikander et al., 2003).
Lastly, voltage-based distribution system state estimators use the formulation of transmission system state estimators, see e.g. (Baran et al., 1994), (Li, 1996), (Lu et al., 1995), (Thornley et al., 2005), (Singh et al., 2009), (Chilard et al., 2009), (Haughton et al., 2012), (Therrien et al., 2013).The advantages of this type of estimators is that it is not limited by topology or by the measurement types, that it has good mathematical foundations and that it is well documented in the literature. A disadvantage is that they are more difficult to understand compared to load allocation algorithms.
2.2 State estimation
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2.2.1 Problem statement
Considering the advantages of the transmission system approach, the state estimation is formulated as a weighted least squares optimization problem and the state of the network is represented by the phase node voltages. Therefore, the state of the network, x , is represented by a vector of the three-phase complex voltage at every node:
T 2
1, , ]
[V V VN
x (2.1)
where all the quantities are three-phase complex vectors represented in abc quantities:
T c i b i a i i V V V
V [ , , ] , N is the number of nodes of the network and Tdenotes transposition. The voltage is expressed in rectangular (or Cartesian) form.
The state of the system is estimated from a set of redundant measurements taken on the network, e.g. voltage or current flow measurements. The state estimation approach assumes that the relation between the measurements and the state variables can be modeled with:
e x h z ( )
(2.2)
where zis the measurement vector (size m x 1), (.)h is a set of nonlinear state-dependent
measurement functions and e is the unknown measurement error vector whose mean value is equal to zero (E(ei)0).
The state is estimated such that it fits best to the redundant measurements available. The weighted least squares estimator minimizes the weighted sum of squared residuals:
)) ( min( arg ˆ J x x with J(x)rTR1r subject to 0 ) ( ) ( x c x h z r (2.3)
where xˆ is the state estimate,J(x) is the weighted sum of squared residuals – also called the cost or the objective function, ris the residual vector (size m x 1), R is the covariance matrix of the
measurement errors (size m x m). The constraintsc(x) (size nc x 1) are used for exact equations (virtual
measurements), e.g. zero injection nodes. This least squares optimization problem will result in the most likely state estimate (maximum likelihood estimator) if the measurement noise follows a normal distribution with covariance matrixR.
Numerical accuracy
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In voltage-based state estimation, the problem can be indeed badly conditioned in the following situations (Monticelli, 1999):
Use of a large number of injection measurements Presence of both long and very short lines
Simultaneous use of large and low weighting factors for different equations
The two first conditions are often met in distribution systems which can lead to bad condition number. The third factor is here limited because of the use of equality constraints instead of highly weighted equations for the so-called virtual measurements c(x).
If the problem is ill-conditioned, small errors in the input parameters will give significant errors in the output because of round off errors. The numerical accuracy can be quantified with the condition number of the solution matrix. More details about the link between condition number and numerical accuracy are given in Appendix A.
If lines with very low impedance are present, e.g. short cable line segments, numerical problems can thus be present. The standard technique to overcome this problem is to remove these small branches and to merge the neighboring buses or to set the lines impedance to a larger value (Therrien et al., 2013).
2.2.2 Solution
The method of Lagrange multipliers is used to solve the constrained minimization problem (2.3). The Lagrange function of this problem is defined by:
)) ( ( ) ( ) ( 2 1 x h z r x c x J L T T (2.4)
with and the Lagrange multipliers associated to the equality constraints.
The state estimate xˆ must satisfy the first order optimality conditions of the Lagrange function:
0 0 0 0 L L r L x L 0 ) ˆ ( 0 ) ˆ ( 0 0 ) ˆ ( ) ˆ ( 1 x h z r x c r R x H x CT T (2.5)
where H(x)is the Jacobian of h(x) with respect to the state variables (size m x n, with n the number of state variables) and C(x)is the Jacobian of c(x) with respect to the state variables (size nc x n).
The condition number of this system of equations is further improved by scaling and by:
13 0 ) ˆ ( 0 ) ˆ ( 0 ) ˆ ( ) ˆ ( 1 x c x h z R x H x C s s T s T (2.6) with and . 2.2.2.1 Linear systems
In the particular case of linear measurement equations and constraints, i.e.:
x C x c x H x h ) ( ) ( (2.7)
The state is calculated via the direct solution of the following linear system of equations:
0 0 ˆ 0 0 0 0 1 z x C R H C H s s T T (2.8) 2.2.2.2 Nonlinear systems
In general, the set of measurement equations is nonlinear. Therefore, the nonlinear termsh(x) and )
(x
c in (2.6) are linearized around a nominal value x and the variations of k H(x)and C(x)are neglected, leading to an iterative solution known as the Newton-Raphson method. At iteration k, the following is computed: ) ( ) ( 0 0 0 ) ( 0 ) ( ) ( ) ( 0 1 k k s s k k k k T k T x c x h z x x C R x H x C x H (2.9) k k k x x x 1 (2.10)
In equation (2.9), the coefficient matrix is called the Hachtel matrix. The stopping criterion for these iterations is the stabilization of the state variables, i.e. at iteration k the following check is made:
) ( max ik 1 k i i x x (2.11)
For instance a value of 10-4 pu for can be chosen. The result of these iterations is the state estimate .
ˆx In order to start the iterations, an initial guess of the state variables ( 0
14
start). But to ease the convergence of the algorithm, the state variables are here initialized using a load flow solution. The second reason to use a load flow as initial solution is that the matrix in (2.9) is singular if a flat start initialization is chosen when unsynchronized current phasor or current magnitude measurements are used (see (Abur et al., 2004) for the problem related to current magnitude
measurements and section 2.3.5 for the unsynchronized current phasor measurements).
2.2.3 Observability
The network will be said to be observable if the measurement set allows the identification of the state variables. Indeed, not every measurement set will allow the identification of the state variables; and if the system is unobservable, additional meters will have to be installed or pseudo-measurements (e.g. load forecasts) will have to be provided.
The minimum condition is that the number of measurement equations and constraints is higher than the number of states variables. But the observability is also function of the measurement types and of their locations.
A method for testing the observability of a network consists in checking if Jacobian matrix C H has full column rank at flat start (Wu et al., 1988); otherwise the solution matrix of (2.9) is singular and the system solution cannot be calculated. Some methods have been developed in transmission systems to ease this computation by using linearized models and by using a decoupled formulation (Abur et al., 2004).
An alternative to the numerical methods is to use a topological observability analysis algorithm. A topological observability analysis, using graph theory, can be found for instance in (Nucera et al., 1991). The advantage compared to numerical method is that it is not subject to numerical errors because the observability is only determined with logical operations.
2.2.4 Accuracy of the state estimate
In addition to the state estimate, the state estimator provides also a measure of the state estimate uncertainty: its covariance matrix,cov(xˆ), can be calculated. This matrix is function of the accuracy of the measurements (R in (2.3)) and of the measurement set (number of measurements, their types and their locations).
15 0 0 0 0 0 0 0 0 ˆ 6 5 3 5 4 2 3 2 1 1 1 z B B B B B B B B B z C R H C H x T T T T T s s (2.12)
with B the inverse of the Hachtel matrix; which is mainly function of the measurement configuration and line parameters but it depends also a little on the load flow conditions. From this, we have:
z B
xˆ 2T . Therefore, the covariance of the state variable is calculated with:
2 2 2 2 2 2 ) cov( ] )) ( ( E(z)) -(z [ ] )) ˆ ( ˆ ))( ˆ ( ˆ [( ) ˆ cov( B R B B z B B z E z B E x E x x E x E x T T T T T (2.13)
If the measurement equations are nonlinear, the Hachtel matrix of (2.9) is computed at the state estimate xˆand (2.13) is used to obtain the covariance matrix of the state estimate. This is an
approximation since the Hachtel matrix is now function of the fitted state variables and since the high order terms of the Taylor series were neglected; but this method is still commonly used because of the ease of computation (Johnson et al., 1992). An accurate but computation intensive solution would be to use a Monte Carlo method.
From the covariance of xˆ, it is possible to compute the covariance matrix of any other variables describing the network state, for instance the current flows in the branches.
If y is a linear function of the state variables:yˆFxˆ, then its covariance can be calculated using the same methodology as above:
T F x F yˆ) cov(ˆ) cov( (2.14)
If the function is nonlinear (e.g. for the power flows), it is linearized around the estimated state: x F x f x x
f(ˆ ) (ˆ) and equation (2.14) is applied.
Therefore, from the state estimation results, the estimator can compute all the other variables describing the network state and their covariance matrix.
2.3 Measurement model
16
systems, load forecasts (also called pseudo-measurements) are also used to make the system observable.
The new type of measurements in power systems are synchronized measurements. Synchronized phasor measurements can be recorded for instance by Phasor Measurement Units (Phadke et al., 1986). These measurements provide more information than the classical ones, they can be sent to the distribution management system at higher frequencies (e.g. up to several times per seconds instead of every several seconds) and the accurate time stamps avoid state estimation errors caused by
asynchronicity errors (Hurtgen, 2011).
A contribution of this thesis is that the algorithm can also handle three-phase unsynchronized phasors measurements. Unsynchronized phasor measurements, as opposed to synchronized phasor
measurements, consist in phasor measurements that do not have accurate time stamps (i.e. time stamping error of more than some microseconds) and could be recorded by power quality meters. Lines model
In this section, the measurement functions (h(x) in (2.2)), i.e. the relations between the error free measurement and the state variables, are formulated for these different types of measurements. The expressions for all the measurement functions assume that the lines are modeled with pi-line equivalents shown in Figure 2-2 and that the line parameters are exactly known. The topology is for the moment assumed to be exactly known as well, topology errors will be considered in section 2.7. This assumption is made because the main source of errors in the estimation will be the measurement errors and not the line parameters errors. This is especially true as uncertain load forecasts are used in distribution system state estimation.
Figure 2-2 Pi-line model of a network branch
As the estimator is three-phase, three-phase line models are used: the line series parameters, the mutual coupling between the phases (defining Y in Figure 2-2) and the capacitances to ground and kl
17
2.3.2 Synchrophasors
The relationship between a current phasor flow from bus k to bus l and the networks node voltages is given below: ) ( k l kl k Skl kl Y V Y V V I (2.15)
with Ykl Gkl jBkl a 3x3 matrix modeling the three-phase branch series admittance, and
Skl Skl Skl G jB
Y its shunt admittance. This equation is developed in real and imaginary parts:
i l r l i k r k kl kl kl kl kl Skl kl Skl kl Skl kl Skl i kl r kl V V V V G B B G G G B B B B G G I I (2.16)
where r and i subscripts stand for the real and imaginary parts of the three-phase phasor that are expressed in abc quantities, e.g:
klcr
Tr b r kl a r kl r kl I I I
I . Therefore, the complex equation (2.15) contains in fact six real equations.
Similarly, for a current phasor injection at bus k:
N Bus kN Bus kl k V V Y Y I 1 ... (2.17)with YklBus (3x3 complex matrix) the element kl of the three-phase bus admittance matrix of the network. This equation can be developed in real and imaginary part similarly to the branch current flow equation.
Voltage phasor measurement equations are related to the associated state variables via a unity matrix. As the state is represented in Cartesian form, all these measurement equations are linear with respect to the state variables. The Jacobian matrix is thus directly obtained. Therefore, the state estimation with phasors-only measurements will converge to the state estimate in a single iteration.
2.3.3 Power measurements
The power measurements (injection or branch flow) are handled here with an equivalent phasor formulation proposed for instance in (Lu et al., 1995).
18 * 1 ˆ meas k meas k equi meas V jQ P I (2.18)
the * superscript denotes the complex conjugate. The equivalent phasor measurements are then processed in state estimation using the measurement equations (2.16) if the measurement is a branch flow measurement or (2.17) if it is an injection measurement.
Since these measurements are converted into equivalent current equations, the weighting of these equations is also updated at each iteration. The weighting of the equivalent phasor is computed by error propagation (equation (2.14)) from the covariance of the power measurement and the voltage estimate and by linearizing (2.18). The state estimation algorithm is thus iterative when power measurements are used (see Figure 2-3). The iterations are stopped when the equivalent current measurements stabilize: 1
maxImeask equi Imeask equi with for instance equal to 10-4pu. The
advantage of this formulation is that the Jacobian matrix of the power measurements does not need to be updated at every iteration.
Figure 2-3 State estimation using equivalent phasor formulation used for power measurements
2.3.4 Voltage magnitude measurements
A voltage magnitude measurement at node l is related to the state variables (expressed in rectangular form) with:
2
p 2 i l p r l p l V V V (2.19)19
2
p 2 i l p r l p r l p r l p l V V V V V (2.20)The derivatives with respect to the imaginary part are obtained similarly.
Current magnitude can be added using a similar methodology. However, because of potential
convergence problems and the existence of multiple solutions when the active power flow direction is undermined (Abur et al., 1997), and because this work focuses on networks with distributed
generation, these measurements are not considered in this work.
2.3.5 Unsynchronized phasors
Some measurement devices record asynchronously three-phase measurements. Thus, they can provide unsynchronized phasor measurements: the magnitude of the three-phases and the phase angle of the three-phases; but the absolute phase of these phasors remains unknown because the sampling clock of the measurement device is not synchronized to a common reference. The advantage of unsynchronized phasor over magnitude measurements is that the angle difference between the three-phases is known, which provides more measurement redundancy.
Such measurements could be for instance obtained from power quality measurement devices. The next figure shows the difference between a synchronized and an unsynchronized phasor measurement.
Figure 2-4. Example of a three-phase phasor, its record by a synchronized phasor measurement device (with measurement error), and its record by an unsynchronized phasor measurement device with a time error of δ degrees
(here δ = 50°)
In order to find a simple and easy solution to the case with unsynchronized phasors, an additional unknown to the problem per device that provides no accurate time stamps is added: an unknown angle shift of that resynchronizes the device clock to the chosen common time reference. This
20
In case of nuns unsynchronized measurement devices in a network of N nodes, the state vector is extended as follows:
Vr VNr Vi VNi nuns
x 1 ... 1 ... 2 ...
(2.21)
with i the synchronization angle associated to device i, and device 1 was selected as the reference. In (2.21), the first 6Nvariables are the three-phase voltage state variables expressed in rectangular coordinates (later denotedx ) and the last v nuns1 variables are the synchronization angles.
An unsynchronized phasor measurement recorded by device i is related to the state variables as follows: i j v s i uns i x h x e h ( ) ( ) (2.22)
where his(xv)is the measurement function for the synchronized phasor measurement of the same type (i.e. synchronized voltage phasor, branch current flow or current injection, see section 2.3.2).
However, the unsynchronized measurement is shifted by an unknown synchronizing operator. For instance, for an unsynchronized current phasor flow measurement on line kl recorded by device i, the measurement equation will be:
i j l k kl k Skl uns kl Y V Y V V e I ( ( )) (2.23)
In the former equations, there are six real equations as in the synchronized case, but a real unknown
i
is added, so the measurement redundancy is lower than in the synchronized case.
The Jacobian of these measurement equations are calculated by the derivation of the above equation:
l j v s i i uns i i j v v s i v uns i e j x h x h e x x h x x h ( ) ) ( ) ( ) ( (2.24) with v uns i x x h
( ) the Jacobian matrix of corresponding synchrophasors measurement equation.
Since the other measurement equations are no function of the synchronizing angles, the associated derivatives are all zero:
0 ) ( l i x h (2.25)
21 Initialization of the synchronization angles
From the load flow solution, the synchronized operators are initialized as follows: they are computed such that the shifted measurements fit best to those obtained from the load flow solution. For example, for an unsynchronized current measurement Imeas and its value obtained from a load flow solution
LF
I , the angle of the device will be initialized with:
c meas c LF b meas b LF a meas a LF I I phase I I phase I I phase 3 1 0 (2.26)
This initialization procedure is necessary for two reasons. First, the method can fail to converge if the synchronizing operators are initialized too far from the true solution (with an error greater than approx. 60°). Second, when using unsynchronized current phasor measurements, the measurement Jacobian
H is singular if a flat start initialization is chosen. This last issue is similarly met when using current magnitude measurements in conventional state estimators (Abur et al., 2004).
Using a load flow solution will therefore give a good starting point provided that the load flow model is close enough to the true network state.
2.3.6 Pseudo-measurements
In distribution systems, there are usually not enough measurements to allow the observability of the system. Therefore, pseudo-measurements are introduced. These measurements consist in fact in load (or distributed generation) models obtained from load forecasts or historical data.
These measurements are handled in a similar way as the power measurements: they are converted into equivalent current phasor injection at each iteration. With this method, any type of voltage dependent load model can be easily included:
) ( 1 k k equi meas f V I (2.27)
Normally, constant power load models are used. For instance, for a three-phase ungrounded load connected at node i, the following constant power load model is used (see Figure 2-5):
0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ . . . * 1 1 1 1 . * 1 1 1 1 . equ meas c i equ meas b i equ meas a i k b i k c i bc k a i k b i ab equ meas b i k a i k c i ac k a i k b i ab equ meas a i I I I V V S V V S I V V S V V S I (2.28)
where S , ab S and ac S are the complex power consumed between phases ab, ac and bc respectively. bc
22
Figure 2-5 Three-phase ungrounded load model
Other types of load connection, e.g. single phase or three-phase grounded loads, can be easily
represented as well. The power consumed can be also function of the voltage (e.g. with a combination of constant impedance, constant current and constant power load model (also called ZIP load model)), but it is in practice difficult because these load models are hardly available.
With the same approach, any type of DG model can be easily included. In normal conditions, nodes with distributed generation units are represented as constant power buses with reactive injection equal to zero (PQ bus).
2.3.6.1 Load forecasting
The load pseudo-measurements are very important parameters of the network model. The load forecasts will strongly impact the state estimation performances, not only on the state estimation accuracy but also on bad data processing and topology error detection functions, as it will be shown later in this chapter.
Load forecasting is not studied in this work, only the network aspects are considered. It is assumed that a load forecasting module exists and gives load estimations and its uncertainty for the current step of state estimation. A short overview of the load forecasting problematic is nevertheless given
hereafter.
Load estimation in distribution is a very challenging task as each MV load may be composed of few customers resulting in a load with a lot of randomness. This has been a research topic since networks exists because load estimates are required for a lot of network analyses: for the network operation but for the network planning as well (Baran, 2001). However, depending on the application, the load must be represented for a region, substation level or the individual secondary substation level (MV/LV transformer). For state estimation, the load estimate for every secondary substation is needed. Several methods to set pseudo-measurements parameters, i.e. load magnitude and standard deviation, have been proposed in the literature; these methods are summarized below, from the lesser effort and lower cost to the most accurate and expensive one.
23
2013). A more accurate method is to compute the load magnitude from the yearly or monthly customers’ billing data.
If the load class (commercial, industrial or residential) is available, the model can be updated over time using typical load curves.
More accurate forecasts can be obtained by doing some load research (Seppälä, 1996), (Alfares et al., 2002), (Wan, 2003), (Hoffman, 2006). Some loads are randomly selected in the feeder and are recorded during a certain amount of time. From these recordings and some other external factors such as the temperature, the weather conditions, the time of the day, a relatively accurate load forecast model function of time and of the external factors is derived. By classifying the loads and analyzing the customers’ billing data, the load forecast model may be extended to the neighboring loads which were not monitored during the load research.
Given the feeder measurements, the state estimator provides a nodal load estimate as well. Therefore, the outputs of the state estimator may be used to update the database of the load forecasting module (Wu et al., 2008), see Figure 2-6. Such feedback of the state estimator has the advantage that the load forecast models will be updated to consider e.g. the load growth over the years. However, this method does not seem to have been implemented and tested in practice. And load research is still necessary because the state estimator alone will not give good nodal loads estimations because it has only access to scarce feeder measurements.
Figure 2-6 Feedback of the state estimator to the load forecasting module module (Wu et al., 2008)
An alternative to these methods is to a use smart metering infrastructure (Arritt et al., 2013), (Wu et al., 2008), (Haughton et al., 2012) which would be very accurate but more expensive. In this case, a dedicated processing of the smart meter data would be required to aggregate the low voltage load measurement to medium voltage loads needed by the state estimator. However, it is not certain that smart metering data will be available for the operators to manage their network.
24
2.3.7 Virtual measurements
These measurements represent exact information about the system but for the ease of formulation, they are represented as measurements. They are typically used to represent zero-injection nodes, whose measurement equations will be the same as for a current phasor injection (equation (2.17)) except that there are considered as zero equality constraint (c(x)in (2.3)).
2.4 Illustrations
In this section simulation results are presented to illustrate the state estimation formulation presented above; the equivalent current formulation and the use of unsynchronized phasors are particularly emphasized. The results are also displayed on a map of the network for the ease of understanding. Lastly, the impact of the meter placement on the state estimation accuracy is explained and illustrated.
2.4.1 Benchmark network
The algorithm is illustrated on the CIGRE medium-voltage (20kV) rural distribution network benchmark derived from a real German MV distribution network (Rudion et al., 2006).
The single line diagram of the network is shown in Figure 2-7. The system is three-phase, three-wire and is composed of 12.7 km of overhead lines and 12.89 km of underground cables. The shortest line segment is 250 m long (segment 6-7), and the longest is 4.9 km (segment 12-13). The switch between node 8 and 14 is normally open. Distributed generators are present at nodes 5, 7, 9 and 10. The distributed generator outputs are given in Table 2-1, they are modeled as constant power source with unity power factor. The total load is 6.4 MW. The exact load repartition at each node and line parameters are given in Appendix B.
Node DG Type P (kW)
5 Battery 600
7 Wind Turbine 1500
9 CHP 522
10 Battery 200
Table 2-1 Distributed generator parameters
This benchmark network was chosen because it was developed by CIGRE to study distributed generation integration in MV distribution networks. Moreover, it exhibits a radial topology typical of classical rural distribution systems, and a weakly meshed topology that will probably be used more often in future distribution systems because of the advantages of such topologies (Nikander et al., 2003).
25
b
U Sb Ib Zb
kV
20 10MVA 289A 40
Table 2-2 Base values for voltage, power, current and impedance
Figure 2-7 Medium-voltage distribution benchmark network with base measurement set
2.4.2 Simulation results
2.4.2.1 Measurement setIn order to illustrate the use of unsynchronized phasor measurements and the equivalent current phasor formulation, a measurement set consisting of unsynchronized phasor measurements and power
measurements (actual or pseudo-measurements) is considered. Synchrophasor measurements will be illustrated in the chapters related to fault analysis (chapters 3 and 4).
Three measurement devices (M1, M2 and M3 on Figure 2-7) record phasors with inaccurate time stamps and device M4 records the power injection of DG9. The measurement set consists of:
M1 records unsynchronized voltage phasors of node 0 and current phasors on line 1 and 0-12,
