*Eastern Interconnection Phasor Project*

Performance Requirements Task Team (PRTT)

Performance Requirements

Part II

Targeted Applications: State Estimation

5/20/2006

*Task Team Leader:*

Damir Novosel, KEMA

*Contributors:*

Hongxia (Lucy)Wu, AREVA (Task Force Leader)

Virgilio Centeno, Virginia Tech

Armando Guzman, SEL

Sakis Meliopoulos, Georgia Tech, CERTS

Ken Martin, BPA

Jim Graffy, BPA

Bruce Fardanesh, NYPA

Henry Huang, PNNL, CERTS

Sam Brattini, KEMA

John Finney, KEMA

Walt Stadlin*,* KEMA

Table of Contents

Glossary of Terms 3

1\. Introduction 6

1.1 Brief review of State Estimation 6

1.2 Organization of this document 7

2\. Phasor Data Applications in State Estimation 8

2.1 Model Validation 8

2.2 State Measurement 9

2.3 State Estimator Enhancement 12

2.4 Hybrid State Estimator 13

3\. Data Accuracy and Quality [19](#data-accuracy-and-quality)

3.1 Total Vector Error (TVE) [19](#total-vector-error-tve)

3.2 Segregate errors of magnitudes and phase angles
[20](#segregate-errors-of-magnitudes-and-phase-angles)

4\. Reliability of Data Delivery [22](#reliability-of-data-delivery)

4.1 Data reliability of GPS-synchronized equipment
[23](#data-reliability-of-gps-synchronized-equipment)

4.2 Data access (PI database) [23](#data-access)

4.3 Communication standard (OPC format) [24](#communication-standard)

5\. Suggested Minimum Requirements 25

5.1 GPS-synchronized equipment 25

5.2 Phasor data concentrator 26

5.3 Databases 27

5.4 Communication network 27

6\. Integration of Phasor Data with State Estimation
[29](#integration-of-phasor-data-with-state-estimation)

7\. State Estimation Metrics 30

7.1 Accuracy 31

7.2 Performance 37

7.3 Robustness 39

7.4 Completeness 42

7.5 PMU Placement for State Estimation 44

8\. References 50

9\. Appendix 53

**Glossary of Terms**

This section provides some useful definitions pertinent to
GPS-synchronized devices, communication protocols and communications
media.

**DFR** -- Digital Fault Recorder

**DDR** -- Dynamic Disturbance Recorder

**SER** -- Sequence of Events Recorder

**PMU** -- Phasor Measurement Unit. A device that samples analog voltage
and current data in synchronism with a GPS-clock. The samples are used
to compute the corresponding phasors. Phasors are computed based on an
absolute time reference (UTC), typically derived from a built in GPS
receiver.

**PDC** -- Phasor Data Concentrator. A logical unit that collects phasor
data, and discrete event data from PMUs and possibly from other PDCs,
and transmits data to other applications. PDCs may buffer data for a
short time period but do not store the data.

**Relay** -- An electromechanical or electronic device applied to the
purpose of power apparatus protection. A relay typically monitors
voltages and currents associated with a certain power system device and
may trip appropriate breakers when a potentially damaging condition is
detected.

**IED** -- Intelligent Electronic Device. A general term indicating a
multipurpose electronic device typically associated with substation
control and protection.

**UTC** -- Coordinated Universal Time (initials order based on French).
UTC represents the time-of-day at the Earth\'s prime meridian (0°
longitude).

**IRIG-B** -- Time transmission formats developed by the Inter-Range
Instrumentation Group (IRIG). The most common version is IRIG-B, which
transmits day of year, hour, minute, and second once per second, over a
1 kHz carrier signal.

**GOES** -- Geostationary Operational Environmental Satellites. Operated
by the National Oceanic and Atmospheric Agency (NOAA). Two GOES
satellites broadcast a time code referenced to UTC. Clocks based on this
transmission are accurate to 100 microseconds.

**GPS** -- Global Positioning System. A satellite based system for
providing position and time. The accuracy of GPS based clocks can be
better than 1 microsecond.

**pps** -- Pulse-Per-Second. A signal consisting of a train of square
pulses occurring at a frequency of 1 Hz, with the rising edge
synchronized with UTC seconds. This signal is typically generated by GPS
receivers.

**kpps** -- One thousand pulses per second. A signal consisting of a
train of square pulses occurring at a frequency of 1 kHz, with the
rising edge of synchronized with UTC milliseconds. This signal is
typically generated by GPS receivers.

**Sampling Rate** -- The number of samples (measurements) per second
taken by an analog to digital converter system.

**Navigation** -- The mode in which GPS receiver has locked onto signals
from three or more satellites thus providing accurate time, as well as
position.

**COMTRADE-file format** -- COMTRADE file format is a standardized ASCII
text or binary file (2 formats), originally designed for Digital Fault
Recorders. It can be used to transfer locally recorded values from a PMU
over to the central data storage. COMTRADE ASCII format is not efficient
for long-term data storage but could be used for event file retrieval.

**PhasorFile** -- A binary storage format that is used by PDC for
long-term storage of SynchroPhasor data. Currently, this format is not
standardized, and may be left in such a state as long as stored data is
made available in an industry standard format (i.e. COMTRADE).

**[Communications Related Terms]{.underline}**

**Unicast** -- UDP transmission from one host to another
(source/destination). When talking about a link on a network, typically
a unicast link is inferred.

**Broadcast** -- Data transmission from one host to many. The
destinations will be all computers on the same sub-network. Every
computer on the sub-network must process the data. Broadcast is not
routable to other network segments.

**Multicast** -- Data transmission from one host to many. Data is
transmitted to a group IP address. Any member of the group can access
the address to receive the data. Anybody can then join in this multicast
group, and when a server sends to the group, everyone in the group will
receive the data. The advantage is that this protocol is routable and
does not overload all computers in the local subnet.

**WAN** -- Wide Area Network, stretching across large geographical
distances. Latency's can be very large, up to many seconds.

**LAN** -- Local Area Network, within a small building or office. Very
low latency between endpoints on the network, often less than a few
milliseconds.

**VLAN** -- A simulated LAN that is spread across a LAN, but uses
special IP addresses so that it appears a "physically" separate LAN.

**Lossless data compression** -- Data stored in a lossless format can be
retrieved exactly as it was created.

**Lossy data compression** -- Data that is stored in a lossy format will
have degraded accuracy when retrieved. However, more data can be stored
in the same amount of space. Lossy data compression should be applied
with caution and is not expected to play major role in synchrophasor
data collection.

**[Communication Protocols Terms]{.underline}**

**IEEE 1344** -- A highly efficient protocol for real time Synchrophasor
data. Typically data is streamed in this format over UDP/IP or across a
serial link.

**BPA/PDCStream** -- An extension of IEEE 1344, widely used by the BPA
PDC and HMI software on the West Coast.

**IEEE C37.118** -- This protocol and its associated standard are
intended to replace IEEE 1344 and the BPA/PDCStream protocols. Typically
data is streamed in this format over UDP/IP or across a serial link.

**OPC DA** -- (Open Process Control Data Access) OPC was created for
industrial automation, for use within a factory, for example. It is
designed to share simple data between computers running only Microsoft
Windows®. There are 3 revisions that are commonly used. Different
revisions are generally not compatible. This protocol is useful for
simple data sharing between computers in a small LAN, but has serious
security and performance issues when deployed across a WAN. OPC uses
TCP/IP the underlying link.

**OPC HAD** -- (Open Process Control Historical Data Access) an offshoot
of OPC DA which allows a client to request stored data. This is a
separate protocol, and different servers/clients must be developed.

**OPC AE** -- (Open Process Control Alarms and Events) an offshoot of
OPC DA, which allows clients to be notified on alarm conditions. As with
OPC HAD, this is a separate protocol, and different servers/clients must
be developed.

**OPC XML DA** -- (Open Process Control XML Data Access) -- An OPC DA
protocol designed for use across a WAN. This protocol uses the standard
Web Services structure, using SOAP and XML. This protocol is simple to
work with and will allow PMU devices that don't run Windows® as an
operating system, to be an OPC server for providing data to any client.
The OPC Foundation is creating a 'Unified Architecture' using the
XML-based structure as the foundation for future development.

**TCP/IP** -- TCP/IP is a low-level protocol for use mainly on Ethernet
or related networks. Most of the higher-level protocols use TCP/IP to
transport the data. TCP/IP provides a highly reliable connection over
unreliable networks, using checksums, congestion control, and automatic
resending of bad or missing data. TCP/IP requires time to handshake new
connections and will block if missing data is being resent.

**UDP/IP** -- UDP/IP is a low-level IP protocol that provides
low-latency communication across Ethernet or related networks. UDP/IP
does not provide any error-control or resending of missing or bad data.
The Application will need to check data for correctness. UDP/IP however,
does not require time for handshaking and will not block, making it
ideal for real-time data communications.

**HTTP** -- HTTP is a protocol made popular by the Internet and web
pages. Web pages are transmitted using HTTP. It has also become the
mechanism for the Web Services Paradigm using SOAP and XML. HTTP uses
TCP/IP as the underlying protocol.

**FTP** -- FTP is the file transfer protocol. It is a simple protocol
where a client can connect and request a file to be downloaded. A
separate data connection is automatically created where the data is then
transferred across the network while the command connection becomes
unavailable. FTP is commonly used to get recorded data from devices.

**VPN** -- (Virtual Private Network). A communication network
constructed by using public wires to connect nodes with procedures to
ensure that only authorized users can access the network and that the
data cannot be intercepted. These procedures typically use encryption
and other security mechanisms.

# Introduction

## Brief review of State Estimation

State Estimation plays a key role in real-time power system operations.
It determines a best estimate of the current actual power system state
(voltage, angle, CB status, tap, etc.), based on available SCADA
measurements, power system model and other data. MW/MVAR flows in the
transmission lines, transformers, and other equipment can then be
derived. State Estimation results are used by subsequent network
contingency analysis, security enhancement, optimization applications,
dynamic security analysis (including voltage and transient stability),
and other applications. It is the foundation of power system grid
security analysis. Also State Estimation provides the system topology to
Simultaneous Feasibility Test, which is an important application in the
power market system. In the past few years, with the advent of
electricity markets and the greater emphasis on improved grid security
due to blackouts, State Estimation has become a critical, 'must-run
successfully' control center function. Fig. 1.1 describes the role of
State Estimation in power system operation.

One of the key state variables in State Estimation is the bus voltage
phase angle. This was not available as a measurement before. Nowadays,
PMUs can provide the direct measurement of synchronized voltage phase
angle at different substation buses, at rates many times faster than the
SCADA scan rates which are typically in seconds. However SE may execute
at 2-5 minutes intervals. Incorporating phase angle measurement in State
Estimation requires little change since the angle is already the key
state. The addition of current phasor data leads to more complex
modifications. Modeling of these new measurements in State Estimation
can be achieved by methods from Monticelli and Ali Abur's books
\[24\]\[28\].

![](media/image1.png){width="4.550694444444445in"
height="2.716666666666667in"}

Fig.1.1. The role of State Estimation in power system operations

There are two major PMU projects currently being conducted in North
America: WAMS in the western area \[21\] \[23\] and EIPP in the eastern
area \[22\]. More than 100 utility companies, ISOs, national labs, and
vendors are participating in these projects. Hundreds of PMUs have been
installed in substations and the data are now available for the network
applications and system monitoring. Utilities are also typically willing
to share synchronized phasor data with neighboring utilities.

Initial research from academics and vendors have shown that PMU data can
provide valuable, 'instantaneous' information about system security (and
imminent instability), and has the potential to improve the reliability
and robustness of State Estimation. However there are some practical
implementation issues involved when the PMU data are introduced in State
Estimation.

## Organization of this document

This document is organized as follows:

-   Chapter 1 gives a brief review of State Estimation and its important
    role in present power system operations.

-   Chapter 2 reviews how phasor data can be used in State Estimation:
    model validation, state measurement, State Estimator enhancement,
    and hybrid State Estimator.

-   Chapter 3 describes the data accuracy and quality

-   Chapter 4 discusses the reliability of data delivery for the
    GPS-synchronized equipment, data access (PI database), communication
    standard (OPC format)

-   Chapter 5 suggests minimum requirement for the GPS-synchronized
    equipment, phasor data concentrator, databases, and communication
    network.

-   Chapter 6 discusses issues when phasor data are implemented in State
    Estimation

-   Chapter 7 describes the state estimation metrics in terms of
    accuracy, performance, robustness, and completeness.

-   Chapter 8 lists the references and Chapter 9 for Appendix

# Phasor Data Applications in State Estimation

State Estimators in modern electric utility are complex tools used to
determine the state of the power system under quasi-steady state
conditions. The key state of the system is defined as the collection of
positive sequence voltage phasors (magnitude and angle) at all nodes in
the system. The conceptual realization of phasor measurement has added a
new dimension to the State Estimation problem by making possible the
direct measurement of the state of the system. Since the initial PMU
algorithms were developed in the mid 1980\'s, several uses of PMUs
related to state estimators have been proposed, tested or implemented.
This section reviews the most relevant proposed applications of phasor
data related to State Estimation:

1.  Model Validation

2.  State Measurement

3.  Enhanced State Estimation

4.  Hybrid State Estimator

## Model Validation

The first practical application that combined the use of PMU data and
State Estimators was model validation. Most of the research and analysis
is performed using system models of varying levels of details and
complexity depending on the application. Until the implementation of the
first PMUs there were very few possibilities of evaluating the accuracy
of the system model for wide-area applications. The availability of
phasor measurement data has made possible the comparison of the static
and dynamic behavior of the system with the one predicted by various
system models. System models are not expected to provide accurate
information for all possible scenarios but gross errors and incorrect
model settings need to be identified and corrected in order to increase
the level of confidence on the information derived from the models.
Unless PMUs are available at every node in the system it is not possible
to perform a model validation without the use of a state estimator data.

In order for the simulations to be compared with PMU measurements, the
initial conditions for the simulation must be as close as possible to
the actual system conditions under which the PMU data is captured. The
first system model validation using PMU data was performed in Georgia in
June of 1992 as part of an EPRI sponsored project on Parameter
Identification Data Acquisition System (PIDM). For these tests, a 500kV
line under light load was open and closed while six PMUs collected data
of the response of the system to the line-switching event. Figure 2.1
shows the location of the six PMUs used for this model validation. Prior
to each switching event the State Estimator outputs were recorded to be
used later to set initial conditions for the simulation of the same
event. Figure 2.2 shows one of the oscillations recorded by the PMUs
compared to the oscillations obtained from the system model of Southern
company. The State Estimator information prior to the test was used to
set the initial conditions for the simulation of the event. . Note from
the figure that the simulation and the measurement are very close for
the two substations of Southern company where the most accurate models
were used. The simulation is not as accurate for the other three PMUs
located at substations in other utilities due to the use of equivalents
to model the external systems.

+------------------------------------+---------------------------------+
| ## ![                              | ## ![](media/ima                |
| ](media/image2.png){width="3.02361 | ge3.png){width="3.0381944444444 |
| 11111111112in" height="3.897916666 | 446in" height="3.89583333333333 |
| 6666667in"} {#section .unnumbered} | 35in"} {#section-1 .unnumbered} |
|                                    |                                 |
| ##                                 | ## Fig. 2.2.                    |
| Fig. 2.1. PMU locations for 1992 v |  Results of 1992 system validat |
| alidation {#fig.-2.1.-pmu-location | ion {#fig.-2.2.-results-of-1992 |
| s-for-1992-validation .unnumbered} | -system-validation .unnumbered} |
+------------------------------------+---------------------------------+

## State Measurement

Since the phasor measurement units provide a direct measurement of the
system state, one of the first PMU applications suggested related to
state estimation is the concept of State Measurement. This futuristic
application will replace the State Estimator's RTUs with PMUs and the
Weighted Least Square estimators with Linear State Estimator which uses
only phasor data \[2\]. To incorporate redundancy on the data set for
this linear estimator PMU's current measurements would need to be added
to the estimator. The measurement vector for this proposed estimator is
given as:

![](media/image4.wmf){width="1.1805555555555556in"
height="0.5277777777777778in"} (2.1)

where V~B~ and I~L~ are the true values of voltages and currents and
ε~B~ and e~L~ are their corresponding measurement error vectors. If the
line current vector is expressed in terms of the bus voltages and system
admittance, Equation (2.1) can be expressed as:

![](media/image5.wmf){width="1.75in"
height="0.5277777777777778in"}![](media/image6.wmf){width="0.8465277777777778in"
height="0.2361111111111111in"} (2.2)

Notice that *I* is the identity matrix. If the covariance of the error
is given by

![](media/image7.wmf){width="1.0694444444444444in"
height="0.5277777777777778in"}

and the measurements are assumed to be uncorrelated, W is a diagonal
matrix and (2.2) can be solved as:

![](media/image8.wmf){width="1.6666666666666667in" height="0.25in"} or
![](media/image9.wmf){width="1.2222222222222223in" height="0.25in"}
(2.3)

where *G* is the constant gain matrix that will only need to be changed
for topology changes in the system. Equation (2.3) is linear and the
state can be estimated with a single matrix computation without the need
for iterations and eliminating convergence problems. Even though this is
a non iterative solution, it may still be subject to ill conditioning,
in which case, the solution can be quite bad.

One of the main implementation problems with this futuristic concept is
the requirement of PMU measurements at every node in the system. Due to
the high cost of having a PMU at each node, some of the studies
performed in the mid 1980's focused on PMU placement and pseudo
measurements for complete or partial observability of the system for
static and dynamic state estimators \[2\]\[3\]\[4\]\[5\]. One of these
studies suggests a gradual placement of PMU and provides a methodology
for PMU placement with a selected depth of unobservability \[5\]. The
study defines depth of unobservability as the distance of an unobserved
bus to its observed neighbors and provides a methodology for a phased
installation of PMUs. Figure 2.3 shows this concept for a degree one of
unobservability. The right placement of three PMUs in the given system
results in only two nodes not been observable, three fully observable
nodes and eight pseudo observable nodes. Fig. 2.4 shows the same concept
applied to the IEEE 57bus system.

![](media/image10.png){width="4.625694444444444in"
height="3.6708333333333334in"}

Fig. 2.3. Conceptual illustration of PMU placement with depth-one
unobservability \[8\]

![](media/image11.png){width="6.0in" height="8.005555555555556in"}

Fig. 2.4. PMU placement with depth-one unoberservability for the IEEE 57
bus system \[8\]

## State Estimator Enhancement

Several studies have been performed on the integration of PMU data into
existing WLS estimators \[6\]\[7\]. The measurement vector for state
estimators with PMU data is given by:

![](media/image12.wmf){width="0.6805555555555556in"
height="0.5277777777777778in"} where
![](media/image13.wmf){width="0.7777777777777778in"
height="0.7777777777777778in"} and
![](media/image14.wmf){width="0.8055555555555556in"
height="0.7777777777777778in"}

If PMUs only provide measure angle differences one of the main concerns
of this implementation is the determination of the swing bus. The
simplest solution to this problem is to require that a PMU be set in the
selected swing bus and that the angles of all PMU be computed with
respect to the swing PMU. The problem with this solution is the high
dependability on the swing PMU whose absence will render PMU data
useless for the estimation process. The selection of which phase angle
differences to use should be dynamic so that loss of one or more
readings does not make it impossible to use the phase angles at all.

The only documented implementation of a state estimator with phasor
measurement was performed at Sevillana de Electricidad in Spain in the
early 1990s, the implementation included a detailed analysis of the
effects of angle measurement errors in the quality of the solution of
the enhanced WLS estimator \[7\]. This study found that to have any
positive effect on the estimator the precision of the phase angle must
be better than 0.3 degrees, or 50 μseconds (50Hz system). For the effect
of the phase angle measurement to be noticeable the precision on the
angle difference must be better than 0.12 degrees (20 μsecond error in a
50Hz system). The study also found that there is little reduction in the
number iterations when phasor measurements are included in the
measurement set. The main results from this study are summarized in
Table 2.1.

It should be mentioned that the phasor measurement units installed at
Sevillana are different to the ones being used in the EIPP project. At
Sevillana the GPS signals were used to time the zero crossing of the
voltage signal and the zero crossing was used to determine angle
differences. Additional filters were performed on the angle measurement
to determine phase angle differences **every 10 seconds**. In addition
these units computed only phase angles, voltage magnitude data was not
part of the synchronized measurement.

Table 2.1 Phasor Measurement effectiveness classification from
Sevillana's study \[7\]

+------------------+----------------+----------------------------------+
| **S              | **Standard     | **Classification**               |
| ynchronization** | Deviation**    |                                  |
|                  |                |                                  |
| **Accuracy**     | **of phasor    |                                  |
|                  | metering**     |                                  |
| **               |                |                                  |
| (microseconds)** | **Error        |                                  |
|                  | (degrees)**    |                                  |
+------------------+----------------+----------------------------------+
| \>30             | \>0.54         | Ineffective, No impact on        |
|                  |                | estimator solution               |
+------------------+----------------+----------------------------------+
| 20 -- 30         | 0.36 -- 0.54   | Somewhat effective, little       |
|                  |                | impact on solution               |
+------------------+----------------+----------------------------------+
| 7 -- 20          | 0.126 -- 0.36  | Effective, impact is similar to  |
|                  |                | that of                          |
|                  |                |                                  |
|                  |                | other measurements               |
+------------------+----------------+----------------------------------+
| \< 7             | \< 0.126       | Very Effective, strong effect on |
|                  |                | solution,                        |
|                  |                |                                  |
|                  |                | superior to that of other        |
|                  |                | measurements                     |
+------------------+----------------+----------------------------------+

There are some recent studies on using the absolute angle measurement
from PMU in State Estimation. Issues such as bad data and data drop out
need to be considered since the absolute angles change very fast.

## Hybrid State Estimator

The hybrid state estimator uses (a) a quadratic model of the system and
(b) measurements consisting of traditional SCADA data and PMU data to
estimate the states of a system. It can be applied to a per phase
equivalent model as well as a three phase model. The description of the
method uses a three phase formulation. The usual per phase equivalent
approach is a subset of the described model.

## 2.4.1 Measurement Data Set {#measurement-data-set .unnumbered}

The measurement types supported are classified into (a) phasor
measurements and (b) non-synchronized measurements. The list of
measurements is given in Table 2.2.

Table 2.2 List of Measurements

  ------------------------------------ -------- --------------------------- --------
  **Phasor Measurements**                       **Non-Synchronized          
                                                Measurements**              

  **Description**                      **Type   **Description**             **Type
                                       Code**                               Code**

  Voltage Phasor, $\overset{\sim}{V}$  1        Voltage Magnitude, $V$      4

  Current Phasor, $\overset{\sim}{I}$  2        Real Power Flow, $P_{f}$    5

  Current Injection Phasor,            3        Reactive Power Flow,        6
  ${\overset{\sim}{I}}_{\text{inj}}$            $Q_{f}$                     

                                                Real Power Injection,       7
                                                $P_{\text{inj}}$            

                                                Reactive Power Injection,   8
                                                $Q_{\text{inj}}$            
  ------------------------------------ -------- --------------------------- --------

Since at each bus the model may have a neutral node as well as a ground
node, the measured phase voltages are always considered as the phase to
neutral voltages. The above measurements are related to the state of the
system via the "model" equations. As for system states we consider the
voltage phasors at all nodes of the system, i.e. the three phase
voltages at all buses of the system, the neutral node voltage as well as
the ground node voltage.

It is important to note that if there is at least one synchronized
measurement, then the state of the system is considered to be the phasor
voltages (real and imaginary part) for all nodes. This is different from
the traditional state estimation, where the phase angle of one voltage
is selected to be zero.

First the state of the system is defined as the node voltage phasors at
each node of the system. A bus k will have three to five nodes, phases
A, B and C, possibly a neutral and possibly a ground node. The state of
the system at this bus is the node voltage phasors. We will use the
following symbols (for a four node bus, i.e. phases A, B, C and neutral
N).

$${\overset{\sim}{V}}_{k,A} = V_{k,A,r} + \text{jV}_{k,A,i}$$

$${\overset{\sim}{V}}_{k,B} = V_{k,B,r} + \text{jV}_{k,B,i}$$

$${\overset{\sim}{V}}_{k,A} = V_{k,A,r} + \text{jV}_{k,A,i}$$

$${\overset{\sim}{V}}_{k,A} = V_{k,A,r} + \text{jV}_{k,A,i}$$

The state for a four node bus k will be defined as follows:

$${\overset{\sim}{V}}_{k} = \begin{bmatrix}
{\overset{\sim}{V}}_{k,A} \\
{\overset{\sim}{V}}_{k,B} \\
{\overset{\sim}{V}}_{k,C} \\
{\overset{\sim}{V}}_{k,N}
\end{bmatrix} = \begin{bmatrix}
V_{k,A,r} + \text{jV}_{k,A,i} \\
V_{k,B,r} + \text{jV}_{k,B,i} \\
V_{k,C,r} + \text{jV}_{k,C,i} \\
V_{k,N,r} + \text{jV}_{k,N,i}
\end{bmatrix}$$

It is also important to note that normally measurements of neutral or
ground voltages are not available. On the other hand these voltages are
very small under normal operating conditions. For this reason, we
introduce one pseudo-measurement of voltage phasor for each neutral and
ground node in the system. The value of this measurement is exactly
zero. The equations for each one of the measurements are given below.
The symbols utilized appear in Figure 2.5.

$$z_{V,k,A} = {\overset{\sim}{V}}_{k,A} = {\overset{\sim}{V}}_{k,A} - {\overset{\sim}{V}}_{k,N}$$

$$z_{V,k,B} = {\overset{\sim}{V}}_{k,B} = {\overset{\sim}{V}}_{k,B} - {\overset{\sim}{V}}_{k,N}$$

$$z_{V,k,C} = {\overset{\sim}{V}}_{k,C} = {\overset{\sim}{V}}_{k,C} - {\overset{\sim}{V}}_{k,N}$$

${\overset{\sim}{I}}_{d1,k,A} = C_{d1,k,A}^{T}\begin{bmatrix}
{\overset{\sim}{V}}_{k,A} \\
{\overset{\sim}{V}}_{k,B} \\
{\overset{\sim}{V}}_{k,C} \\
{\overset{\sim}{V}}_{m,A} \\
{\overset{\sim}{V}}_{m,B} \\
{\overset{\sim}{V}}_{k,C}
\end{bmatrix}$,

Similar equation can be obtained for phases B and C.

$$V_{k,A} = \left| {\overset{\sim}{V}}_{k,A} \right| = \sqrt{V_{k,A,r}^{2} + V_{k,A,i}^{2}}$$

$$P_{d1,k,A} = \text{Re}\left\{ {\overset{\sim}{V}}_{k,A}\left( C_{d1,k,A}^{T}\begin{bmatrix}
{\overset{\sim}{V}}_{k,A} \\
{\overset{\sim}{V}}_{k,B} \\
{\overset{\sim}{V}}_{k,C} \\
{\overset{\sim}{V}}_{m,A} \\
{\overset{\sim}{V}}_{m,B} \\
{\overset{\sim}{V}}_{k,C}
\end{bmatrix} \right)^{} \right\}$$

$$Q_{d1,k,A} = \text{Im}\left\{ {\overset{\sim}{V}}_{k,A}\left( C_{d1,k,A}^{T}\begin{bmatrix}
{\overset{\sim}{V}}_{k,A} \\
{\overset{\sim}{V}}_{k,B} \\
{\overset{\sim}{V}}_{k,C} \\
{\overset{\sim}{V}}_{m,A} \\
{\overset{\sim}{V}}_{m,B} \\
{\overset{\sim}{V}}_{k,C}
\end{bmatrix} \right)^{} \right\}$$

![](media/image15.wmf)

Fig. 2.5 Measurement Definition -- Three Phase Model

To facilitate the definition and the measurements and to devise a scheme
for interfacing with the three phase quadratic power flow program, each
measurement is defined with the following set:

$$S_{\text{meas}} = \begin{Bmatrix}
m_{\text{type}} & n_{\text{device}} & n_{\text{bus}} & n_{\text{phase}}
\end{Bmatrix}$$

where:

$m_{\text{type}}$: measurement type defined as in Table 2.2

$n_{\text{device}}$: power device ID, plus manufacturer and IED (relay,
RTU, etc.) ID

$n_{\text{bus}}$: bus name

$n_{\text{phase}}$: measurement phase, A, B or C

The above set allows complete correspondence between measurement and
system state. This sets are utilized as follows. For each IED, the
measurement pointers are defined for each channel of the IED. A sample
file for a specific IED is given in Figure 2.6.

> IED-ID string
>
> TIME_TAG iDayNumber dSeconds
>
> VOLTAGE_PHASOR 1234 GLEN161 A-N 93.1 3.5 kV
>
> VOLTAGE_PHASOR 1234 GLEN161 B-N -46.3 -80.6 kV
>
> VOLTAGE_PHASOR 1234 GLEN161 C-N 46.1 79.6 kV
>
> CURRENT_PHASOR 1534 GLEN161 A 0.833 0.081 kA
>
> CURRENT_INJECTION_PHASOR 1534 GLEN161 A 0.000 0.000 kA
>
> VOLTAGE_MAGNITUDE 1234 GLEN161 A-N 93.4 kV
>
> REAL_POWER_FLOW 1534 GLEN161 A 77.55 MW
>
> REACTIVE_POWER_FLOW 1534 GLEN161 A 7.55 MVAr

Fig. 2.6 Example File of Measurement Pointers

### Hybrid Three-Phase State Estimator

This section presents the hybrid three-phase state estimator. This state
estimator uses standard SCADA data and synchronized data together with a
full three phase system model to perform state estimation. The
mathematical procedure is described next.

The measurements are assumed to have an error that is statistically
described with the meter accuracy. The present state estimator supports
the measurements that are listed in Table 2.2. Each one of these
measurements has the following mathematical model.

Phasor measurements:

$${\overset{\sim}{z}}_{\nu} = {\overset{\sim}{V}}_{k,A} - {\overset{\sim}{V}}_{k,N} + {\overset{\sim}{\eta}}_{\nu}$$

$${\overset{\sim}{z}}_{\nu} = {\overset{\sim}{I}}_{d1,k,A} + \eta_{\nu} = C_{d1,k,A}^{T}\begin{bmatrix}
{\overset{\sim}{V}}_{k,A} \\
{\overset{\sim}{V}}_{k,B} \\
{\overset{\sim}{V}}_{k,C} \\
{\overset{\sim}{V}}_{m,A} \\
{\overset{\sim}{V}}_{m,B} \\
{\overset{\sim}{V}}_{k,C}
\end{bmatrix} + {\overset{\sim}{\eta}}_{\nu}$$

Pseudo-measurements for neutrals and grounds:

$${\overset{\sim}{z}}_{\nu} = 0 + j0 = {\overset{\sim}{V}}_{k,N} + {\overset{\sim}{\eta}}_{\nu}$$

Non-synchronized measurements:

$$z_{\nu} = \left| {\overset{\sim}{V}}_{k,A} - {\overset{\sim}{V}}_{k,N} \right|^{2} + 2\eta_{\nu} = \left( V_{k,A,r} - V_{k,N,r} \right)^{2} + \left( V_{k,A,i} - V_{k,N,i} \right)^{2} + 2\eta_{\nu}$$

$$z_{\nu} = P_{d1,k,A} + \eta_{\nu} = \text{Re}\left\{ {\overset{\sim}{V}}_{k,A}\left( C_{d1,k,A}^{T}\begin{bmatrix}
{\overset{\sim}{V}}_{k,A} \\
{\overset{\sim}{V}}_{k,B} \\
{\overset{\sim}{V}}_{k,C} \\
{\overset{\sim}{V}}_{m,A} \\
{\overset{\sim}{V}}_{m,B} \\
{\overset{\sim}{V}}_{k,C}
\end{bmatrix} \right)^{} \right\} + \eta_{\nu}$$

$$z_{\nu} = Q_{d1,k,A} + \eta_{\nu} = \text{Im}\left\{ {\overset{\sim}{V}}_{k,A}\left( C_{d1,k,A}^{T}\begin{bmatrix}
{\overset{\sim}{V}}_{k,A} \\
{\overset{\sim}{V}}_{k,B} \\
{\overset{\sim}{V}}_{k,C} \\
{\overset{\sim}{V}}_{m,A} \\
{\overset{\sim}{V}}_{m,B} \\
{\overset{\sim}{V}}_{k,C}
\end{bmatrix} \right)^{} \right\} + \eta_{\nu}$$

The state estimation problem is formulated as follows:

$\text{Min}\ J = \sum_{\nu \in \text{phasor}}^{}\frac{{\overset{\sim}{\eta}}_{\nu}^{}{\overset{\sim}{\eta}}_{\nu}}{\sigma_{\nu}^{2}} + \sum_{\nu \in \text{non} - \text{syn}}^{}\frac{\eta_{\nu}\eta_{\nu}}{\sigma_{\nu}^{2}}$
(2.4)

It is noted that if all measurements are synchronized the state
estimation problem becomes linear and the solution is obtained directly.
In the presence of the non-synchronized measurements and in terms of
above formulation, the problem is quadratic, consistent with the
quadratized power flow. Specifically, using the quadratic formulation,
the measurements can be separated into phasor and non-synchronized
measurements with the following form:

$$z_{s} = H_{s}x + \eta_{s}$$

$$z_{n} = H_{n}x + \left\{ x^{T}Q_{i}x \right\} + \eta_{n}$$

In above equations, the subscript s indicates phasor measurements while
the subscript n indicates non-synchronized measurements. The best state
estimate is given by:

Case 1: Phasor measurements only.

$\hat{x} = \left( H_{s}^{T}\text{WH}_{s} \right)^{- 1}H_{s}^{T}\text{Wz}_{s}$
(2.5)

Case 2: Phasor and non-synchronized measurements.

${\hat{x}}^{\nu + 1} = {\hat{x}}^{\nu} + \left( H^{T}\text{WH} \right)^{- 1}H^{T}W\begin{bmatrix}
z_{s} - H_{s}{\hat{x}}^{\nu} \\
z_{n} - H_{n}{\hat{x}}^{\nu} - \left\{ {\hat{x}}^{\nu^{T}}Q_{i}\hat{x^{\nu}} \right\}
\end{bmatrix}$ (2.6)

where:

$W = \begin{bmatrix}
W_{s} & 0 \\
0 & W_{n}
\end{bmatrix}$, $H = \begin{bmatrix}
H_{s} \\
H_{n} + H_{\text{qn}}
\end{bmatrix}$

# Data Accuracy and Quality 

GPS-synchronized equipment has the capability to provide a data
acquisition system with the following precision:

1.  1\. Time tagging with precision better than 1 microsecond (or
    equivalently 0.02 degrees of phase at 60 Hz).

    2\. Magnitude precision of 0.1% or better.

This precision cannot be achieved for the overall system in any
practical application, i.e. in the substation environment. In addition,
depending on the implementation approach and equipment used, the
accuracy of the collected data and the reliability of the data
availability may differ. Typical GPS synchronized equipment (PMU's) is
very accurate. However, the inputs to this equipment are scaled down
voltages and current via instrument transformers, control cables,
attenuators, etc. We collectively refer to it as the instrumentation
channel. The instrumentation channel components are typically less
accurate. Specifically, potential and current instrument transformers
may introduce magnitude and phase errors that can be magnitudes of order
higher than the typical PMU accuracy. Although, high accuracy laboratory
grade instrument transformers are available, their application in
substation environment is practically and economically infeasible.

Data accuracy can be addressed as Total Vector Error (TVE) \[9\] or
segregate errors of magnitudes and phase angles.

## Total Vector Error (TVE)

The Total Vector Error (TVE) definition that IEEE C37.118 \[9\]
describes can be rewritten as follows:

![](media/image16.wmf){width="1.5965277777777778in"
height="0.6111111111111112in"}

where:

![](media/image17.wmf){width="0.3611111111111111in"
height="0.2777777777777778in"} is the ideal phasor

![](media/image18.wmf){width="0.5416666666666666in"
height="0.2777777777777778in"} is the measured phasor

Notice that the measured phasor can be a voltage phasor or a current
phasor. Figure 3.1 shows the graphical representation of the ideal,
measured and error phasors. The error phasor is:

![](media/image19.wmf){width="0.375in"
height="0.2777777777777778in"}=![](media/image20.wmf){width="1.0138888888888888in"
height="0.2777777777777778in"}

The circle shown in Fig. 3.1 encompasses all possible phasor
measurements for a specified TVE.

![](media/image21.wmf){width="3.698611111111111in"
height="2.4756944444444446in"}

Fig. 3.1 Ideal, measured and error phasors.

The circle encompasses all possible phasor measurements for a specified
TVE. For example, the TVE in percentage for a device that has magnitude
error of 0.1 % and phase error of 0.1° is:

TVE % =![](media/image22.wmf){width="2.138888888888889in"
height="0.5555555555555556in"}

The above TVE calculation includes magnitude and phase errors. TVE for
magnitude error only is 0.1 % and TVE for phase error only is 0.1745 %.

## Segregate errors of magnitudes and phase angles

PMU measurements have the following characteristic. The magnitude error
is typically different than the phase error. For most PMU units, the
phase error is much smaller than the magnitude error. Utilizing the
total phasor error results in missing information and compromises the
performance of the state estimator. Within the formulation of the hybrid
state estimator, the phase and magnitude errors are segregated by simply
introducing an additional datum. This datum is treated as a pseudo
measurement. The definition of the pseudo measurement is:

$$z_{\text{pseudo}} = \left( \text{tan}(\delta) \right)\text{Re}\left\{ \overset{\sim}{V} \right\} - \text{Im}\left\{ \overset{\sim}{V} \right\} + \eta$$

The standard deviation of the error $\eta$ depends on the
characteristics of the PMU. It expresses the error of the phase
measurement. This formulation segregates the magnitude and phase errors
of PMU data.

BPA made a number of performance tests on PMUs \[35,36\]. The first of
these were on the original PMUs produced at Virginia Tech. In 1995 these
tests were repeated on the first commercial PMUs. Both PMU types were
found to have excellent steady state and transient response
characteristics. Steady state accuracy with standard scaling was around
0.5%. It could be improved with calibration to 0.1% over the full load
range. Table 3.1 shows a sample of steady state phase and frequency
results. The phase angle measurements showed a constant offset error, so
the deviation figures were 0. Perhaps the most impressive result was the
low measurement noise which was 3-10 times lower than the error. This
allowed determining measurement error with a single point rather than
averaging several readings.

Table 3.1 Sample phase and frequency test results for the commercial PMU
\[37\]

  ------------------ ------------------ ------------------- -------------
  Test Phase (°)     \% Error           Std Dev (°)         Max Dev (°)

  30                 0.024              n/a                 n/a

  45                 0.021              n/a                 n/a

  200                0.002              n/a                 n/a

  Test Freq (Hz)     \% Error           Std Dev (Hz)        Max Dev (Hz)

  59.12              0.002              0.002               0.007

  60                 0                  0.0002              0.001

  60.32              0                  0.0025              0.006
  ------------------ ------------------ ------------------- -------------

The dynamic response tests were equally impressive. Step and ramp
changes in amplitude, phase, and frequency were injected using a digital
model power line. All changes were well within specification and
expectation. The PMU responded to step changes within a cycle with no
overshoot. The frequency measurement tracked changes within 1.5 cycles.
Figure 3.2 shows the response with 59-61-59 Hz frequency ramp.

![](media/image2.wmf)

Fig. 3.2. Frequency measurement response to a 59 to 61 Hz continuous
ramp (sawtooth).

# Reliability of Data Delivery

The reliability of data delivery depends on two requirements: (a)
ability of the GPS-synchronized equipment to output correct data
continuously (for example, some event may interrupt phasor computations
for a short time), and (b) ability of the communication layers to
deliver data without error or loss and on time.

Standards for item (a) above do not exist. Standards for item (b) exist
in various forms and in general are very complex. Discussion of these
issues is provided below. For the purpose of setting a goal regarding
reliability of data delivery, the following requirements will be placed
for layer 1 applications:

1.  1\. Correct data streaming from GPS-synchronized equipment will not
    be interrupted more than once in any 10 minute interval and no
    interruption can exceed 0.5 seconds.

    2\. The reliability of the communication layer should be better than
    0.999 with the ability to locally store data and avoid data loss.

System performance must be evaluated by considering what each element in
the system does. The PMU makes the actual measurement and transmits the
data digitally. Communications move the measurement to the PDC. The PDC
correlates, stores, and re-distributes it. Once in digital form, the
measurements do not degrade. They may be lost, misplaced, or delayed at
various stages of transmission and handling. Consequently, performance
measures for the system after the PMU are data loss, component and
system reliability, and operation speed.

A fully functional PDC records operating statistics including data loss,
signal loss, and PMU time synchronization failures. To do this the PDC
checks each packet for errors and keeps a record of the number of
packets with checksum errors as well as the total number received. If
packets are not received for a short period of time, it records a signal
loss. This is usually a communication failure but could also indicate a
PMU failure. Loss of synchronization must be detected by the PMU as it
is not always evident in the data. The BPA PDC records both short term
and long term sync losses for system monitoring. A typical performance
summary is shown in Table 4.1, which shows the BPA phasor system
performance reliability over a 2 month period. Signal Reliability is %
of time data is continuously received, and sync reliability is the % of
time the PMU is synchronized with GPS \[37\].

A PDC or complete communication system failure would cause loss of all
data. The only PDC failures recorded at BPA have been due to program
'bugs' that stalled the system. The PDC has a hardware watchdog timer
for this possibility, so if the core program fails, it will
automatically re-start and be fully functional within 2 minutes.
Non-core functions, such as recording disturbance files on disc, can
fail and not automatically restart, but the core input program will
continue. Very few failures of core or non-core programs have been
recorded during the last 7 years of continuous operation, and no
significant data loss has occurred. BPA employs several independent
communication systems, so no complete failures have occurred. There has
been some data loss from a particular substation due to the failure of a
single channel.

Table 4.1 Phasor System Performance Reliability

  --------------- ------------- ---------------- ----------------------------
  Station         Reliability                    Notes
                  (%)                            

  (PMU)           Signal        Sync             

  GCoul           97.52         99.974           PMU fail 2 days

  JDay            99.929        99.996           Normal, modem

  Malin           99.997        93.74            PMU clock failure

  Colstrip        99.82         100              Comm sys problems

  BigEddy         99.99         99.988           Normal, fiber, digital

  MValley         99.983        99.74            PMU clock problems

  Keeler          99.996        99.95            Normal, modem
  --------------- ------------- ---------------- ----------------------------

Many aspects of operation speed could be considered. Each system element
must operate at sufficient speed to process data without loss. More
speed does not improve loss or reliability, but can reduce delay.
Latency, the delay from event to data availability, is very important
for control applications. Delays in the order of milliseconds can be
critical. Latency can be determined for various stages in the system.
The first stage is time of measurement. The PMU computes phasors over
some number of power line cycles which will delay the apparent time of
event due to averaging. It will have additional delays due to filtering
and signal processing. The second delay is the time of transmission
between the PMU and the PDC. Third is the input processing time of the
PDC. Last is the time to output the data to a controller. (A control
system will have additional delays for controller, communication, and
actuators). For state estimation, latency may not be a critical issue if
phasor data are included into a traditional state estimator which
typically runs once every few minutes.

The follow sections present discussion and evaluation of various options
that affect reliability of data delivery.

## Data reliability of GPS-synchronized equipment

Present GPS-synchronized measurement equipment has different methods of
data generation priorities. For example dedicated GPS-synchronized
equipment may have an independent channel for each measurement and
computation of phasors frequency and rate of change of frequency that is
uninterrupted from other calculations. Other equipment may use
multiplexing and the priority of the phasor, frequency and rate of
frequency change computational algorithms may be affected by other
functions. If the priority of such computations is suspended, the
accuracy of the data will be also suspended.

## Data access

Present GPS-synchronized measurement equipment has different methods of
data access and different levels of data availability. For example one
manufacturer outputs only the phasors without access to the data of all
three phases or point-on-waveform data. Some devices have RS232 ports,
other Ethernet, etc. Some devices are interfaced to data concentrators
at the substation and the data concentrator is charged with
communications. With the EIPP scheme, each GPS synchronized measurement
device should communicate with the PDC and each PDC should communicate
with other PDCs. In other words there are several levels of
communications.

## Communication standard

Within WECC, much work has been performed that resulted in standards for
communicating and transferring data. The C37.118 IEEE Standard defines
the data formats for PMU to PDC and PDC to PDC communications. WECC has
also defined a standard protocol for PDC to PDC communications. These
standards are usable today and should be retained.

There is also activity in defining communications protocols for more
general applications. The IEC Technical Committee 57 Working Group 03
(TC57 WG03) has the responsibility of developing a standard for
telecontrol, teleprotection and associated telecommunications for
electric power systems. The group has created the IEC 60870-5, a group
of five utility-specific protocol standards. The IEC standard is being
accepted by all major manufacturers of intelligent equipment for utility
applications.

It is important to note that there is a way to merge the C37.118
standard within the IEC 60870.

The issue of network and protocols is a very complex issue and a moving
target. There is a great deal of activity that has resulted in several
interoperability projects and demonstrations. On the other hand BPA/PNNL
has invested a great deal of effort in developing
communications/network/protocols that are close to some standards but
not interoperable with the new IEC standard. The issue of
standardization should be extensively discussed and decisions should
take place early in the game. One way to approach it is to develop
filters/converters at the PDC level that will make the EIPP
interoperable with other IEDs. It is important to recognize that the
deployment of relays with GPS-synchronization is presently occurring and
they will proliferate in the future. If we tap on this resource, it will
dramatically boost the EIPP project. As the industry is moving in this
direction, the two major issues will be: (a) interoperability (the relay
manufacturers are addressing this issue) and (b) throughput.

# Suggested Minimum Requirements 

This section provides the minimum requirements for the various
components of the phasor measurement network. It is based on the
discussions and analysis of the issues presented in the previous
sections. The suggested minimum requirements are deemed adequate for the
layer 1 applications of this system, namely streaming of phasor data and
event capturing. The requirements would be addressed at the following
aspects: timing accuracy; measured quantities; data sample rates; phasor
data rates; phase angle accuracy; magnitude accuracy; TVE (Total Vector
Error); allowable latency; data formats; data transfer protocols;
database; etc.

## GPS-synchronized Equipment 

Table 5.1 summarizes the minimum requirements for phasor measurement
units.

Table 5.1 GPS-synchronized Equipment Suggested Minimum Requirements

+---------------+------------------------------------------------------+
| **Sp          | **Suggested Minimum Requirement**                    |
| ecification** |                                                      |
+---------------+------------------------------------------------------+
| Sy            | Data synchronized with UTC time from a GPS receiver  |
| nchronization | with accuracy of 1 microsecond or better.            |
| Mechanism     |                                                      |
+---------------+------------------------------------------------------+
| Measured      | Voltage phasors, Current phasors, Frequency (f),     |
| Quantities    | rate of frequency change (df/dt), digital status     |
|               | indications                                          |
+---------------+------------------------------------------------------+
| Frequency     | At least 10 Hz bandwidth (55-65 Hz) within 3 dB; -40 |
| Response      | dB rejection at frequencies above the sampling       |
|               | Nyquist frequency, and -60 dB at power system        |
|               | harmonic frequencies                                 |
+---------------+------------------------------------------------------+
| Raw Data      | Minimum of 1440 samples per second                   |
| Sampling Rate |                                                      |
+---------------+------------------------------------------------------+
| A/D           | 12 Bits                                              |
| Conversion    |                                                      |
| Resolution    |                                                      |
+---------------+------------------------------------------------------+
| Phasor Data   | Support IEEE 1344, upgrade to IEEE C37.118 in 2006:  |
| Rate          | 30 samples per second                                |
+---------------+------------------------------------------------------+
| Streaming     | IEEE1344, upgrade to C37.118 in 2006                 |
| Data Format   |                                                      |
+---------------+------------------------------------------------------+
| Event         | Over-Current, Under/Over-Voltage,                    |
| Triggering    | Under/Over-Frequency, Under/Over df/dt, status       |
|               | change                                               |
+---------------+------------------------------------------------------+
| Naming of     | New Standard under development (WG Report: File      |
| Event Files   | naming convention for time sequence data)            |
+---------------+------------------------------------------------------+
| Event Data    | COMTRADE or BPA PhasorFile                           |
| Formats       |                                                      |
+---------------+------------------------------------------------------+
| Event Data    | Phasors, Optionally waveforms, Individual phase      |
| Types         | phasors                                              |
+---------------+------------------------------------------------------+
| Event Data    | 10 cycles of pre-trigger data, 1 second total        |
| Record        | duration                                             |
| Lengths       |                                                      |
|               |                                                      |
| (Waveform     |                                                      |
| Data)         |                                                      |
+---------------+------------------------------------------------------+
| Event Data    | 30 seconds of pre-trigger data, 120 seconds total    |
| Record        | duration                                             |
| Lengths       |                                                      |
|               |                                                      |
| (Phasor Data) |                                                      |
+---------------+------------------------------------------------------+
| Event Data    | 1000 events                                          |
| Retention     |                                                      |
+---------------+------------------------------------------------------+
| Continuous    | 14 days                                              |
| Data          |                                                      |
| Retention     |                                                      |
|               |                                                      |
| (Phasor Data) |                                                      |
+---------------+------------------------------------------------------+
| Data Link     | Ethernet (Optional leased line, dial-up, microwave)  |
+---------------+------------------------------------------------------+
| Protocol      | UDP, (Optional TCP/IP)                               |
+---------------+------------------------------------------------------+

##  {#section-2 .unnumbered}

## Phasor Data Concentrators (PDC) 

A Phasor Data Concentrator is a logical unit that collects phasor data,
and discrete event data from PMU's and other PDC's, and transmits data
to other applications. PDC's should have storage capability to buffer
data for a reasonable time to allow data alignment and other vital
tasks. Thus, a PDC is capable of receiving, aligning, storing and
transmitting GPS-synchronized data. Table 5.2 below summarizes the
minimum requirements for a phasor data concentrator. The minimum
requirements enable applications of streaming phasor data and event
capturing.

Table 5.2: Phasor Data Concentrator Suggested Minimum Requirements

+------------------+---------------------------------------------------+
| Specification    | Suggested Minimum Requirement                     |
+------------------+---------------------------------------------------+
| Input Data       | IEEE1344, upgrade to C37.118 in 2006 if available |
| Format           |                                                   |
|                  | Optional: OPC                                     |
+------------------+---------------------------------------------------+
| Output Data      | IEEE 1344. Upgrade to C37.118 in 2006. (Optional: |
| Format           | PDC Stream, PDCxchng)                             |
+------------------+---------------------------------------------------+
| Data Alignment   | Adopt BPA standard\*                              |
+------------------+---------------------------------------------------+
| Output Data Rate | It should support IEEE1344 and C37.118. Default   |
|                  | value: 30 samples per second                      |
+------------------+---------------------------------------------------+
| Streaming        | User Defined Configuration                        |
| Channels         |                                                   |
+------------------+---------------------------------------------------+
| Continuous Data  | 32 Days                                           |
| Retention        |                                                   |
+------------------+---------------------------------------------------+

\* We recommend adaptation of the BPA standard described in reference.

It is important to note that it is possible that a PDC may receive data
from PMUs from different manufacturers. Aligning data from different
manufacturer PMUs may be a complex task that requires knowledge of the
characteristics of each unit. For application level one, the alignment
of data is done on the basis of the time tag that each PMU data has.
This may result in misalignments of several microseconds. For streaming
data applications and event capturing applications, this misalignment is
not critical. For other applications it may be critical.

## Databases 

A database resides locally with the PDC to read, record, store, and
manage phasor data. Table 5.3 below summarizes the minimum requirements
for a phasor database. The PDC software should be flexible to
accommodate user selections.

Table 5.3: Phasor Database Suggested Minimum Requirements

+------------------+---------------------------------------------------+
| Specification    | Suggested Minimum Requirement                     |
+------------------+---------------------------------------------------+
| Data Recording   | Circular Buffer (new data overwrites oldest data) |
| Organization     | -- Length of buffer should be user selected       |
+------------------+---------------------------------------------------+
| Data Retention   | Archive important data                            |
+------------------+---------------------------------------------------+
| Data Compression | Lossless Compression                              |
+------------------+---------------------------------------------------+
| Input Data       | OPC, PDCstream, COMTRADE. IEEE 1344. Upgrade to   |
| Format           | C37.118 in 2006                                   |
|                  |                                                   |
|                  | Optional: (Optional PDCxchng, Phasorfile, XML)    |
+------------------+---------------------------------------------------+
| Output Data      | OPC. Optional: IEEE 1344, C37.118, PDCstream,     |
| Format           | COMTRADE, PDCxchng, Phasorfile, XML               |
+------------------+---------------------------------------------------+
| Continuous       | Yes                                               |
| Recording        |                                                   |
+------------------+---------------------------------------------------+

## Communication Network 

The network comprises the physical communication paths for phasor data
exchange. Table 5.4 summarizes the minimum requirements for a network.

Table 5.4: Communication Network Suggested Minimum Requirements

  -------------------------------- --------------------------------------
  Specification                    Suggested Minimum Requirement

  PMU to PDC                       Continuous link , leased line network.

  PDC-PDC                          T1 leased line

  DB-DB                            Internet (Optional NERCNet)

  Bandwidth PMU to PDC             36800 baud (Optional 10/100 T-Base)
                                   14,400 BPS

  Security                         VPN/Internet

  Latency                          5 seconds

  Protocols                        See section 4 in \[38\]
  -------------------------------- --------------------------------------

Data communications are usually organized as multilayer entities. The
lowest layer upon which all others rest is the physical layer. Six
layers exist above the physical layers, where various protocols are
implemented. Detailed analysis and evaluation of each communication
layer is beyond the scope of this document. However, it is useful to
touch on the requirements of the bottom layer -- the physical layer.
There are several options for implementing the physical communications
layer of the data transfer:

• Serial asynchronous communication channels (RS-232, 9.6 to 64 kbps)

1.  • 10/100 Mbps Ethernet network interface

    • SONET

The RS-232 standard is relatively simple and has been effectively used
for many years in transferring data from PMU's to data concentrators.
The main drawback of the RS-232 based communications is limited
bandwidth. Table 5.5 illustrates the bandwidth requirements of PMU data
transfers as a function of number of channels, and data format. The
communications channel bandwidth is expressed in kilo bits per second
(kbps). The table assumes a rate of 60 phasors per second for each
channel. The data from each sampling instant are organized into packets
which include the phasor values as well as additional information such
as PMU identification and status codes as defined in the IEEE C37.118
Standard.

Table 5.5: Required Communications Bandwidth for PMU Data

+-----------+---------------------+-----------------------------------+
| Number of | Required Bandwidth  | Required Bandwidth                |
|           |                     |                                   |
| Phasors   | (16 Bit Integer     | (32 Bit Floating Point Format)    |
|           | Format)             |                                   |
+-----------+---------------------+-----------------------------------+
| 2         | 15.3 kbps           | 21.1 kbps                         |
+-----------+---------------------+-----------------------------------+
| 4         | 19.2 kbps           | 28.8 kbps                         |
+-----------+---------------------+-----------------------------------+
| 6         | 23.0 kbps           | 36.5 kbps                         |
+-----------+---------------------+-----------------------------------+
| 12        | 34.6 kbps           | 59.5 kbps                         |
+-----------+---------------------+-----------------------------------+
| 24        | 57.6 kbps           | 105.6 kbps                        |
+-----------+---------------------+-----------------------------------+
| 100       | 203.5 kbps          | 397.4 kbps                        |
+-----------+---------------------+-----------------------------------+

Comparing the required bandwidth to that of a standard 56kbps RS232 port
it is concluded that RS232 is usable for up to 24 channels for 16 bit
integer phasor representation, or 12 channels for 32 bit floating point
phasor representation. Thus, for a higher number of channels a different
communications hardware layer must be selected. For example, it is
estimated that a 10Mbps Ethernet will support up to 1500 phasors
originating from 150 different PMU's at a rate of 30 samples/sec, while
a 100Mbps Ethernet will support 15,000 phasors originating from 1500
PMU's.

It is expected that due to bandwidth requirements, data links beyond the
PMU to local PDC will require faster communication hardware layer such
as 10/100Mbs Ethernet. It is important to note that the Ethernet network
to be used for real-time phasor data communications must be free of data
transfer latencies. It is possible to deploy such networks with proper
equipment and protocol selection. The details of this process are beyond
the scope of this document and are best handled by IT staff.

# Integration of Phasor Data with State Estimation

As stated above, SCADA measurement is the main real-time data source to
State Estimation and PMU data are different from the common SCADA data.
Many practical implementation issues occur when the PMU data are
introduced in State Estimation \[19\]:

1.  PMU data conditioning. PMU data quality needs to be checked before
    they are used by state estimation. Similar to SCADA data problems,
    PMU data may be static over some time, with flipped signs, have
    scaling problems, etc.

2.  The angle measurements available to SE may be absolute or relative
    angles. If absolute angles, the reference is in fact the UTC time
    and the GPS satellites are usually reliable. But bad data, data
    drop-outs and other issues during communication need to be
    considered. This is especially important for the absolute angle
    measurements since they change very quickly. If relative angles,
    users should be careful with selection of a reference PMU and
    re-assignment of the reference in case it fails. Although the
    location of the reference PMU has no impact on SE, the PMU data are
    all relative to the same reference. A reference PMU (maybe a pseudo
    PMU) has to guarantee to work all the time.

3.  Calibration of PMU data from different vendors. Research has found
    that PMU vendors have different measurement values of the same data,
    which may be caused by the device or the algorithm. In such cases,
    measurements will need to be calibrated and offsets are used to
    correct the data.

4.  Calibration of PMU data from different utilities. PMU calculates
    positive sequence data based on the three-phase (ABC) measurements.
    Utilities may have different name conventions for their phases. Also
    the substation names, node names, and transmission line names. These
    differences need to be considered during the modeling.

5.  Synchronization of the PMU (received 20-30 times a second) and SCADA
    data (updated every 4 seconds or even longer). The general State
    Estimation program (which typically runs every 2-3 minutes) uses one
    snapshot of SCADA data. SCADA data time skew is a big problem for
    the present State Estimation. With the accurate time stamp added to
    the PMU data, synchronized PMU data set can be used in State
    Estimation. Notice that phasors rotate at a rate that is
    proportional to the frequency deviation from nominal, so any time
    skew among the phasors will render as useless the phase angle
    differences. Since only a small portion of the substations are
    equipped with PMUs, the majority of SCADA data are still not
    synchronized. Determining the best time that PMU data to be used
    with other SCADA data needs a lot of system case studies.

6.  Tuning of the PMU data weights for optimal online performance. It
    should be noted that PMUs still rely on PTs/CTs to collect the data.
    These PTs/CTs are also used by other devices such as RTUs to measure
    the voltage and current, which are sent to SCADA. The PMU data
    weights in State Estimation need to consider two factors, PT/CT and
    PMU errors. Compared to other SCADA data, PMU data have the
    advantage for synchronization and can be assigned a higher weight.

# State Estimation Metrics 

This section is an excerpt from the report "Metrics for Determining the
Impact of Phasor Measurements on Power System State Estimation" that was
prepared by KEMA, Inc. for the Eastern Interconnect Phasor Project in
March 2006.

It is desired that a commercial state estimator excel in four principal
areas:

-   **Accuracy** -- The state estimator's principal function is to
    determine actual system conditions using noisy measurements. It is
    desired that estimated quantities be as close as possible to their
    true values.

-   **Performance** -- The outputs from the state estimator are needed
    in a timely fashion by other applications in a control center
    environment.

-   **Robustness** -- In practice, commercial state estimators must
    operate in the presence of measurement errors that violate
    uncertainty assumptions, network topology that differs from that
    reported by the SCADA system, and network model parameters that are
    inconsistent with observed power flow measurements. A good state
    estimator is robust in that it is able to detect and correct for
    these problems.

-   **Completeness** -- The state estimator is required to provide
    estimates of power system quantities for as much of the network as
    possible.

The sections that follow propose metrics for each of these categories.
The purpose of these metrics will be to provide a basis for measuring
the effects phasor measurement units will have on the results of
commercial state estimators. This places the requirement on any proposed
metric that it should not make any assumptions about or place any
restrictions on the algorithms used to perform the estimation. While the
literature is replete with comparisons of different estimation
algorithms, commercial state estimators wrap proprietary algorithms, and
metrics of their performance thus must necessarily take a 'black box'
perspective. In addition, a good metric should be relatively easy to
design and test for. Difficult tests are less likely to be executed.

With all these criteria in mind, an initial set of metrics was
identified and developed. The set was reviewed, augmented, refined and
culled by a panel of helpful reviewers[^1]. The final recommended set of
metrics and supporting comments owe much to their contributions.

Because nearly every control area or electricity market is connected to
neighboring transmission systems that influence system behavior, it is
nearly always necessary to extend the network model used for state
estimation beyond the primary regions of interest into areas where
modeling is typically less precise and telemetry less pervasive.
Accordingly, it is expected and accepted that estimation of power system
quantities will be better for the immediate network than for outlying
regions. In this document, these networks shall be referred to as the
internal and external networks, respectively.

The terms internal and external are intentionally loosely defined, as an
appropriate partition of the network model into these two components
will vary by state estimator application. The metrics introduced in the
remainder of this document will refer often to the internal network
exclusively, with the assumption that the internal network corresponds
to the regions where state estimator performance is needed to be the
best. This is not a prescriptive limitation, however, and all metrics
identified within this report can be applied to the internal, external,
or whole networks as needed.

## Accuracy 

The intended use of a state estimator solution must be considered when
defining the criteria for accuracy metrics. In control centers and
market operations centers today, the state estimator is required to
deliver a power flow solution that, as closely as possible, represents
current conditions in the field. As a tool for helping to ensure system
security, it is important that the state estimator solution indicate to
operators actual or near violations where they exist without triggering
alarms from false positives.

For market operations, the state estimator output is required to provide
a power flow solution. This solution is commonly used to provide total
system loss estimates, load bus allocations, and generator injections,
all of which have direct effect on financial settlements with market
participants. Further, the calculation of ex post locational clearing
prices by some markets \[45\] involves linearization around the
quiescent operating condition provided by the state estimator solution.

### Real-Time Accuracy Metrics

Unfortunately, it is very difficult to judge the accuracy of the state
estimator during real time operation due to inability to know the actual
system conditions. What can be ascertained in real-time, however, is
whether the state estimator solution (including derived quantities)
satisfies important necessary conditions to be a power flow solution.

The sum of both real and reactive power injections and flows into each
bus must be identically zero to satisfy network power flow equations.
For any given estimate of the state, the estimated real and reactive
power flows through branches and shunts is uniquely (and easily)
determined. The sum of these branch and shunt flows necessarily must be
exactly offset by the estimated real and reactive net injections from
generation and load.

Only for buses known to have no injection can a non-zero net injection
estimate be known to be incorrect and labeled as a mismatch. Where the
number of generators and loads is known to be exactly one (1) at a bus,
the estimated injection for that apparatus may be uniquely assigned. In
this case, it is not proper to consider the difference between the
estimated injection and available local injection telemetry as a
mismatch indicating inaccurate estimation; the difference is only a
measurement residual the statistics of which can be examined in the
context of bad data identification.

When multiple devices exist, network equations do not provide a
mechanism to determine the split of the estimated net injection among
the devices; if individual injection estimates are required (as they are
for real time market clearing, for example) then a heuristic allocation
scheme must be applied. If the net injection to be assigned among the
devices at the bus is consistent with the known capabilities of the
devices, a full allocation of the injection should be straightforward.
In other cases, (e.g. allocation of a net injection of 300MW to two
generating units, each known to be limited to 100MW in capacity), the
algorithm designer will have to choose between device level consistency
and nodal zero net injection.

With some reservation, the following are proposed as metrics related to
injection mismatch:

  -------------------- -------------------------------------------------------------------------------------------------------- ------------------------
                       $$M_{P\text{mismatch}} = \underset{b \in B\text{int}}{\text{max}}\left| \sum_{j}^{}P_{j}^{b} \right|$$   (7.1)

                       $$M_{Q\text{mismatch}} = \underset{b \in B\text{int}}{\text{max}}\left| \sum_{}^{}Q_{j}^{b} \right|$$    (7.2)
  -------------------- -------------------------------------------------------------------------------------------------------- ------------------------

These metrics measure real and reactive power mismatch for the internal
network, defined by the set of buses *Bint*. The quantities included in
the sums for (7.1) and (7.2) are all the estimated real and reactive
network flows and estimated injections at each bus. That is, the
quantities used to calculate (7.1) and (7.2) should exactly match those
corresponding to the 'power flow' solution from the state estimator used
as input to subsequent processes and calculations. These metrics do not
measure accuracy, only the ability of the estimator to meet an important
necessary condition for a solved power flow solution.

### Controlled Test Accuracy Metrics

Other metrics for state estimation accuracy can be applied by operating
the state estimator in a controlled test environment where the 'true'
system conditions are determined from a solved power flow and the
'actual' measurements are created by perturbing quantities derived from
the power flow solution with noise. Such experiments are routinely
performed by state estimator vendors during the refinement and tuning of
their algorithms. In short, the test procedure follows this pattern:

1.  For a given network model, system conditions and topology, solve a
    power flow and extract voltage magnitudes, angles, branch real and
    reactive flows, injections, etc.;

2.  Create measurement data to be submitted to the state estimator by
    perturbing the corresponding quantities from the power flow
    solution;

3.  If desired, introduce topology or parameter 'errors' to alter the
    network model to be used by the state estimator;

4.  Produce the state estimator solution, including derived quantities;

5.  Evaluate the state estimator's performance by comparing quantities
    from the power flow solution to those derived from the state
    estimator solution.

The most obvious (and convenient) way to generate an accuracy metric is
to choose a power flow solution quantity of interest (e.g. bus voltage
magnitude) and define a norm-like calculation on the difference between
the 'true' value (from the power flow solution) and the estimated value,
derived from the state estimate.

Before discussing appropriate quantities for comparison it is worthwhile
to examine some implications of the form of the metric. Consider, for
example, a network model with *L* branches and three candidates to
measure accuracy of real power flow *Pf* on these branches[^2]:

  --------------------- ------------------------------------------------------------------------------------------------------------------- -----------------------
                        $$\sum_{j = 1}^{L}\left| \text{Pf}_{j}^{\text{true}} - \text{Pf}_{j}^{\text{est}} \right|$$                         (7.3)

                        $$\sum_{j = 1}^{L}\left( \text{Pf}_{j}^{\text{true}} - \text{Pf}_{j}^{\text{est}} \right)^{2}$$                     (7.4)

                        $$\underset{j = 1\ldots L}{\text{max}\ }\left| \text{Pf}_{j}^{\text{true}} - \text{Pf}_{j}^{\text{est}} \right|$$   (7.5)
  --------------------- ------------------------------------------------------------------------------------------------------------------- -----------------------

 

The first of these (7.3) is the 1-norm of the estimation error, and is
proportional to the average error in branch flow estimation. The second
(7.4) is the square of the 2-norm (Euclidean norm) of the estimation
error. The third (7.5) is the infinity norm of the estimation error,
i.e. the 'worst case' error. All three seem reasonable, in that larger
values indicate worse performance, and zero error results in a zero
calculation.

Each of (7.4) and (7.5) can return very different 'scores' for nearly
equivalent network models with the same state estimator solution and
same power flow solution. The top half of Fig. 7.1 shows a simple
(lossless) branch with a 0.2 p.u. error in estimation of real power
flow. In the bottom half of the diagram, additional buses have been
added to split the line into three line segments; the actual and
estimated flows remain the same. With three line segments instead of
one, the contribution to the 1-norm (7.3) is three times as much,
although both models represent equivalent subsets of the network.

Only the infinity norm, (7.5), is immune to these effects, in that it
always reports the 'worst case' estimation error. Unlike the first two,
however, one lone bad estimation error can result in a poor score,
incorrectly suggesting the estimation was poor.

In conclusion, it is perhaps important to keep in mind that metrics
similar to those shown above usually lack a meaningful physical
interpretation, and are somewhat sensitive to model construction.
Accordingly, while they are useful to compare differences in algorithm
or differences in measurements for a fixed network model, their use in
other circumstances should be approached with caution.

With these caveats in mind, two simple metrics are now introduced that
judge the state estimator's ability to estimate the most important
quantities defining a power flow solution: voltages and flows.

An important attribute of the state estimator is its ability to predict
real and reactive power flow on branches. By combining these as one
complex quantity ($\overrightarrow{S} = P + \text{jQ}$), the following
metric for complex power flow estimation accuracy is proposed:

  ------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------
                $$\text{Macc}_{S\text{flow}} = \left( \sum_{j}^{}\frac{\left| {\overrightarrow{S}}_{j,\text{from}}^{\text{true}} - {\overrightarrow{S}}_{j,\text{from}}^{\text{est}} \right|^{2} + \left| {\overrightarrow{S}}_{j,\text{to}}^{\text{true}} - {\overrightarrow{S}}_{j,\text{to}}^{\text{est}} \right|^{2}}{\text{MVA}_{j}^{2}} \right)^{\frac{1}{2}}$$   (7.6)

  ------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------

In (7.6), the summation index *j* ranges over all branches in the
internal network; as indicated in the equation, error terms for complex
flow at each end of the branch should be included. The term *MVA* in the
denominator is a limit of the branches capacity; its inclusion in
recognition that a 10MW estimation error, for example, is more
operationally significant for a small 69kV line than for a large 500 kV
branch. Importantly, note that each squared term is the magnitude of a
difference of complex numbers, not the difference of magnitudes.

A commonly reported accuracy metric for state estimation in many fields
of application, including power system state estimation, is a norm of
the error of the state estimate. For nearly all power system state
estimator implementations, nearly all the elements of the state vector
consist of bus voltage magnitudes and angles. While it is reasonable to
compare voltage magnitude errors and voltage angle errors separately, it
is straightforward to combine both in a norm metric that captures the
effect of both:

  --------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------
                        $$\text{Macc}_{V} = \left\| {\overrightarrow{V}}^{\text{error}} \right\|_{2} = \left( \sum_{j}^{}\left| {\overrightarrow{V}}_{j}^{\text{true}} - {\overrightarrow{V}}_{j}^{\text{est}} \right|^{2} \right)^{\frac{1}{2}}$$   (7.7)

  --------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------

In (7.7), $V_{i}^{\text{true}}$ and $V_{i}^{\text{est}}$ are the 'true'
and estimated complex phasor voltage at the *j*th bus of the internal
network reported on a per unit basis. It is important, of course, that
the same reference bus be used for both 'true' and estimated voltages.

Different weighting factors than those used in (7.6) and (7.7) may prove
more useful for specific applications. If the investigator finds it
necessary to adjust the weightings for a specific purpose, it is
recommended he/she also report test statistics using the formulae as
specified here.

The accuracy metrics (7.6) and (7.7) were designed and chosen to test
the ability of the state estimator to match a controlled test model in
the aggregate; it is suggested that a state estimator that returns good
(low) values on these metrics and on the mismatch metrics (7.1) and
(7.2) has done a good job of producing a power flow solution matching
system conditions. The use of the 2-norm is a compromise between the
1-norm, related to the average estimation error over estimated
quantities, and the infinity norm, which while useful for 'worst case'
analysis punishes state estimators which produce only a handful of bad
estimates. The 2-norm is comprehensive in that it sums over all
quantities, and punishes bad estimates with the quadratic term.[^3]

### Other Candidate Measures of Accuracy

The value of the objective function is nearly always readily available
upon completion of state estimation. Certainly for fixed inputs
(measurements, topology and model parameters), an improvement in the
objective function indicates superior state estimator performance. It is
tempting to compare objective function values with and without phasor
measurements to measure the impact of the phasor measurements on the
state estimator performance.

It is straightforward to show, however, that a weighted least squares
estimator capable of finding the global optimum will always return a
larger objective function value when new measurements are added.
Consider the optimal objective function value *J*~1~ for a weighted
least square (WLS) implementation with *m* (non-PMU) measurements
*z~i~,* where *i = 1...m:*

  ---------------------- ----------------------------------------------------------------------------------------------------------------------- --------------------
                         $$\begin{array}{r}                                                                                                      (7.8)
                         J_{1} = \underset{\hat{x}}{\text{min}}\sum_{i = 1}^{m}\left( \frac{z_{i} - h_{i}(\hat{x})}{\sigma_{i}} \right)^{2} \\   
                         s\text{.}t\text{.}\ c(\hat{x}) = 0                                                                                      
                         \end{array}$$                                                                                                           

  ---------------------- ----------------------------------------------------------------------------------------------------------------------- --------------------

 

In (7.8), the vector $\hat{x}$ is the estimate of the state,
$h_{i}(\hat{x})$are the nonlinear power flow equations describing the
*m* measurements, and $c(\hat{x}) = 0$are (optional) constraints
representing, for example, equality constraints included for
zero-injection buses. Adding only *p* new measurements (from PMUs or
elsewhere) and reëstimating always yields an objective function value
*J~2~* that is at least as great as *J~1~*:

  --------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------
            $$\begin{matrix}                                                                                                                                                                                                                (7.9)
            J_{2} & & \underset{\hat{x}}{\text{min}}\left( \sum_{i = 1}^{m}\left( \frac{z_{i} - h_{i}(\hat{x})}{\sigma_{i}} \right)^{2} + \sum_{j = m + 1}^{m + p}\left( \frac{z_{j} - h_{j}(\hat{x})}{\sigma_{j}} \right)^{2} \right) \\   
             & & \underset{u}{\text{min}}\sum_{i = 1}^{m}\left( \frac{z_{i} - h_{i}(u)}{\sigma_{i}} \right)^{2} + \underset{v}{\text{min}}\sum_{j = m + 1}^{m + p}\left( \frac{z_{j} - h_{j}(v)}{\sigma_{j}} \right)^{2} \\                 
             & & J_{1} + \underset{v}{\text{min}}\sum_{j = m + 1}^{m + p}\left( \frac{z_{j} - h_{j}(v)}{\sigma_{j}} \right)^{2} \\                                                                                                          
             & & J_{1}                                                                                                                                                                                                                      
            \end{matrix}$$                                                                                                                                                                                                                  

  --------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------

 

The inequality shown in (7.9) is derived as follows: The first line
(equality) is the minimum over set $\hat{x}$of the combined sum and is
the value of the objective function when all *m* + *p* measurements are
considered. The second line states that this minimum is greater than or
equal to the sum of the minima over the two sets of measurements (the
sets *u* and *v* are of the same dimension as $\hat{x}$). The third line
recognizes the first (global) minimum in line two as exactly $J_{1}$
from (7.8) above. The last line recognizes the second minimum to be
nonnegative and draws the inequality relationship.

Accordingly, the objective function value cannot be used as a metric for
accuracy to compare state estimator performance with different
measurement sets. It is straightforward to show that other commonly used
objective functions also must increase with added measurements. For this
reason, despite its ease of application, the objective function should
not be used as a metric when comparing the impact of new or different
measurements (e.g. from PMUs) on the state estimator solution.

An accuracy metric for controlled tests cited in the literature is the
Error Estimation Index (EEI) \[31\], given as

  ------------------- ----------------------------------------------------------------------------------------------------------------- ------------------
                      $$\text{EEI} = \sum_{i = 1}^{m}\left( \frac{z_{i}^{\text{true}} - z_{i}^{\text{est}}}{\sigma_{i}} \right)^{2}$$   (7.10)

  ------------------- ----------------------------------------------------------------------------------------------------------------- ------------------

 

where $\sigma_{i}$is the actual standard deviation of the Guassian,
zero-mean random noise used to pollute the noise-free measurement
$z_{i}^{\text{true}}$in creating the noisy measurement $z_{i}$. The
introduction of the normalizing factor $\sigma_{i}$in the denominator
effectively serves to discount measurement estimation errors for noisy
measurements. Recalling that the stated objective of the accuracy metric
is to measure the ability to produce an accurate power flow solution,
the EEI suffers in comparison to the metrics recommended above for two
reasons. First, the introduction of the term $\sigma_{i}$eliminates from
the calculation any power system quantities that are not in the
measurement set. While improving measurement accuracy is one goal of a
state estimator, as important is the ability to provide accurate
visibility of unmetered quantities. Second, the importance of an
accurately estimated quantity to the end user of the state estimator
bears no relation to the measurement error variance; the normalizing
factor $\sigma_{i}$is not appropriate in this context.

A similar candidate metric is the performance index (PIP) used in \[33\]
to determine the ability of the state estimator to accurately discern
real power flow measurements:

  ---------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------
             $$\text{PIP} = \frac{\sum_{i = 1}^{m_{\text{pf}}}\left( \text{Pf}_{{}_{i}}^{\text{true}} - \text{Pf}_{i}^{\text{est}} \right)^{2}}{\sum_{i = 1}^{m_{\text{pf}}}\left( \text{Pf}_{{}_{i}}^{\text{true}} - \text{Pf}_{\text{ei}}^{\text{meas}} \right)^{2}}$$   (7.11)

  ---------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------

 

where *Pf* denotes real power flow on *m~pf~* branches. For good
estimation, the estimate of each flow will lie closer to the true than
will the measured value, and the entire metric will be less than one.
The introduction of the actual flow measurements in the denominator,
however, eliminates the possibility of including estimation accuracy for
unmetered lines. For this reason, (7.11) is not recommended as an
alternative to (7.6).

## Performance

While it is desirable that the state estimator program be as accurate as
possible, there are additional performance requirements observed in
practice that relate to the ability of the state estimator to deliver to
downstream processes a stable solution in reasonable and predictable
time.

### Convergence

The precision, if not the accuracy, of a state estimator that uses an
iterative solution scheme can be judged by its ability to converge. The
confidence that users of real-time state estimators have in the value of
the solution is diminished if significant changes in state variables or
objective function are still occurring at the terminal iteration.[^4]
The following three metrics are either known to be in use today in
control center environments or are minor variations thereof:

  --------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------- ------------------
                                                                                                                                                                                $$\text{Mconv}_{\text{obj}} = \ \left| 1 - \frac{J^{k_{\text{term}}}}{J^{k_{\text{term}} - 1}} \right|$$   (7.12)

                        $$\text{Mconv}_{V} = \underset{i \in B_{\text{int}}}{\text{max}}\left| 1 - \frac{V_{i}^{k_{\text{term}}}}{V_{i}^{k_{\text{term}} - 1}} \right|$$                                                                                                                   (7.3)

                        $$\text{Mconv}_{\theta} = \underset{i \in B_{\text{int}}}{\text{max}}\left| \theta_{i}^{k_{\text{term}}} - \theta_{i}^{k_{\text{term}} - 1} \right|$$                                                                                                              (7.14)
  --------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------- ------------------

For each of these, *k~term~* denotes the terminal iteration. The
metric$\text{Mconv}_{\text{obj}}$ shown in (7.12) measures the relative
change in objective function value *J* at the last iteration, while the
metric $\text{Mconv}_{V}$ shown in (7.3) measures the largest final
relative change in bus voltage magnitude over the buses of interest.
Note that $\text{Mconv}_{\theta}$uses the absolute difference to avoid
problems when the angle is near zero, which will occur near the system
reference bus.

A clarifying example of use of these metrics is given by the following
statement: 'The real-time state estimator solution shall be deemed
acceptable if at termination the power mismatch metric
$M_{P\text{mismatch}}$ is no larger than 50 MW and both of the
convergence metrics $\text{Mconv}_{V}$ and $\text{Mconv}_{\theta}$ are
no larger than 0.002'.[^5] This statement very closely matches the
criteria used by a large North American ISO to monitor their real-time
state estimator performance.

Another convergence metric is suggested by a necessary condition for the
state estimator to find the optimum solution, namely
$\frac{\partial J}{\partial\hat{x}} = 0\text{.}$ If a conventional
weight least squares algorithm is used, the objective function is as
follows:

  -------------------- ----------------------------------------------------------------------------------------------- ------------------
                       $$J(\hat{x}) = \sum_{j = 1}^{m}\left( \frac{z_{j} - h_{j}(\hat{x})}{\sigma_{j}} \right)^{2}$$   (7.15)

                                                                                                                        
  -------------------- ----------------------------------------------------------------------------------------------- ------------------

and the necessary condition for optimality is given by the vector
equation

  -------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------
                       $$g(\hat{x}) = \frac{\partial J}{\partial\hat{x}} = - 2\sum_{j = 1}^{m}{\left( \frac{z_{j} - h_{j}(\hat{x})}{\sigma_{j}^{2}} \right)\frac{\partial h_{j}(\hat{x})}{\partial\hat{x}}} = 0$$   (7.16)

  -------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------

Accordingly, it may be of interest to
report$\left\| g({\hat{x}}^{k_{\text{term}}}) \right\|_{2}$as a scalar
measure of the satisfaction of (7.16) at the terminating iteration
*k~term~*. However, the burden introduced by the complexity of the
calculation does not warrant the recommendation of this measure and a
convergence metric for general use. Algorithm designers and researchers
may find merit in its use, however.

### Timing

The faster that state estimation software executes successfully, the
more realistic a picture of evolving system conditions the operators
will have. Ideally, the estimated state would be available as soon and
as often as the telemetry is updated. A candidate timing metric is the
average execution time in seconds for state estimation, taken over a
significant sample of cases.

Measuring process timing precisely and consistently on real-time systems
operating on computers with many shared processes, however, can be
difficult. Recall that the motivation for this study is the need to
measure the benefits that inclusion of phasor measurement units have on
state estimation. It is the majority opinion of the contributors to this
study that any effect on execution time from the augmentation of the
measurement set with phasor measurements will be not meaningful.
Accordingly, and because in nearly all control center environments
algorithmic execution time is not perceived to be a limiting factor on
system performance, no timing metric is recommended for broad use at
this time.

## Robustness 

Robustness is the ability of the state estimator algorithm to perform
its function in other than ideal conditions. In particular, commercial
state estimators are required to perform in the presence of bad
measurement data, bad network topology information, and bad or uncertain
parameters used to describe the power system network model. In addition,
it is of interest to know the sensitivity of overall estimation accuracy
to the accuracy of the measurement system. Missing measurements are
addressed in a subsequent section on completeness.

### Bad Measurement Identification and Detection

An important property of any state estimator in use is the ability to
identify and reject bad measurements, *i.e.* measurements for which the
observed data has error that is inconsistent with the modeling
assumptions. While bad measurement (and topology, and parameter)
rejection will impact a state estimator's ability to score well on
accuracy tests with properly designed experiments, it is also a simple
matter to construct tests to measure bad measurement identification
explicitly. A flexible metric for this purpose is given as follows:

  ------ ----------------- ------------------------------------------- --------
  * *    *Mmeas~σ~*(*p*,   \% of cases the state estimator correctly   (7.17)
         *a*) =            identifies all *p* bad measurements created 
                           by adding a constant term of *aσ* or *--aσ* 
                           (with equal likelihood) to the measurement, 
                           where *σ*  is the assumed standard          
                           deviation of the measurement error.         

  ------ ----------------- ------------------------------------------- --------

Most commercial state estimators are easily capable of flagging single
or isolated bad measurements; accordingly, the parameter *p* in (7.17)
allows the test designer to build more difficult test cases where
multiple measurements are bad. Multiple sets of input data should be
created to use this metric, and the same data sets should be used when
comparing algorithms or comparing measurement sets. Finally, it is
important that the parameter *a* in (7.17) be large enough so as not to
generate measurements that could alternately be viewed as either bad or
good but unlikely.

The description of (7.17) requires the tester to measure the algorithm's
ability to identify 'correctly' all *p* bad measurements. Two acceptable
standards for correctness are:

1.  Explicit identification of exactly the *p* bad measurements;

2.  Identification of the *p* bad measurements, but also with the
    identification of other measurements as bad (false positives).

The former of these is obviously preferred to the latter, but even the
latter may be considered acceptable; a state estimator that
'accidentally' discards a few good measurements is usually still able to
produce a very good state estimate. When using (7.17) the tester should
be explicit as to what defines correct bad measurement detection when
reporting results.

Bad meter calibration can be simulated by perturbing measured values
with an affine or quadratic transformation. Typically, multiple
measurement scans are required to detect non-random calibration errors
using a generalization of state estimation known as remote meter
calibration. While the use case of meter calibration via generalized
state estimation is beyond the scope considered here, it is a
straightforward exercise to create metrics to measure performance.

It is worthwhile considering a metric for the detection of bad data
(necessary but not sufficient for bad data identification). Consistent
with the assumptions required for WLS estimation to be maximum
likelihood estimation, the measurement errors are usually assumed to be
Guassian with zero mean and known variance. This means that for data
that are not 'bad' that each measurement error when normalized by
dividing by its standard deviation will have a standard normal
distribution

  -------------------- ---------------------------------------------------------------------- ------------------
                       $$e_{i}^{N} = \frac{e_{i}}{\sigma_{i}}\text{\textasciitilde}N(0,1)$$   (7.18)

                                                                                               
  -------------------- ---------------------------------------------------------------------- ------------------

The sum of squares of *m* independent standard normal variables is a
random variable with a $\chi_{m}^{2}$ distribution having mean *m* and
variance 2*m*. If these same variables are coupled by *n* constraints,
their sum of squares with *m-n* degrees of freedom has a
$\chi_{m - n}^{2}$distribution with mean *m-n* and variance 2(*m-n*). It
follows that the sum of normalized measurement errors follows such a
distribution:

  -------------------- ----------------------------------------------------------------------------------------------------------------------------------------- ------------------
                       $$J(\hat{x}) = \sum_{i = 1}^{m}{\left( \frac{z_{i} - h_{i}(\hat{x})}{\sigma_{i}} \right)^{2}\text{\textasciitilde}\ \chi_{m - n}^{2}}$$   (7.19)

                                                                                                                                                                 
  -------------------- ----------------------------------------------------------------------------------------------------------------------------------------- ------------------

Clearly large normalized measurement errors will make the sum
$J(\hat{x})$ large. A single instance of this sum available after state
estimation can be compared to the $\chi_{m - n}^{2}$distribution to
determine the probability *p* the a random draw of the distribution can
be as large as the observed sum:

  -------------------- --------------------------------------------------------------- ------------------
                       $$p = \text{Pr}\left( \chi_{m - n}^{2} > J(\hat{x}) \right)$$   (7.20)

                                                                                       
  -------------------- --------------------------------------------------------------- ------------------

Many commercial state estimator programs compare *p* to a confidence
threshold to decide whether the measurement set contains bad data. If
the threshold remains fixed, then *p* can be used as a metric to
forecast whether the state estimator algorithm can detect (if not
identify) bad data. Because the inclusion of such calculations is
non-trivial, and because most commercial vendors will already be using
χ^2^ analysis in the determination of (7.17), the application of the
technique in a metric is judged not warranted.

### 7.3.2 Bad Topology Detection {#bad-topology-detection .unnumbered}

Perhaps the greatest obstacle to reliable and accurate power system
state estimation is the not uncommon occurrence of the actual network
topology differing from the assumed network model because of inaccurate
knowledge of switch settings. When compared to bad measurement
detection, for which there are reasonably straightforward and commonly
accepted hypothesis tests and strategies for bad measurement
identification and elimination, network topology identification
algorithms are computationally more difficult, more recent in design,
and therefore less pervasive in commercial application.

Very often the SCADA system relies upon manual operator entry of switch
settings to define branch connectivity, and these data are prone to
inaccuracy. Operating with an incorrect topology, the state estimator
struggles to find a system state vector consistent with measured values.
Accordingly, recent advances in commercial state estimation algorithms
known alternately as generalized state estimation or topology estimation
have been focused on the determination (and correction) of incorrect
open/closed breaker settings. Topology estimation can be embedded within
the state estimation algorithm or packaged as a distinct process.

The suggested metric is similar to those proposed for bad measurement
detection:

  ------------------- ------------------------------------ --------------
  *Mtop*(*p*) =       \% of cases the state estimator      (7.21)
                      correctly identifies all *p*         
                      topology errors.                     

  ------------------- ------------------------------------ --------------

### 7.3.3 Parameter Estimation {#parameter-estimation .unnumbered}

The power system model also may have substantially incorrect modeling
parameters for branch reactance, resistance, and line charging
susceptance that adversely impacts the state estimator's ability to
determine an accurate state estimate. Some commercial state estimators
today are capable of parameter estimation, where questionable model
parameters are treated as augmentations of the state vector so that they
may be estimated. If the software permits the detection and estimation
of questionable values, parameter 'errors' can be introduced into the
test case and a metric that is similar to (7.17) may be used.

## 7.4 Completeness  {#completeness .unnumbered}

In addition to refining the accuracy of measurements, the state
estimator is required to provide reasonable estimates of measurable
power system quantities where measurement devices do not exist. The
extent to which this is possible at any given time is measured by
observability; the robustness of observability with respect to
measurement loss is redundancy.

### 7.4.1 Observability {#observability .unnumbered}

One of the primary goals of state estimation is to provide the
estimation of measurements on branches and at buses where no telemetry
exists. It is a standard requirement for commercial state estimation
software to determine the unobservable areas of the network and to
generate a minimal set of pseudo measurements that restores system
observability.

Much published work exists (e.g. \[25\] and \[29\]) on cost-effective
methods to increase system observability by PMUs, using techniques very
much the same as those used for optimal meter placement. Most optimal
placement techniques attempt to restore to full rank matrices associated
with either the topology or the gain matrix[^6]. Accordingly, it is
tempting to propose metrics for observability based upon the rank of
various matrices. However, doing so tends to impose an algorithmic
approach upon the estimation algorithm, which is not appropriate for the
purpose of this study.

Instead, it is proposed to let the estimation algorithm use its own
methods for determining observability and report the number of
in-service buses that are unobservable and require pseudomeasurements:

  ---- -------------- ----------------------------------------- ---------
       *Mobs =*       \# of in-service buses of interest that   (7.22)
                      are unobservable, i.e. are contained      
                      within unobservable islands.              

                                                                
  ---- -------------- ----------------------------------------- ---------

The phrase 'of interest' in (7.22) allows the metric to be applied in a
manner consistent with use of the state estimator (e.g. the internal
network). While identification of the unobservable portion of the
network is nontrivial, this functionality is considered standard on
modern commercial state estimation programs. Accordingly, (7.22) is
recommended because its application in a testing environment is expected
to be straightforward.

### 7.4.2 Redundancy {#redundancy .unnumbered}

Redundancy relates to the effect upon estimation of the loss of one or
more measurements. The most important impact of loss of measurements is
the loss of system observability; a measurement that is required to
maintain observability is said to be a critical measurement. A simple,
but incomplete, measure of a measurement set's redundancy is the number
of critical measurements. The measure is incomplete, because it
considers only loss of observability from single measurements, whereas
remote terminal unit (RTU) failure or communications link failure may
result in loss of several measurements at the same time. Although a
measure for redundancy considering and combining multiple levels of
redundancy has been proposed \[43\], it is not known to be in use and is
not recommended at this time.

For a critical measurement, the measurement residual defined as the
difference between the measurement value and the estimated measurement
value $r_{i} = z_{i} - h_{i}(\hat{x})$ is zero. That is, any error in
the measurement is transferred directly to the estimated measurement.
For any given measurement, a useful measure of redundancy is the ratio
of the variance of the measurement residual to the variance of the
measurement error. As redundancy improves, the sensitivity of the
estimated measurement to measurement errors will decrease, and the ratio
of the variance of the residual variance to the measurement error
variance will approach unity. A suggested measure of redundancy for the
entire system is the minimum ratio over all measurements:

  ------ ---------------------------------------------------------------------------------------------- ---------
         $$\underset{j = 1\text{.}\text{.}\text{.}m}{\text{min}}\frac{\rho_{j}^{2}}{\sigma_{j}^{2}}$$   (7.23)

                                                                                                        
  ------ ---------------------------------------------------------------------------------------------- ---------

Where $\rho_{j}^{2}$ and $\sigma_{j}^{2}$ are the residual variance and
measurement error variance of the *j*th measurement, respectively. If
weighted least squares estimation is used with the error variances used
for weighting (in the usual manner), then the quantity $\rho_{j}$ can be
found by the relationship

  ------ ------------------------------------------------------------------------------------------------------------------------ ---------
         $$\rho_{j}^{2} = \sigma_{j}^{2} - \text{diag}_{\text{jj}}\left( H\left( H^{t}R_{z^{- 1}}H \right)^{- 1}H^{t} \right)$$   (7.24)

  ------ ------------------------------------------------------------------------------------------------------------------------ ---------

* *

where *H* is the (state dependent) Jacobian of the power flow equations
and *R~z~* is the covariance matrix of the measurement errors.

Because the above and similar measures of redundancy are either
algorithm dependent or difficult to implement in a testing environment,
no redundancy metric is recommended at this time. Redundancy measures
can play an important role in sensor placement algorithms (see for
example \[28\] Ch. 7, \[24\] Ch. 4).

## 7.5 PMU Placement for State Estimation {#pmu-placement-for-state-estimation .unnumbered}

Recall that the motivating factor behind the creation of the metrics
summarized in this section is to define a framework for evaluating the
impact of phasor measurements on state estimator results. In general,
most utilities and control areas today have few (if any) PMUs in place,
and also have telemetry in place to provide observability for the
internal network. Accordingly, when considering the location of new PMUs
to increase state estimator performance in a cost effective manner, the
problem to be solved is one of incremental measurement placement --
finding the best places to add 'meters' to a measurement system that is
already fairly well metered.

For the incremental measurement placement problem, it is likely that
accuracy in the state estimator solution is deemed the metric for which
improvement is wanted most. For systems with very few meters, improving
observability may be the most important criteria. Considerable effort
has been devoted to meter placement to improve observability (see for
example \[44\]\[40\]) resulting in techniques applicable to sensors of
all types, including PMUs. More recently, researchers have devoted
specific attention to algorithms for improving observability using PMUs
in particular (see for example \[18\]\[25\]\[29\]).

Most algorithms for improving accuracy of the state estimator solution
via incremental meter placement do so by attempting to reduce
uncertainty in the estimate of the state vector; \[40\] is a good
example. The inverse of the gain matrix,
$G^{- 1} = \left( H^{T}R_{z}^{- 1}H \right)^{- 1}$is easily shown to be
the covariance matrix of the state error vector $x - \hat{x}$ when
weighted least squares estimation is used, and the Jacobian *H* is
evaluated at the final solution. Quite sensibly, many published
algorithms identify candidate measurements for addition by looking for
ways to reduce the large diagonal elements of $G^{- 1}$ (computationally
efficient using sparse inverse techniques). In \[40\], for example, a
subset of available measurements are identified that are proximal to
buses corresponding to large state error variances; from each
measurement in that subset the diagonal elements of $G^{- 1}$ are
reevaluated with the measurement added and the ones with the greatest
variance reductions are ultimately selected.

In other fields where state estimation is regularly performed
(especially target localization and tracking via image sensors (good
recent examples are \[42\]\[49\]) the information theoretic concept of
entropy is used identify optimal sensor placement. First introduced by
Shannon (\[47\][^7]), entropy (also known as information entropy,
Shannon entropy, and differential entropy) is a scalar measure of the
uncertainty associate with a multivariate random distribution. For the
estimated state vector $\hat{x}$, the Shannon entropy is given by the
integral involving the multivariate density function $f(\hat{x})$as

  ------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------
         $$\begin{array}{r}                                                                                                                                                                                                                 (7.5)
         H(\hat{x}) = - \int_{- \infty}^{\infty}{f(\hat{x})\text{ln}(f(\hat{x}))\text{dx}} \\                                                                                                                                               
         \  = - \int_{- \infty}^{\infty}\cdots\int_{- \infty}^{\infty}{f({\hat{x}}_{1},{\hat{x}}_{2},\ldots,{\hat{x}}_{n})}\text{ln}(f({\hat{x}}_{1},{\hat{x}}_{2},\ldots,{\hat{x}}_{n})d{\hat{x}}_{1}d{\hat{x}}_{2}\ldots d{\hat{x}}_{n}   
         \end{array}$$                                                                                                                                                                                                                      

  ------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------

An algorithm based on reduction in entropy to site new meters for state
estimation can be found in \[46\], the key concepts of which are now
summarized here. If all the 'usual' assumptions regarding Guassian
measurement errors are made and weighted least squares estimation is
assumed, then it is straightforward to show that the Shannon entropy is
given by

  ------ ------------------------------------------------------------------------------------------------------------------------------- ---------
         $$H(\hat{x}) = ½n\left\lbrack 1 + \text{ln}(2\pi) \right\rbrack + ½\text{ln}\left| \text{det}\left( G^{- 1} \right) \right|$$   (7.26)

                                                                                                                                         
  ------ ------------------------------------------------------------------------------------------------------------------------------- ---------

where *n* is the number of states. If $\text{LU} = G$ is a decomposition
of *G* into lower (*L)* and upper (*U)* triangular square matrices, then
the last term in (7.26) can be simplified:

  ------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------
         $$\text{ln}\left| \text{det}(G^{- 1}) \right| = \text{ln}\prod_{i = 1}^{n}\left| U_{\text{ii}} \right| = - \sum_{i = 1}^{n}\text{ln}\left| U_{\text{ii}} \right|$$   (7.27)

  ------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------

Determining the reduction in entropy associated with adding a new
measurement requires only to update the diagonal elements of the matrix
*U* resulting from the change to *G* and calculating the sum shown in
(7.27).

An example shows how entropy might be used to site PMUs to improve
measurement accuracy. Consider the IEEE 300 bus test system[^8], which
is sufficiently large enough to be nontrivial and small enough to
facilitate algorithmic investigation. A base measurement set was created
for this system as follows:

1.  Real and reactive power flow measurements were added at one end of
    each of the 411 lines. For each measurement, a standard deviation of
    1.39%[^9] of full scale rating was used, where full scale rating was
    approximated as 150% of the MVA observed for the base case power
    flow.

2.  Real and reactive power injection measurements were added for each
    of 69 generating units. For each measurement, a standard deviation
    of 1.39% of full scale rating was used, where full scale rating was
    approximated as 150% of the injection MVA observed for the base case
    power flow.

3.  Voltage magnitude was measured at every 10% of out of 10 of 300
    buses. For each measurement, a standard deviation of 0.89% of full
    scale rating was used, where full scale rating was approximated as
    1.0 p.u.

The measurement scheme is, of course, purely fictional. The difficulty
in ascertaining measurement errors for actual devices in the field is
very real, however. Very often the weights used in commercial state
estimators are chosen and modified to achieve consistent algorithmic
convergence, and not based upon careful calculation of expected
measurement error variances.

To measure the impact of PMU location on entropy reduction, in sequence
one PMU was added to each bus, and the change on entropy was estimated
using (7.27). At the same time, the percentage reduction in the expected
variance of the phase angle error at the PMU bus was calculated by
observing the change in the corresponding diagonal entry of the state
error covariance matrix$G^{- 1}$.

The calculation results are shown as scatter plots in Figs.7.2-4 for
three different levels of PMU accuracy (standard deviations of 0.001,
0.1 and 1.0 degrees, respectively). Each graph has two sets of data: (1)
entropy reduction vs. % bus angle variance reduction, and (2) entropy
reduction vs. original case bus angle variance. For each scenario, the
voltage phasor measurement error standard deviation was set equal to
0.0089 p.u., matching the assumption made for the other voltage
magnitude measurements.

Several observations can be made from the results:

1.  Adding a PMU always reduces entropy. This is necessarily true;
    adding a new sensor, even one that is known to have very poor
    accuracy, always improves estimation with the addition of new
    information.[^10]

2.  For very accurate PMUs (standard deviation = 0.001 degrees), the
    state estimator is expected to eliminate nearly all (100%) of the
    uncertainty in the angle variance where the PMU is located. This is
    entirely expected, of course.

3.  The reduction in total state uncertainty as measured by the
    reduction in Shannon entropy is decreased as the PMU accuracy is
    decreased.

4.  For accurate PMUs (0.1 degrees), the relationship between local
    uncertainty reduction (angle variance reduction) and total system
    state uncertainty reduction (entropy reduction) by PMU is nearly
    monotonic. Even for inaccurate PMUs (1.0 degrees), the relationship
    shows a positive correlation.

5.  For extremely accurate (0.001 degrees) and very accurate (0.1
    degrees) PMUs, the sites (buses) where PMU addition results in
    maximal entropy reduction correlate very strongly to the buses where
    the estimated system without PMUs exhibits the largest expected
    phase angle variance. This is seen by observing that in Fig. 7.2 and
    Fig. 7.3 the scatter plots of entropy reduction versus angle
    variance are nearly monotonic.

The observed correlations between entropy reduction and both local bus
angle variance reduction and original bus angle variance are perhaps not
surprising; they suggest that the effects of a newly added PMU are very
local, and will significantly improve overall estimation accuracy only
when the knowledge of the 'local' state is poor. If this hypothesis
proves true in the general case (it would be good to derive appropriate
theory to support the example calculations here), a simple guideline for
placing PMUs to improve state estimation accuracy is suggested: place
them where corresponding diagonal entries of $G^{- 1}$ are large.

Enthusiasm for any sensor placement techniques that require numerical
analysis of any matrices dependent on measurement error statistics (e.g.
the gain matrix $G = H^{T}R_{z}^{- 1}H$) should be tempered because most
utilities and control areas do not maintain good knowledge of the
observed accuracy of existing telemetry. The recommended solution is
remote meter calibration (e.g. \[17\]\[39\]), a variant of parameter
estimation that permits both measurement accuracy determination and
correction.

![](media/image2.wmf){width="5.240277777777778in"
height="3.3055555555555554in"}

Fig. 7.2. Effect of PMU Placement (PMU Accuracy = 0.001 Degrees)

![](media/image3.wmf){width="5.240277777777778in" height="3.625in"}

Fig. 7.3. Effect of PMU Placement (PMU Accuracy = 0.1 Degrees)

![](media/image4.wmf){width="5.25in" height="3.6347222222222224in"}

![](media/image5.wmf){width="5.25in" height="3.6347222222222224in"}

Fig. 7.4. Effect of PMU Placement (PMU Accuracy = 1.0 Degrees)

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    **9 Appendix SuperCalibrator \[38\]**

Background: GPS-synchronized equipment receives inputs from instrument
transformers, control cables, attenuators, etc. Many utilities will use
CCVTs for instrument transformers. While GPS-synchronized equipment are
more accurate devices than typical utility SCADA and other monitoring
equipment, the overall precision of GPS-synchronized measurements is
limited by the instrumentation channel. For example, while
GPS-synchronized equipment has the capability of measuring the phase
angle to a precision of 0.01 degrees, the instrumentation channel may
introduce a phase error that may be ten times larger. In particular,
CCVTs introduce an error to the measurement that is appreciable at the
power frequency as compared to the precision of PMU equipment and very
large with respect to PMU equipment at other frequencies. Depending on
the application, this performance may be unacceptable.

Conceptually, the overall precision issue can be resolved with
sophisticated calibration methods. This approach is quite expensive and
faces difficult technical problems. It is extremely difficult to
calibrate instrument transformers and the overall instrumentation
channel in the field. Laboratory calibration of instrument transformers
is possible but a very expensive proposition if all instrument
transformers need to be calibrated. In the early 90\'s I directed a
research project in which we developed calibration procedures for
selected NYPA's high voltage instrument transformers \[1\]. From the
practical point of view, this approach is an economic impossibility.

We propose a viable and practical approach to correct for these errors.
The approach is based on an estimation process at the substation level
for correcting for these errors. Specifically, we propose to develop a
methodology and a software product that will perform as a
"supercalibrator". This product may reside at the PDC level, and it will
operate on the streaming data. The process is expected to be fast and
therefore it will introduce only minor time latencies. The procedure
will maintain the data format including the time tags of the data.

A brief description of the methodology follows. A detailed model of the
substation (from which PMU data are originating) will be constructed.
This model will be an integrated model, i.e. it will include the three
phase model of the substation, the model of the instrumentation channels
that feed inputs to the PMUs and the model of the PMUs. As the data
stream, each set of data at a specific time tag will be processed via a
general state estimation process. The procedure will provide the best
estimate of the data as well as performance metrics of the estimation
process. The most important metric will be the expected value of the
error of the estimates. The software will continuously display these
metrics and it will also store the computed metrics. The best estimate
of the data will be used to regenerate the streaming data flow.

It is important to note that the proposed super-calibrator is also a
tool for remote calibration. Since these equipment are digital and since
the super-calibrator will determine what the "reading" of each device
should be, a calibration factor can be inserted into each channel of the
GPS-synchronized equipment. This very simple method is also very
effective.

[^1]: Special thanks go to Ali Abur, John Allemong, and the state
    estimation and PMU experts at ABB, AREVA, Siemens and PNNL. Their
    suggestions have been invaluable in determining metrics that are
    both useful and reasonable to apply to modern commercial state
    estimators.

[^2]: Here, and elsewhere in this section the superscripts *true* and
    *est* denote 'true' and 'estimated' quantities, respectively. 'True'
    quantities are determined from the power flow solution, while
    estimated quantities are calculated from power flow equations
    evaluated using the estimated state.

[^3]: There are compelling reasons (e.g. robustness) to use the infinity
    norm in designing the state estimation algorithm; see Chapter 6 of
    \[24\] for an excellent treatment.

[^4]: Whereas most state estimators use iterative algorithms and require
    convergence before terminating, other criteria such as iteration
    limits, time limits, or detection of cycling may cause 'early'
    termination.

[^5]: This example is a rewording (using metrics defined in this
    document) of the condition the Midwest ISO published as the
    acceptable criteria for the estimator prior to market start. Found
    in a document simple titled 'Metric Interpretive Guidance 03
    Completed and Verified' at
    <http://www.midwestmarket.org/publish/Document/2df21_101f7800020_-7f840a48324a?rev=1>
    (link valid as of 09 Jan 2006), the original quote is 'State
    Estimator Solution must converge to .002 per unit voltage magnitude
    and/or angle and maximum power mismatch of 50MW State Estimator
    model within the Midwest ISO Market footprint.'

[^6]: The gain matrix, usually denoted by *G* in the state estimation
    literature, defines the relationship between the gradient of the
    objective function and updates to the state estimate for iterative
    algorithms. If weighted least squares estimation is employed with
    higher order terms ignored, *G* takes the state ($\hat{x}$)
    dependent form $G(\hat{x}) = H(\hat{x})^{T}\text{WH}(\hat{x})$where
    *H* is the gradient of the power flow equations and *W* is a
    diagonal weighting matrix.

[^7]: A good complement to this paper for those interested in the
    history of entropy and estimation is \[48\].

[^8]: This system, originally assembled by a IEEE Power Engineering
    Society Test Systems task force in 1993 under the direction of Mike
    Adibi, is currently available for download from the University of
    Washington Power System Test Case Archive maintained by Richard
    Christie at http://www.ee.washington.edu/research/pstca/.

[^9]: The measurement error range estimates for voltage magnitude and
    real and reactive power magnitude have been adopted from \[39\].

[^10]: The assumption is that the estimation algorithm (WLS, in this
    case) is aware of the accuracy of the newly added sensor and weights
    the measurement accordingly. If the weight is too high for the
    actual accuracy of the meter, or if the error statistics of the
    meter violate the assumptions of matching a zero mean Gaussian
    assumptions, the inaccuracy in the estimate may be increased.