Content uploaded by Olav Krause
Author content
All content in this area was uploaded by Olav Krause
Content may be subject to copyright.
Generalized Static-State Estimation
Approach and key features
Olav Krause
School of Information Technology
The University of Queensland
Brisbane, Australia
o.krause@uq.edu.au
Sebastian Lehnhoff
Institute for Information Technology (OFFIS)
University of Oldenburg
Oldenburg, Germany
Sebastian.lehnhoff@uni-oldenburg.de
Abstract—State Estimation is the foundation of any management
of electric power networks and thus forms a vital part of any
Smart Grid reflecting the operational limitations of the primary
infrastructure. Traditional Static-State Estimation algorithms
are not suitable for under-determined networks, even is the
under-determination is only partly. This is addressed by pre-
analysis of the networks topology and measurement distribution.
In this paper the authors present the approach and key features
of a novel Static-State Estimation technique which is inherently
able to cope with mixed levels of determination the supervised
network.
Static-State Estimation, Newton-Raphson, Obersavability
analysis
I. INTRODUCTION
A vital component of future Smart Grids are the actively
interacting distribution networks, which are enabled by
Information and Communication Technology to actively
supervise and monitor the operational state of their primary
energy delivery infrastructure. Main aim is to enable the
distribution networks to actively manage their existing transfer
capabilities. On transmission system level, this has been done
for decades now, technologically based on Static-State
Estimation developed in the late 1960s and published early
1970s [1-8]. This technique is used in control centers to
interpret a vast amount of measurement data from the network,
to identify false and erroneous measurement data and to
estimate the actual, not directly observable internal state of the
transmission system. Based on this estimated internal state a
couple of high-level functions are performed, like e.g. short
circuit calculations, reliability analysis and - among the most
important – contingency analysis and mitigation. Although the
extent to which contingency analysis is performed in real-time
by computer based analysis varies, it still is the primary
function of the network operation personnel to prevent network
overstrains, which effectively is personnel-based contingency
analysis and mitigation.
Extending these high-level functionalities into lower
voltage level networks is obviously challenged by the vast
number of networks existing in those network layers. This sets
the frame for Autonomous Grid Supervision and Management
schemes, since the personnel-intensity of existing techniques
limits their usability to rather low numbers of networks.
Despite challenges of available communication channels,
economic considerations and mathematical issues related to the
determination of the actual transfer capabilities, the availability
and distribution of measurement devices is one of the most
crucial ones for Autonomous Grid Supervision and
Management. In current transmission systems measurement
devices were and are purposefully installed with respect to the
Static-State Estimation’s need, which – in its current form –
requires the network to be fully observable. There exist a
number of pre-analysis techniques, estimating the observable
part of a network, in case it is not fully observable [10, 11],
[13-15]. Traditional State Estimation is then deployed on a part
of the network model, for which complete determination can be
assumed [11]. This not only requires additional functionality to
be implemented, it also removes useful information about not
fully observable system state variables and - due to its
importance to assure full observability of the identified sub-
network - tends to be more conservative than actually
necessary.
In this paper the authors demonstrate the general approach
and key figures of a generalized Newton-Raphson Power Flow
Analysis and show that it integrates the fields of over-
determined) Static-State Estimation with exactly-determined
Power Flow Analysis and extends it into the field of under-
determined cases. Simulation results presented in this paper
demonstrate the novel approach’s ability to solve over-, under-
and exactly determined cases, as well as mixtures of them. It
does not require network model simplifications, like in [14].
Instead the novel Static-State Estimation technique naturally
accepts under-determined regions within the monitored
network and thus makes pre-analysis for exclusion of
unobservable state variables obsolete. Furthermore, it also
assures that all observable system state variables are identified
and their state estimated, since it removes the need for a
conservative limitation to sub-networks identified by a pre-
analysis. Furthermore it will be demonstrated, that the
presented technique is capable of solving multiple observable
network regions at the same time on a single complete network
model and identifying and establishing observability for
regions only lacking phase reference.
In the following three sections the authors will elaborate the
essence and key differences between traditional Static-State
Estimation, Newton-Raphson Power Flow Analysis and the
novel Generalized Newton-Raphson technique.
I. BACKGROUND
Since the introduction of Static-State Estimation into Power
System Analysis, it was perceived as being closely related to
Power Flow Analysis, which is solving for the internal state of
an electric power network in an exactly determined case.
Although the principle problem is the same – deriving the
(most probable) internal state from a set of measurements,
indirectly observing the internal state – two different
mathematical approaches and solving strategies are used in
Static-State Estimation and Power Flow Analysis.
In both cases the internal state of the power system has to
be determined from input data, whch can be represented is a set
of real or pseudo measurements , or collated
into the input vector , like in (1)
(1)
Assumptions underlying both, Static-State Estimation and
Power Flow Analysis are mono-frequent operation at nominal
frequency and all transients ceased. Under these assumptions
the internal state of a power system can be uniquely described
by the vector of its nodes’ complex valued voltages (see (2)).
(2)
Key to all analysis techniques is the known functional
relationship between the internal state vector and the
theoretical values , corresponding to the available input
data . Since for many of the used input parameters and their
corresponding functional relationship with to the internal state
vector there exists no analytic inverse function, all of the
Static-State Estimation and Power Flow Analysis techniques
are iterative techniques, trying to readjust the assumed internal
state to match the input data with the corresponding
theoretical values .
(3)
As such, Static-State Estimation (SSE), Newton-Raphson
Power Flow Analysis (NR) and Generalized Newton-Raphson
(G-NR) share common concepts and mathematical constructs.
Although many of them are directly corresponding, different
nomenclatures evolved for SSE and NR. Since the authors see
the G-NR mathematically closer to NR rather than to SSE, the
nomenclature for G-NR is derived from NR (see Table I for
key correspondences).
TABLE I
KEY CORRESPONDENCES OF NOMENCLATURES
SE NR G-SE
state vector
input vector
theoretical values
jacobian of theoretical values
Standard variance n/a
mean square error
n/a n/a
II. TRADITIONAL STATIC-STATE ESTIMATION
Traditional Static-State Estimation bases on the direct
minimization of an objective function with a gradient
approach. Key to traditional Static-State Estimation is the
explicit formulation of the estimation error, as being the error
between theoretical values and their corresponding input
date for an assumed internal state (see (4)),
(4)
The individual deviation, or error, of the theoretical
value from its corresponding input date can be
calculated as in (5).
(5)
Instead of minimizing the absolute errors, error is usually
minimized relative to their standard deviation .
(6)
Based on the error, relative to the assumed standard
deviation of the date’s error, the objective function is
defined as (7).
(7)
With being a diagonal matrix containing the assumed
standard deviations of the data sources in its main diagonal
(see (8)) (7) can be restated as (9).
(8)
(9)
To find the minimum of , a tangential approach is used,
whit the optimality criterion of the gradient of being
to be equal to in the minimum. From (9) can be derived
as in (10).
(10)
With the partial derivatives of the theoretical values
assembled into the matrix , like stated in (11), (10) can be
restated as (12), or more compact as (13).
(11)
(12)
(13)
The iterative cycle of the traditional Static-State Estimation
then tries to find the system state vector for which
(see (14)) holds. Again, in traditional Static-State Estimation
a gradient approach is deployed to asymptotically approach .
(14)
To create an iterative Newton cycle – Gauss-Newton in this
case – the partial derivatives of are needed, which can be
calculated with . This results in first and second-order
terms, where the second order terms are omitted to allow the
formulation of a shortened Taylor-Series (see (15)), where
indicates the iteration number:
(15)
Omitting the residual of the Taylor series (16) can be
derived from (14) and (15).
(16)
The reduced order used in (16) can be derived from
(13) as (17).
(17)
The key issue of the traditional Static-State Estimation
iterative cycle as in (16) is the possible non-existence of
. exists only if is full-ranked, non-
singular, which is only true if has a rank of , being
the number of complex state variables of the network model.
This is only the case if the system is at least exactly
determined, leaving traditional Static-State Estimation to fail in
case of the system being even only partially unobservable due
to a lack of input data for certain parts of the system.
To ensure is of rank usually a pre-processing of
the system and its data coverage is deployed and system state
variables assumed to be undeterminable due to assumed
insufficiency of data coverage are removed from the model
[11]. Another approach is to – in case of a partially
unobservable system – to deploy system model reduction to the
point that the reduced system model becomes fully observable
[14]. The first approach comes at the risk of being to
conservative, removing state variables which were observable
and to lose information about unobservable state variables
which still have some kind of functional relationship to the
input data. The second approach distorts the power system
model and makes it hard to interpret results with respect to the
original, non-reduced network model.
III. TRADITIONAL POWER FLOW ANALYSIS
Traditional Power Flow Analysis is exclusively for exactly
determined cases and thus does not consider any possible
errors in the input data, since there is no redundancy in the
input data that could allow detection of errors.
Since the underlying assumption of the traditional Power
Flow Analysis is that the mismatch between input data and
corresponding theoretical values, derived from the assumed
system state, can be asymptotically reduced to zero, the
gradient approach is deployed directly to the equation,
corresponding to (5) of the traditional Static-State Estimation.
Unlike in Static-State Estimation the allowed input data is
rather restricted to assure exact determination of all state
variables. Aside from one “reference bus” always required to
be represented by a fixed voltage magnitude and argument,
input data has to be provided for all other nodes, corresponding
to the state variables, providing either active power and voltage
magnitude – so model generators with voltage regulators – or
active and reactive power. This scheme ensures exact
determination of all state variables, but not solvability. The
following equations refer to a case in which all nodes except
the reference node are represented by active and reactive
power. With the internal state vector being denominated as ,
the input data as and the theoretical values as
(18)
To find with the Newton-Raphson technique again a
gradient approach is used for which the first the Jacobian
matrix of is needed (see (19) and (20)).
(19)
(20)
Assuming, that (18) can be solved, the iterative cycle is
defined as in (21).
(21)
The main issue with (21) is the existence of: , which
may only exists when the system is exactly determined and
thus requires rigid limitations to the allowable input data.
Furthermore it is not able to detect or compensate for any
errors due to the lack of redundancy in the input data and
requires all state variables to be exactly determined by the
input data. As will be shown in the following section all these
limitations can be overcome by a modification of (21).
IV. GENERALIZED STATIC-STATE ESTIMATION
Since the Generalized Static-State Estimation does not pose
any limitations or restrictions to what kind of data is used as
input data and how they are distributed across the power
system model, the authors chose a more general nomenclature
and refer to input data as measurements and corresponding
theoretical values (see ). Since the Generalized Static-
State Estimation technique is closer to the Newton-Raphson
technique than to the traditional Static-State Estimation, the
authors adopted the Newton-Raphson nomenclature where
possible and thus refer to the internal state vector as .
(22)
(23)
Like the traditional Static-State Estimation, the Generalized
State Estimation aims at minimizing the input data side
estimation error (22) to it Least Mean Square Error.
(24)
Unlike the traditional Static-State Estimation, the
Generalized Static-State Estimation does not require explicit
statement of an objective function, but converges into the Least
Mean Square Error solution naturally. This is done with a
gradient approach, requiring the Jacobian matrix
as in (25).
(25)
Although it is not assumed that the measurement side
estimation error can be eliminated, the Generalize Static-
State Estimation still uses the principle formulation of the
Newton-Raphson technique to define its iterative cycle.
(26)
The key difference between the Newton-Raphson technique
and the Generalized Static-State Estimation is the replacement
of the inverse with the pseudo-inverse of the
Jacobian matrix and the convergence criterion being the non-
improvability of the result, thus . The pseudo-inverse
can be stated for any matrix, independent of its rank, rank
deficiencies and its dimensions and always exists. This
removes all limitations to the input data and to the
observability of state variables. Analyzing the pseudo-inverse
in the convergence point reveals why (26) converges into
the Least Mean Square Error solutions
Based on the Singular Value Decomposition of the
pseudo-inverse can be stated as in (27), where is the
rank of matrix (see [15]).
(27)
The Singular Value Decomposition identifies left and right
range and null-space of and consequently also of .
Left and right range are spanned by the first left and right
singular vectors – and respectively - , which are
functionally related by the corresponding singular values .
These ranged identify the parts of measurement and state space
that are functionally linked by the input data. Thus they
identify the observable state variables, since a state variable
that is completely determined by the right range and does not
have a right null-space component is through fully
determined by the input data.
With having only non-zero components in the left
null-space in the point of convergence, it is perpendicular to the
left range of in the point of convergence and thus the
minimum distance between and the left range of ,
assuming the left range of is convex in the vicinity of
(see [15]).
Since Euclidean norm and Mean Square Error share the
same definition, in the point of convergence is the lowest
possible Mean Square Error.
The overage of state variables by the input data can then be
calculated as the angle between the state variables’ state space
vector an the right range of . An angle of means full
coverage, while higher values indicate insufficient coverage.
The variables and respectively indicate the angle of state
variable ’s magnitude and phase angle.
V. EVALUATION AND CONCLUSIONS
To demonstrate the validity of the presented approach a
series of experiments were performed on a five nodes, four
wires network. Table II shows the input data corresponding to
the internal state, the Estimator was supposed to determined.
TABLE II
SYSTEM INPUT USED IN THE FOLLOWING TESTS
3
The first test case is a replication of a typical Newton-
Raphson setup, in which node one represents the reference
node and all others are represented by active and reactive
power balances. As can be seen in Table III, the Generalized
Static-State Estimator correctly determines the internal state
and correctly indicates that all state variables are fully covered
by the input data.
TABLE III
SYSTEM INPUT AND ESTIMATED SYSTEM STATE OF EXACTLY DETERMINED
NEWTON-RAPHSON SCENARIO
1
X X - - -
- - X X -
- - X X -
- - X X -
- - X X -
2
3
4
5
In the second case all possible input data from Table II are
provided into the Generalized Static State Estimator to create
an over-determined case. Again all state variables are
determined correctly and identified as being fully covered by
the input data.
TABLE IV
SYSTEM INPUT AND ESTIMATED SYSTEM STATE OF OVER-DETERMINED
SCENAR IO
1
XXXXX
XXXXX
XXXXX
XXXXX
XXXXX
2
3
4
5
In third test case full information is provided only for the
first three nodes with no information about the last two ones.
The state variables corresponding to the first four nodes are
determined correctly and identified as being fully covered by
the input data. The state variable of node 5 could not be
determined but was correctly indentified to not being covered
by the input data.
TABLE V
SYSTEM INPUT AND ESTIMATED SYSTEM STATE OF MIXED-DETERMINED
SCENAR IO
1
X X X X X
X X X X X
X X X X X
- - - - -
- - - - -
2
3
4
5
For the fourth test case a situation with two separate regions
is created in which the second zone lacks a reference angle.
The first zone is solved correctly, while for the second zone,
consisting of node 4 and 5, voltage magnitudes and phase angle
difference where correctly determined. The Generalized Static-
State Estimator correctly identifies insufficient model coverage
of the voltage phase angles at node 4 and 5.
TABLE VI
SYSTEM INPUT AND ESTIMATED SYSTEM STATE OF MIXED-DETERMINED
SCENARIO WITH PARTIAL LACK OF PHASOR REFERENCE
1
X X X X X
X X X X X
- - - - -
- - - - -
X - X X X
2
3
4
5
To demonstrate the algorithm’s ability to solve
unconnected observable regions on a single network model
simultaneously, a pseudo-measurement (P) is added to node 4
to provide a reference phase angle. With this additional input
data, the algorithm correctly identifies all state variables as
being fully covered by the modified input data. Due to page
limitation the process of identifying such unconnected yet
observable regions cannot be discussed here.
TABLE VIII
SYSTEM INPUT AND ESTIMATED SYSTEM STATE OF MIXED-DETERMINED
SCENARIO WITH PARTIAL LACK OF PHASOR REFERENCE, HEALED BY PHASE
ANGLE PSEUDOMEASUREMENT AT NODE 4
1
X X X X X
X X X X X
- - - - -
- P - - -
X - X X X
2
3
4
5
In this paper the principle technique of the novel
Generalized Static-State Estimation is demonstrated and
compared to the two main related techniques, the traditional
Static-State Estimation and the Newton-Raphson Power Flow
analysis. The validity of the novel approach was demonstrated
through a series of examples, covering all mayor deployment
scenarios.
VI. REFERENCES
[1] Schweppe, F.C., Wildes, J., “Power System Static-State Estimation, Part
I: Exact Model”, IEEE Transactions on Power Apparatus and Systems
(PAS-89), Issue 1, January 1970, pp. 120-125
[2] Schweppe, F.C., Rom, D.B., “Power System Static-State Estimation,
Part II: Approximate Model”, IEEE Transactions on Power Apparatus
and Systems (PAS-89), Issue 1, January 1970, pp. 125-130
[3] Schweppe, F.C., “Power System Static-State Estimation, Part III:
Implementation”, IEEE Transactions on Power Apparatus and Systems
(PAS-89), Issue 1, January 1970, pp. 130-135
[4] Larson, R.E., Tinney, W.F., Peschon, J., “State Estimation in Power
Systems Part I: Theory and Feasibility”, IEEE Transactions on Power
Apparatus and Systems (PAS-89), Issue 3, March 1970, pp. 345-352
[5] Larson, R.E., Tinney, W.F., Hajdu, L.P. Piercy, D.S., “State Estimation
in Power Systems Part II: Implementation and Application”, IEEE
Transactions on Power Apparatus and Systems (PAS-89), Issue 3,
March 1970, pp. 353-363
[6] Smith, O.J.M., “Power System State Estimation”, IEEE Transactions on
Power Apparatus and Systems (PAS-89), Issue 3, March 1970
[7] Schweppe, F.C., Handschin, E.J., “Static state estimation in electric
power systems”, Proceedings of the IEEE, Vol. 62, No. 7, July 1974
[8] Rao, N.D., Roy, L., “A Cartesian Coordinate Algorithm for Power
System State Estimation”, IEEE Transactions on Power Apparatus and
Systems (PAS-102), Issue 5, May 1983, pp. 1070-1082
[9] Krumpholz, G.R., Clements, K.A., Davis, P.W., “Power System
Observability: A Practical Algorithm Using Network Topology”, IEEE
Transactions on Power System Apparatus and Systems (PAS-99), Issue
4, July/August 1980, pp. 1534-1542
[10] Quintana, V.H., Simoes-Costa, A. Mandel, A., “Power System
Topological Observability Using a Graph-Theoretical Approach”, IEEE
Transaction on Power Apparatus and Systems (PAS-101), Issue 3,
March 1982, pp. 617-626
[11] Abur, A., Gómez Exósito, A, “Power System State Estimation – Theory
and ‘Implementation”, ISBN: 0824755707, Betrams, March 2004.
[12] Clements, K.A., Krumpholz, G.R., Davis, P.W., “Power System State
Estimation with Measurement Deficiency: An Algorithm that
Determines the Maximal Observable Subnetwork”, IEEE Transactions
on Power Apparatus and Systems (PAS-101), Issue 9, September 1982,
pp. 3044-3052
[13] Monticelli, A., Wu, Felix F., “Network Observability: Theory”, IEEE
Transactions on Power Aparatus and Systems (PAS-104), Issue 5, May
1985, pp. 1042-1048
[14] Wolter, M., “Grid Reduction Approach for State Identification of
Distribution Grids”, Transactions of the IEEE PES General Meeting
2009, 2009, pp. 1-8
[15] Adi Ben-Israel, Thomas N.E. Greville, “Generalized Inverses – Theory
and Applications”, second edition, ISBN 0-387-00293-6, Springer-
Verlag New York, 2003
[16] E. Handschin, F.C. Schweppe, J. Kohlas, A. Fiechter, "Bad Data
Analysis for Power System State Estimation", IEEE Transactions on
Power Aapparatus and Systems (PAS-94), Issue 2, 1975, pp. 329-337
VII. BIBLIO GRAPHY
Olav Krause (M’05) is lecturer at The University of Queensland, Australia.
He received his doctorate and diploma degree in Electrical Engineering at the
TU Dortmund University, Germany, in 2009 and 2005 respectively.
Sebastian Lehnhoff (M’05) is an Assistant Professor of Energy Information
Systems at the University of Oldenburg and a Director of the Energy R&D
Division of the OFFIS – Institute for Information Technology, Germany. He
received his Ph.D. and diploma degree in Computer Science at the TU
Dortmund University, Germany, in 2009 and 2005, respectively.
