System Measurement State
Any physical system is described by using the mathematical model which describes the behaviour of the whole system using equations. But most of systems do not focus on the modelling uncertainties which might present in the physical system or they do not provide real time information to monitor the system. As there are many uncertainties present in the mathematical model and is not always perfect, they cannot find out the actual physical state of the system (Maybeck, 1979). The state estimation has to be applied for these reasons. Dynamic system is not only driven by the control input but also by the disturbances or noise which is not easy to determine the state of the system.(Welch and Bishop, 2001) This can impact the output which does not show the exact state of the system. Moreover, many measurement devices in the network may be corrupted by noise and the measurement received does not provide the complete data to the system (Maybeck, 1979). All the physical system cannot be measured to find out the state of the system. Even the direct measurement can be done; it may have some errors due to various reasons. A state estimator is an important tool for system monitoring as it processes a redundant set of measurements to obtain the best complete estimate of the current system state (Abur, A et al., (2006) ). The main purpose of the state estimator is to identify measurement errors and remove them if possible or to filter out the noise. (Abur, A. and Exposito, A. G., (2004).
State estimation is the useful tool for monitoring and controlling of the system. The state estimator determines the state of the system which is effected by noise. With the help of those current state of the sytem, the whole system can be monitored. If there is some defects in the system, then the differnt types of action should be taken to recover the sytem in the normal position to ensure the security of the system.
The increase in system automation, the desire for fast and reliable system reconfiguration
and the proliferation of the underlying system and its architectures has produced the need for fast system controllers capable of monitoring and controlling the performance of the system for various operating points. These control and automation algorithms are based on measurements and estimates of the system states that are provided by state estimation techniques. Hence state estimation has an immense importance in the world of automation.
The physical system behavior is very often monitored through a telemetry system. Only a limited number of set of measurements are recorded directly and these do not provide a complete picture of the system state. Nevertheless, the combination of this incomplete information together with the knowledge of the topology of the underlying network and other available information can be enough to calculate the remaining unmeasured sets of measurements. This calculation of these unmeasured sets of measurements is called state estimation.
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the measurements are effected by error or noise, either by inaccuracy of the meter or instrument not working properly which leads to the error in the measurements which is not the actual measurement of the system. these measurement with error are adjusted so that they are close to the actual measurements. these adjusting process is done by optimization process by minimizing the difference between the measurements and the measured values. If the optimization problem is well defined, then the calculated values of measurement are close to the real values. To determine the state of the system, optimization is the only one solution.
State estimation only work when there is sufficient measurements and it filters the incoming data by identifying erroneous measurements which is removed by further processing with optimization function.
State estimation is the useful tool for monitoring and controlling of the system. The state estimator determines the state of the system which is affected by noise. With the help of those current state of the sytem, the whole system can be monitored. If there is some defects in the system, then the differnt types of action should be taken to recover the sytem in the normal position to ensure the security of the system.
Review of State Estimation
in general, the state estimation fall into one of the two categories, either static or dynamic. The diagram below shows the hierarchy of the various state estimation algorithms (Elsaesser, M. 1996).
The parameters describe the physical behavior of the system. The parameters of the system cannot be directly measured. With the help of other measurement in the system the parameter can be determined. An estimator tries to estimate the unknown parameter using the measurements. But the measurement always comes with uncertainties or errors (Heijden, F et al. ,2004). Estimating the parameter of the system is not useful until the uncertainties or errors are removed from the parameters. Parameter estimation uses different techniques such as Minimum mean square error estimator (MMSE), Maximum likelihood estimation, Least Square Fitting, Regression to remove the errors in the measurements (Heijden, F et al. ,2004). The variables defining the system change over time. It is necessary to track the variables or to know the current value of the variables to monitor the system for security and control purpose. State estimation is the technique use to find the state of the unknown variables (parameters) of the system with the available measurements by filtering the errors present in the measurement by minimizing the error present in the measurement to get the values close to the actual values of the measurement.
State estimation can be used in static as well as dynamic system. Static state estimation refers to the process of obtaining the state of the system at a point in time. This can be achieved by measurements present in the system. In static state estimation the state is usually fixed (it does not change with time) and it is possible when the network topology and the parameters are perfectly known (ref book theory and implementation). In the case of dynamic system, ordinary differential equations are used to describe the behavior of the system. The parameters estimated in the static estimation do not change but in dynamic state estimation the parameters being estimated changes during the estimation process (Crassidis, J. L. & Junkins, J. L ,2004). Both static and dynamic system can be either linear or non linear. Depending upon these factors (static, dynamic, linear and non linear model), different state estimation techniques are used to find the state of the system.
fixed There are different techniques used to estimate the state of both systems is illustrated in the following section.
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Linear and dynamic:
The Kalman filter is an efficient recursive filter that estimates the state of a dynamic system from a series of incomplete and noisy measurements . The Kalman filter exploits the dynamics of the target, which govern its time evolution, to remove the effects of the noise and get a good estimate of the location of the target at the present time (filtering), at a future time (prediction), or at a time in the past (interpolation or smoothing (wiki).
[edit] Non-linear filters
The basic Kalman filter is limited to a linear assumption. However, most non-trivial systems are non-linear. The non-linearity can be associated either with the process model or with the observation model or with both.
[edit] Extended Kalman filter
In the extended Kalman filter, (EKF) the state transition and observation models need not be linear functions of the state but may instead be (differentiable) functions.
[edit] Criticism of the extended Kalman filter
Unlike its linear counterpart, the extended Kalman filter is not an optimal estimator. In addition, if the initial estimate of the state is wrong, or if the process is modeled incorrectly, the filter may quickly diverge, owing to its linearization. Another problem with the extended Kalman filter is that the estimated covariance matrix tends to underestimate the true covariance matrix and therefore risks becoming inconsistent in the statistical sense without the addition of "stabilising noise".
Having stated this, the extended Kalman filter can give reasonable performance, and is arguably the de facto standard in navigation systems and GPS.
[edit] Unscented Kalman filter
When the state transition and observation models - that is, the predict and update functions f and h (see above) - are highly non-linear, the extended Kalman filter can give particularly poor performance [JU97]. This is because only the mean is propagated through the non-linearity. The unscented Kalman filter (UKF) [JU97] uses a deterministic sampling technique known as the unscented transform to pick a minimal set of sample points (called sigma points) around the mean. These sigma points are then propagated through the non-linear functions and the covariance of the estimate is then recovered. The result is a filter which more accurately captures the true mean and covariance. (This can be verified using Monte Carlo sampling or through a Taylor series expansion of the posterior statistics.) In addition, this technique removes the requirement to analytically calculate Jacobians, which for complex functions can be a difficult task in itself.
Predict
The Kalman ¯lter(KF) is one of the most widely used methods for tracking and estimation due to its simplicity,
optimality, tractability and robustness. However, the application of the KF to nonlinear systems can be di±cult.
The most common approach is to use the Extended Kalman Filter (EKF) which simply linearises all nonlinear
models so that the traditional linear Kalman ¯lter can be applied. Although the EKF (in its many forms) is a
widely used ¯ltering strategy, over thirty years of experience with it has led to a general consensus within the
tracking and control community that it is di±cult to implement, di±cult to tune, and only reliable for systems
which are almost linear on the time scale of the update intervals.
In this paper a new linear estimator is developed and demonstrated. Using the principle that a set of discretely
sampled points can be used to parameterise mean and covariance, the estimator yields performance equivalent to
the KF for linear systems yet generalises elegantly to nonlinear systems without the linearisation steps required
by the EKF. We show analytically that the expected performance of the new approach is superior to that of the
EKF and, in fact, is directly comparable to that of the second order Gauss ¯lter. The method is not restricted
to assuming that the distributions of noise sources are Gaussian. We argue that the ease of implementation and
more accurate estimation features of the new ¯lter recommend its use over the EKF in virtually all applications.
--
http://www.cs.unc.edu/~welch/kalman/media/pdf/Julier1997_SPIE_KF.pdf
nonlinear
extended Kalman and particle filter
Generally, nonlinear problems are difficult (if possible) to solve and are much less understandable than linear problems. Even if not exactly solvable, the outcome of a linear problem is rather predictable, while the outcome of a nonlinear is inherently not. (wiki)
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Particle filter
They are often an alternative to the Extended Kalman filter (EKF) or Unscented Kalman filter (UKF) with the advantage that, with sufficient samples, they approach the Bayesian optimal estimate, so they can be made more accurate than either the EKF or UKF. The approaches can also be combined by using a version of the Kalman filter as a proposal distribution for the particle filter.
Wiki
The Kalman .lter is essentially a set of mathematical equations that implement a
predictor-corrector type estimator that is optimal in the sense that it minimizes the
estimated error covariance—when some presumed conditions are met.
http://www.cs.unc.edu/~tracker/media/pdf/SIGGRAPH2001_CoursePack_08.pdf
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The mathematical model of the state estimation is based on the relationship between state variables and the measurements, given as:
z = h(x) + e-1
Where,
z - (m x 1) is the measurement vector
x - (n x 1) is the state vector
h - (m x n) is the Jacobian matrix
e is the measurement error, normally, is a random variable with zero mean Gaussian distribution.
define Jacobian matrix:
Measurement vector z is noisy due to various reasons depending on the type of network. State estimation minimizes the noise to find out the state vector x. Data available to describe the network are redundant. State estimation uses these data which contain noise to predict the accurate state of the noise.
The error in the data can be in various forms. Measurement noise is the noise which occurs while taking the actual measurement of the different devices. This type of noise is not related with different measurement taken and is assumed as Gaussian distribution (Elsaesser,M, 1996). Large errors or bad data may cause meter failure, incorrect calibration, etc. The robust state estimator has to be used to deal with these types of error or bad data. (Wu, F. ,1990)
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Different types of state estimation techniques
Weighted Least Square Method
In measurement -state relationship (1), the errors as well as the measurement Z are the random variables and the value of state x can be determined. One of the way to estimate the state of the system is by Maximum Likelihood The errors are independent random variables with Gaussian distribution and mean zero. The variance s2 of the measurement error e determines the certainty of the particular measurement. Large variation means that the corresponding measurement is not accurate. The measurement z is Gaussian distributed with mean h(x) and variance s2. The density function of z can be written as
Where, m is the number of measurements and R=diag(s2).
The maximum likelihood estimate which maximizes the density function of the system measurement minimizes the quadratic term in the exponent that is weighted square residual.
The most common state estimation technique is Weighted Least Square method. WLS minimizes the weighted sum of the square of the residuals.
min J(x)=[z-h(x)]T R-1[z-h(x)](2)
x
the condition for solving equation (2),to estimate x is given below::
-(3)
Where, is the Jacobian matrix of the measurement function.
The nonlinear equation (3) can be solved by iterative method where the liner equation is solved in each iteration to computer the correction:
-(4)
Where is non singular matrix
If the sequence of the point xk generated by iterative method converges, the solution of (3) is converged given is non singular.
Newton's method guarantees the local convergence for given by
The ijth element of is
The WLS ignores the second derivative and chooses
Therefore, equation (4) can be written as
Where
G(x) is the gain matrix. If the set of measurements are sufficient and the network is well distributed, the Jacobian matrix H(x ) will have full rank and consequently G(x) is non singular, then the network is observable i.e. state estimation is possible in that network.
Least Absolute Value
WLS is a common state estimator which gives the optimal estimates with minimum variance from the set of measurement with of the when Gaussian noise is present. But in online computer control system data are not always Gaussian distribution and may contain errors such as reverse sign of measurement, zero reading etc (Elsaesser,M., 1996). WLS is efficient on noise measurement. When it has to deal with bad data then it gives poor result.
The alternative of WLS was proposed based on minimizing of the modulus of measurement inconsistency called Least Absolute Value function. LAV method works by minimizing the following linearized cost function.
Min x C=WT |Dz-J(x)Dx|
Where W is the measurement weighing vector.
Linear programming is used to minimize the limearized cost function. The solution is found by intersection of hyper planes generated by data with the smallest absolute errors and reduces the worse element of data. Therefore, LAV is efficient in bad data rejection but it is less effective in noise rejection error and it takes more time and memory.
Hybrid method
To overcome the problem of WLS and LAV, Hybrid method has been proposed which is the combined method of WLS and LAV. It adjust the weighing vector to adjust significant individual measurement to detect and reduce the significance of bad data still remaining to reduce the noise.(Bullock, J.,1997). The algorithm proposed by Powell which is effective in both noise measurement and bad data rejection(Oliver, M.I. et al.,1987).
tate estimation is a technique used in various network management systems to eliminaite the anomalies present in the underlying system.(Abbas). This thesis describe such network management system.
State estimation of power distribution network
Background:
The Northeast blackout in United States in 1965 , the electric utility caused many researchers to realize that the old practices and tools were not suitable and the serious efforts were required to develop new approaches to attain high level of reliability and security (Liacco, T. ,1990). The main issue was the monitoring and security of the system. They came with an idea to build new control centre using real time digital computers system which collect more information about the power system to monitor the system. Up to that time, the only real time power system data available was the status of the circuit breaker, system frequency the set of real power measurement needed for generation control where as the actual requirement for the new control centre was to gather cyclically every few seconds of all the analogue measurements and beaker position from the entire network under observation. With all the information in the real time database, secure monitoring system is implemented. Monitoring and controlling the power system is conducted through SCADA (Supervisory control and data acquisition).
It was thought that input data needed by the security program is provided by SCADA, which is updated periodically in the real time database, could provide an accurate system view. in general, data in power system include power flow, power injections, voltage magnitude, current magnitude and phase angle. These data may be affected by measurement noise due to various reasons. A more powerful tool is required to remove the erroneous data and provide information for proper implementation to SCADA.
Finally, Fred Schweppe proposed the idea of state estimation to resolve the problem in SCADA and is the important part of EMS (Energy Management System). SE maintains integrity of real time database, provides correct information to the operator with complete, consistent and accurate view of the entire network. (define SE). With the initiation of State Estimation, the practicality of implementing security monitoring function was assured and the transition from SCADA to the EMS was possible. (Liacco, T. ,1990)
Schweppe introduced Weighted Least Square method which is dominant in practical implementation for state estimation since then the state estimation in power system has been active in research area. The WLS minimizes the weighted sum of the squares of the residuals (thesis).
Introduction of power system:
Power systems are operated by system operators from the area control
centers. The main goal of the system operator is to maintain the system in the normal secure state as the operating conditions vary during the daily operation. Accomplishing this goal requires continuous monitoring of the system conditions, identification of the operating state and determination of the necessary preventive actions in case the system state is found to be msecwe. This sequence of actions is referred to as the security analysis of
the system. The first stop of security analysis is to monitor the current state of the system. This involves acquisition of measurements from all parts of the
The power system consists of transmission, distribution and generation system. Transmission system may contain large number of sub station which is connected by transmission lines, transformers and other devices. Generators inject power to the station and are observed by the loads at the substation. The output voltage is generally low so the transformer is used to increase the voltage level for efficient transmission.
The operating condition of the power system is determined at any time if the complex phasor voltage at every system bus is known. The power system can have one of the three possible states: normal, emergency and restorative (Liacco T.,1974),. The system is in normal state if the all loads can be supplied power by the generators without any operational constraints which includes limits on transmission line flow and upper and lower limits on bus voltage magnitude. If the system can remain in normal state following the occurrence of each contingency from the list of critical contingencies, it is said to be secure state otherwise it is insecure operating state. If the system is normal and insecure, the preventive action must be taken before if goes to the emergency state. Due to unexpected event the operating condition may change and cause violation of some of operating constraint, then this state of the system is in emergency state and immediate corrective action should be taken to bring the system back to normal. When the system state is in emergency state, corrective action may avoid collapse of the system by disconnecting various loads line etc. as a result the operating limit violation is eliminated as the system is back in the normal state and this state is called the restorative state.
Security analysis in the power system:
Power system is operated by the system operators and its main goal is to maintain the system in normal and secure state. To accomplish this goal, it requires continuous monitoring of the system condition; recognition of operating condition and determination of the corrective action if the system state is found to be insecure. For the security analysis the first step is to monitor the current state of the system. This involves measurement from all part of the system and processes them to determine the current state. The measurement includes bus voltage, current magnitude etc. the raw data are processed by the state estimator to filter the measurement noise. Define ES
State estimator solution will provide an optimal
State estimation in power system
The mathematical model used in power system state estimation is based on relationship between state variables and measurements, given as:
Z=HX+e
Where Z is the available system measurement. Measurement vector of power flow. Power injection and voltage.X is the system variable. It represent the state vector of bus voltage magnitude and phase angle. E is the measurement error which is usually random variable with zero mean Gaussian distribution. H is the Jacobian matrix.
The WLS estimator determines the X by minimizing the weighted least square error:
J(X)=[Z-HE]T W[Z-HX]
Where W is the diagonal matrix whose element are real power measurement weighing matrix.
The WLS problem is solved by iterative method where the least square problem is solved at each iteration. There are mainly three methods: Normal Equation, Orthogonal Transformation and Hybrid method, to solve state estimation problem and to minimize J(X) to obtain estimate X (Monticelli,A. & Wu, F.F.,1986).
Measurements received at the control center will include line power
Hows, bus voltage and line current magnitudes, generator outputs, loads,
circuit breaker and switch status information, transformer tap positions,
and switchable capacitor bank values. These raw data and measurements
are processed by the state estimator in order to filter the measurement noise
and detect gross errors. State estimator solution will provide an optimal
estimate of the system state based on the available measurements and on
the assumed system model. This will then be passed on to all the energy
management system (EMS) application functions such as the contingency
analysis, automatic generation control, load forecasting and optimal power
now, etc. The same
System operator control the power system from the area of control centre and it's main purpose is to maintain the system in normal state. To accomplish this task it needs continuous monitoring of the system conditions, identification of the operating state and determination of the necessary preventive action in case of the system is found to be insecure. If the system is normal and insecure, the preventive action must be taken before if goes to the emergency state. Due to unexpected event the operating condition may change and cause violation of some of operating constraint, then this state of the system is in emergency state and immediate corrective action should be taken to bring the system back to normal to avoid system to collapse.
Powers system is operated by system operator from the area control centre.
Power systems are operated by system operators from the area control
centers. The main goal of the system operator is to maintain the system in
the normal secure state as the operating conditions vary during the daily
operation. Accomplishing this goal requires continuous monitoring of the
system conditions, identification of the operating state and determination
of the necessary preventive actions in case the system state is found to be
msecwe. This sequence of actions is referred to as the security analysis of
the system.
The first stop of security analysis is to monitor the current state of the
system. This involves acquisition of measurements from all parts of the
Sate estimation has made it possible the change of power system control design from SCADA to EMS which is characterize by the application of advance function of security monitoring and control
But no one has realized that the measurement can have some errors due to various reasons and because of the errors SCADA failed to implement secure monitoring system in the power system.
2.1. Introduction
System security is that aspect of system operation that allows the operators to determine whether an unforeseen event such as a line or generator outage or a fault will cause equipment to be outaged thereby causing other equipment to overload and cause more outages, etc. The resulting situation can result in a low frequency, low voltages, severe overloads on transmission lines and transformers, or an instability. Any of these can remove a large number of pieces of equipment and leave a large number of electric customers without power. The costs in lost revenue, customer losses and equipment damage can be extremely large.
In the United States, we usually trace the emphasis on system security in EMS to the large scaleblackouts that happened in the mid 1960's when very large areas of the US and Canada were left without electric power. Engineers realized that the security of the power system had to be managed properly so as to avoid serious problems.
In addition to the need for monitoring system security, there appeared a major shift in the late 1960's and early 1970's in the basic design of an EMS. Originally the generation control and scheduling system was thought of as one system. The supervisory control system used to monitor and control circuit breakers and other equipment in the transmission substations was considered a second separate system. Realizing that both the supervisory system and the generation control system used communication links and computers meant that there would be an advantage to
combining them. The result is the modern EMS which combines Supervisory Control and Data Acquisition (SCADA) capabilities along with generation dispatch, scheduling and control capabilities. Modern EMS's now had the ingredients necessary to provide operators with advanced security analysis capabilities. This feature is considered absolutely necessary in operating a power system as it allows operations personnel to make the most efficient use of the transmission system by loading it up to its limit without placing it in an insecure state.
2.2. Monitoring: Alarm Processing and State Estimation
Secure operation requires that operators monitor the system for existing conditions that might be cause for action (such as voltage or power flow limit violations, abnormal switching as a result of a fault, etc.) as well as make predictions of system conditions that might occur if an event such as a fault or a line or generator happens.
Monitoring the power system takes place in two ways. The basic process of taking measurements in the substations, transmitting the values to the central computer and comparing those values to stored limits is known as alarm processing. In addition to the processing of “analog” type alarms, the EMS also monitors the status of various binary devices (breaker and switch open/closed status, temperature under limit/over limit indicator, etc.) together these indicators make up tens of thousands of “points” that must be monitored and displayed to the operators.
The main emphasis the design of alarm processors in recent years has been to bring these points to the operators attention very rapidly and this has meant that operators can be overloaded with hundreds of new alarms to read in a few seconds. To help in this case, knowledge based alarm processors have been developed that can filter out all but the most important alarms and then present summary alarms so the operator can deduce the situation quickly.
When it comes to the transmission system, however, there is a need to further process the real time measurements so that a complete mathematical model of the system can be built. This is done using a power flow model and a state estimation algorithm which can read many redundant measurements and calculate the statistically most probable set of states (voltage phase angles and magnitudes) existing on the network. In addition to providing the states, the state estimator has the
ability, given the right set of measurements, to detect and identify measurement that are bad. The bad measurements are removed and reported to the operators so they can be recalibrated.
The result is a much greater trust in the readings of the measurements themselves. Present computer technology allows one to design a state estimator that runs at the speed that 5data is gathered and thus would always present state estimated data to the operators.
Problems in the state estimation systems in present use usually center on the equivalent network used to model the neighboring systems. Since there are few instances where utility computer systems have access to information about the neighboring systems, the results depend upon using poorly formulated forecasts of what is happening to the neighbor's system - and this can lead to nonconvergence of the estimator algorithm or to unusable results.
2.3. Static Security Assessment
Once a state estimate is complete the operators have a model of the power system as it presently exists. The next effort is to test that model for a large number of outages to determine if the system can recover from the outage without problems. The outage events or contingencies can be modeled using a power flow program by running the contingencies one at a time. However, even with the most powerful computers this is a difficult process since the single contingencies plus multiple contingencies may number in the thousands and the total time for all of the power flows on all the contingency cases would take hours. Since the operators need to know which contingency cases are going to give trouble, there must be a way to select the troublesome cases from among all the cases which report no trouble. Such “contingency selection” or “contingency screening” algorithms have been developed and work
quite well. They work by a combination of fast approximate power flows and other analytical techniques that make a prediction of how much limit violation a contingency will produce without solving the entire power flow.
Once a limit violation is predicted and verified by running a power flow on that case the operator must take action to relieve the violation. This may take the form of generation shift, switching the transmission system, or in extreme cases load shedding. Of course if the overload is one that may be tolerated for a significant period of time, the operators may do nothing and “ride it out”. Often, the operator must run a calculation to determine the best means of avoiding a contingency and this involves placing the contingency as a constraint in an OPF.
2.4. Security Constrained Optimal Power Flow
Knowing that an OPF can accommodate a constraint that will guarantee that a contingency overload is eliminated has led to an elaborate program that includes the contingency analysis and an OPF wherein all contingencies are tested, and all overloads are converted to constraints and placed into the OPF. After solution it must be iterated through the contingency analysis again to be sure it has found all bad cases. The end result is a dispatch which guarantees that all contingencies tested are not going to result in trouble.
2.5. Dynamic Security Assessment
One of the grand challenges in the past five years has been to develop a means to asses whether the current power system can withstand a severe fault condition that would result in a system instability. The problem here is similar to the static security analysis in that there are thousands of fault cases that need to be tried, and the time to solve all of them is prohibitive. In fact, research groups working on this problem have targeted a selection procedure that would eliminate all but
fifty cases that would then have to be solved. However, even fifty cases that must be solved in great detail presents a very challenging problem - since each case may take many minutes to solve.
Most researchers in this area are suggesting that advanced computer architectures such as vector processors or multiple processor machines must be used.
2.6. Future Developments in Security Assessment
One future development in the area of power system security that must be addressed, as mentioned before, is the need to provide a cost vs. risk measure for correcting for a contingency (be it a static problem or a dynamic problem). One can readjust the system to protect against contingencies and run its operating cost up so much that it is too expensive given the level of risk avoidance needed.
2.7. Security Analysis and Open Transmission Access
When the transmission system is to be operated as an open system there is a different problem in maintaining system security. First, there is the need to allow independent generating facilities to obtain access to the transmission system in a safe manner. That is, they must contact the transmission system operator and “reserve” transmission capacity for their transaction. The reservation process will necessitate the testing of the system for system security while modeling the proposed transaction.
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