1. Introduction

State estimation is an analytical application designed to determine the probable operating state of a power system transmission network [1]. This methodology, originally introduced by Schweppe in the 1960s [2], has since become a cornerstone in the monitoring and analysis of power systems. Traditional State Estimation (SE) algorithms reliably estimate the transmission system’s operating state when it operates under steady-state conditions with no significant frequency deviations [3], which was historically a reasonable assumption.

However, traditional SE algorithms are not without limitations. The precision of the State Estimator results heavily depends on the single-phase model and the availability and precision of system measurements, such as voltage magnitude, power flow measurements on branches, and real and reactive power injections at each bus [1, 2, 3]. Over time, the transmission system has expanded significantly, becoming approximately 5–6 times larger, and it no longer consistently operates in a steady-state condition due to the penetration of renewable energy and implementation of the power market [3]. To illustrate this point, consider load growth between 2020 and 2025: the average increase in the U.S. was around 3%, whereas Texas is projected to see a 30% increase. This rapid evolution in grid size and load dynamics has diminished the reliability and accuracy of traditional state estimator approaches.

Coinciding with this rapid growth, the power system is undergoing substantial modernization, with new energy generation methods being deployed. These include solar and wind power, often integrated with energy storage solutions like batteries. Since these devices rely on power electronics, they respond to changes in the power system extremely quickly. Concurrently, the availability of measurements for the power system has increased; today’s systems have a greater percentage of real-time measurements for the state estimator compared to the 1960s, which relied more on real-time and pseudo-measurements [1, 4, 5].

1.1 Importance of state estimation in power systems

Today, State Estimator is an essential application for real-time operations of the power grid. Its critical importance is highlighted by the following significant past events:

The 2011 Southwest United States Blackout, for instance, originated from a single transmission line failure and rapidly escalated into a cascading grid collapse, resulting in an estimated $100 million economic impact. This event powerfully demonstrates the profound societal consequences of grid failures [6].
More recently, the near collapse of the ERCOT Grid in Texas during Winter Storm Uri underscored the inherent fragility of power systems [6]. Reports indicate the ERCOT system was merely 4 minutes and 37 seconds away from total system failure [7], leading to restoration times measured in days rather than hours or minutes.

These types of events emphasize the necessity of monitoring, analyzing, and reporting power system conditions within minutes, enabling grid operators to take swift action to prevent catastrophic failures.

Beyond grid reliability, the accuracy of state estimation also carries significant financial implications. Many regions determine energy transaction prices based on locational pricing, which in turn is derived from planned and existing real-time operating state of the transmission grid [8]. This necessitates that the state estimator consistently provides accurate and reliable results. What was once primarily viewed as a reliability tool has now also become a crucial financial instrument.

1.2 Problems suffered by traditional state estimators

Despite significant advancements in measurement and communication technologies, the core algorithmic foundation of state estimation has remained largely unchanged in the power industry over the past 50 years. Traditional methods—especially Weighted Least Squares (WLS) and Least Absolute Value (LAV)—were built on assumptions that are increasingly incompatible with modern power system realities. These include the use of a single-phase model, P/Q/V measurement sets, non-simultaneous measurements, and steady-state operation at a single frequency [2, 3]. Such assumptions imply balanced three-phase conditions and perfect network models based on positive sequence notation, which rarely hold true in today’s power grids [3].

Traditional estimators also assume Gaussian-distributed, independent measurement noise and perfect knowledge of network topology and parameters. However, in practice, measurement errors are often non-Gaussian, and system models frequently contain inaccuracies. Factors such as high penetration of renewables, dynamic operating conditions, cyber threats, and hardware failures introduce complexities that traditional estimators were not designed to handle [6].

Furthermore, traditional estimators lack inherent mechanisms to distinguish between reliable and erroneous data. WLS attempts to fit all available measurements, including bad ones, which leads to biased estimates. LAV, though more robust to outliers, still fails when bad data occur at leverage points—measurements with high influence due to their location or assigned weight. These estimators are particularly vulnerable to corrupted data, which can dominate the solution and go undetected.

Another core limitation lies in their reliance on linearization. Traditional state estimators use iterative Gauss-Newton-type methods that require well-conditioned Jacobian matrices and good initial guesses to converge. In stressed or ill-conditioned systems—such as during contingencies or with topology errors—these methods may diverge or converge to physically implausible solutions. This vulnerability was notably observed in the 2003 blackout, where a topology error caused the state estimator to crash, and operators had to manually intervene due to the system’s failure to provide a valid solution.

Bad data detection in traditional frameworks is generally performed after the initial estimate using residual-based methods, such as the largest normalized residual test. However, these techniques depend on the availability of a converged solution and are ineffective in the presence of severe data corruption or leverage points. In many cases, especially under significant errors, the estimator cannot even produce an initial solution, thus leaving no foundation for subsequent bad data identification [6, 9].

While it is often assumed that increasing the number of measurements improves estimator performance, this can introduce more bad data or communication-related anomalies. Modern measurement infrastructures use internet-based protocols, where failures in a shared communication path can simultaneously affect multiple RTUs. As a result, large portions of the power system can experience missing or delayed measurements. Additionally, measurements from devices such as Intelligent Electronic Devices (IEDs) and Phasor Measurement Units (PMUs) are received at varying rates and time scales, leading to synchronization and skew challenges that traditional estimators are ill-equipped to address.

Even when data is available, traditional estimators can produce physically unrealistic results. A notable example is the generation of negative loads, where load buses appear to be generating power—an artifact of unconstrained optimization without physical validation. These anomalies complicate downstream applications like contingency analysis, voltage stability assessment, and pricing.

In summary, the foundational weaknesses of traditional state estimators include:

Obsolete modeling assumptions [2, 3],
Sensitivity to bad data and leverage points [6, 9],
Dependence on convergence and initialization [10],
Ineffectiveness in detecting hidden or structured data errors, and
Inability to enforce physically realistic constraints.

These challenges highlight the urgent need for structurally robust and adaptive estimation frameworks capable of operating reliably in the face of complex, uncertain, and dynamic system conditions.

Traditional state estimators have been widely adopted and successfully implemented by many power utilities. Some companies, even without regulatory obligation, have expressed a strong interest in running their state estimators in real-time environments. However, despite these efforts, limitations in the accuracy, robustness, or convergence of traditional estimators have prevented some utilities from fully integrating them into their real-time operations. This highlights the performance gap that still exists between current state estimation tools and the stringent demands of real-time grid monitoring and control.

1.3 Traditional state estimators for power transmission grids

Among traditional state estimators, weighted least square state estimator and the least absolute value state estimator are the most widely applied in power systems, which are also the two state estimators received the most extensive research and investigation since the first date when the state estimator was introduced to power systems in the 1960s.

The ability to analyze and study the behavior of the transmission system in real-time has become a critical requirement as power system complexity escalates. As problems with traditional state estimation methods persist and as the transmission system becomes more and more complex with instantaneously changing operating states, new algorithms for state estimation are imperative.

Traditional Power System State Estimation (SE) is the core application providing real-time “snapshots” of complex bus voltages, crucial for grid operators. It enables critical functions like contingency analysis and optimal power flow. The process models the network via

z=hx+e

where x is the state vector (voltages/angles), z represents SCADA measurements (P, Q, V, flows), and e is assumed Gaussian noise [2, 4, 5, 11].

Below is a summary of main types of traditional state estimators.

1.3.1 Weighted Least Squares (WLS)

The Weighted Least Squares (WLS) state estimator is the most widely used algorithm in power system control centers. It estimates the system’s bus voltages by minimizing the sum of weighted squared differences between actual measurements (e.g., power flows, injections, and voltages) and their corresponding model-based estimates. WLS provides a statistically optimal estimate under the assumption of Gaussian measurement noise. WLS estimator minimizes the weighted least squares cost

J=z−hxR2=z−hxTR−1z−hx

using Gauss-Newton iterations. This ensures quadratic convergence with good initial guesses. Efficiency is enhanced by techniques like fast-decoupled WLS and equality-constrained WLS for embedding hard constraints. Post-solve, bad-data processing (e.g., χ², largest nominalized residuals (LNR) tests) flags and isolates faulty measurements for re-estimation [2, 11].

Historically, state estimators assumed steady-state operation, balanced single-phase modeling, Gaussian noise, and sequential SCADA scans, making it practical for 1960s computing. However, rising PMU penetration, renewable variability, and cyber threats now necessitate more robust, dynamic, and distributed extensions to this foundational algorithm [1, 9].

1.3.2 Least absolute value (LAV)

The Least Absolute Value (LAV) state estimator was introduced as a more robust alternative to WLS, particularly in scenarios where measurement errors deviate from the Gaussian assumption or when outliers and bad data are present. Unlike WLS, which minimizes the sum of squared residuals, LAV minimizes the sum of absolute residuals:

J=z−hx1

This formulation reduces the influence of large residuals, thereby improving resilience to bad or corrupted data. However, LAV leads to a non-differentiable and non-quadratic optimization problem, which is more challenging to solve than WLS. Early implementations used linear programming or iteratively reweighted least squares, but convergence could be slow or unreliable for large-scale systems. Despite these difficulties, LAV has been shown to outperform WLS in the presence of gross measurement errors, and has motivated continued research into robust and efficient solution techniques for practical application in power system state estimation [11].

The main obstacles preventing the widespread application of the LAV state estimator in power systems are its slow computational speed and its vulnerability to leverage points. The LAV estimator tends to produce near-zero residuals for bad leverage points, making them extremely difficult to detect. Despite significant research efforts, the leverage point problem remains an open challenge.

1.4 Motivation for advanced solutions

As can be seen from the discussion so far, as the Power System continues to grow, the need for better and faster analysis applications will continue to be required. And this means the bridge between the ‘real world’ and the analytical world will continue to be required. This bridge will continue to be the state estimator. To keep up with grid modernization challenges and rapid changes in the operating states of the Power System, new solutions will be required that can analyze the current and predict the future state of the Power System with increasing levels of accuracy and dependability. As the financial impact of failure of these applications increases, the importance of operating the Power System becomes critical and failure or incorrect results will not be tolerated. The future of these applications will depend on newer and better methods; some are currently in their infancy, and others are yet to be developed.

1.4.1 Brief introduction to static state estimator plus (SSE+)

Gou and Shue [9] proposed a new state estimator, Static State Estimator (SSE+), designed to address several fundamental limitations inherent in traditional power system state estimators. The key advantages of SSE+ are summarized as follows:

Nonlinear equation handling: Unlike traditional estimators that rely on linearizing nonlinear measurement equations (e.g., via Gauss-Newton iterations), SSE+ solves the original nonlinear formulation directly. This approach avoids approximation errors introduced by linearization and results in a more accurate and stable estimation process.
Robustness to bad data: Traditional WLS estimators are sensitive to bad measurements, which can arise from gross measurement errors, communication failures, or incorrect topology or parameter information. SSE+ is designed with built-in bad data rejection capabilities, effectively mitigating the impact of such errors on the final state estimate.
Improved convergence: SSE+ avoids many of the convergence issues that affect both WLS and LAV estimators, particularly in stressed or ill-conditioned systems. By employing a new objective function formulation and leveraging compressive sensing principles, SSE+ can converge to meaningful solutions even when traditional estimators fail.
Basis set and sparse residual utilization: SSE+ introduces a novel use of sparse linear residual equations derived from the measurement model to identify reliable measurement subsets, termed the basis set. This allows for targeted estimation and outlier exclusion without relying on iterative residual tests or reweighting procedures.

Overall, SSE+ represents a significant advancement in state estimation, offering improved accuracy, robustness, and convergence across a wide range of operating conditions.

2. Drawbacks of traditional state estimators

Despite their longstanding role in Energy Management Systems (EMS) operations, traditional state estimators like WLS and LAV suffer from several limitations that undermine their reliability, particularly in adverse or error-prone environments.

2.1 Fundamental assumptions of WLS state estimator

The Weighted Least Squares algorithm is a traditional optimization-based technique for power system state estimation. It operates by minimizing the sum of weighted squared measurement residuals, defined as the discrepancy between measured values and those predicted by the state variables:

Jx=∑i=1mwizi−hix2

where zi is the measured value, hix is the nonlinear function of the state variables x, and wi is the weight assigned based on measurement accuracy.

This formulation is grounded in statistical estimation theory and assumes that measurement errors are zero-mean, independent, identically distributed, and follow a Gaussian (normal) distribution. Under these assumptions, the WLS estimator yields the maximum likelihood estimate of the system state.

The success and accuracy of WLS rely on several critical assumptions [12]:

Measurement noise is Gaussian: The algorithm is statistically optimal only if measurement noise follows a normal distribution. If errors are heavy-tailed or include outliers, WLS becomes sensitive and can produce biased or erroneous state estimates.
Independence and homoscedasticity: WLS assumes that measurement errors are uncorrelated and that their variances are known and fixed. Correlated measurement errors or inaccurate weight assignment can significantly degrade estimator performance.
Accurate system model: The functions hx depend on the system model, including network topology and branch parameters (e.g., line impedances, transformer tap settings). WLS assumes this model is perfectly known. In practice, topology or parameter errors (e.g., switch status mistakes or outdated line parameters) can introduce modeling errors, leading to systematic estimation bias or even divergence.
Good initial guess: For nonlinear WLS formulations, a good initial estimate is often required for iterative solvers such as Gauss-Newton or Newton-Raphson to converge.

While WLS has been widely adopted in Energy Management Systems (EMS) due to its mathematical simplicity and efficient computation, its performance can deteriorate in real-world scenarios where the above assumptions are no longer valid, which are normally caused by different types of bad data, gross measurement errors, and system model inaccuracies. These limitations have motivated the development of more robust state estimation techniques that can tolerate such deviations from ideal assumptions.

2.2 Vulnerability to bad data

Traditional WLS is especially vulnerable to bad data because it implicitly assumes that all measurements are reliable and minimizes the sum of the weighted squared residuals. When bad data are present, the estimator compromises by adjusting the state vector to fit both accurate and erroneous measurements. This often results in errors being spread across the entire estimated state rather than being isolated. Consequently, WLS estimates can suffer from:

Biased voltage magnitudes and angles,
Inaccurate power flow calculations,
Misidentification of system events or contingencies, and
Misleading situational awareness for system operators.

Even a small number of undetected bad measurements can significantly distort the estimate—especially in stressed or weakly observable systems. While residual-based techniques, such as the largest normalized residual test, are often applied post-estimation to detect bad data, they come with notable limitations:

They can fail to detect errors located at leverage points—measurements that exert disproportionate influence on the estimate (further discussed in Section 2.4).
They rely on statistical thresholds and may be ineffective when errors are correlated or when the noise deviates from Gaussian assumptions.
They operate as post hoc tools, providing no influence on the estimation process unless bad data are actively removed and the estimator is rerun.

In response to WLS’s shortcomings, the Least Absolute Value (LAV) estimator was introduced as a robust alternative. While LAV is more resistant to large residuals due to its use of the L1-norm cost function, it has a critical limitation: its breakdown point is zero when bad data appear at leverage points [13]. As shown by Rousseeuw and others, LAV fails to reject even a single bad leverage point, which can dominate the solution and render the estimate unreliable. Therefore, despite its improved handling of gross errors compared to WLS, LAV remains vulnerable to strategically placed or high-impact bad measurements.

Furthermore, iterative bad data removal, such as excluding suspicious measurements followed by re-estimation, can become computationally expensive and still miss coordinated or hidden attacks. Because the power system is a nonlinear system, WLS and LAV state estimators may consider an error-free measurement as bad data, which mistakenly has the largest normalized residual. These challenges underscore the need for more robust estimation frameworks that not only reduce sensitivity to bad data but also incorporate mechanisms for rejecting unreliable measurements during the estimation process itself.

2.3 Convergence challenges under stressed conditions

Under stressed operating conditions, such as periods of high system loading, voltage instability, contingencies, or following a disturbance, the convergence behavior of the traditional WLS state estimator often deteriorates. One of the primary technical reasons is that the Jacobian matrix of the nonlinear measurement equations, which is central to the Newton-based iterative solution, becomes ill-conditioned or nearly singular. This occurs when small changes in the state variables produce very small or poorly scaled changes in the measurement residuals, indicating a loss of numerical sensitivity and observability.

Since WLS typically relies on Newton’s method or the Gauss-Newton approximation to solve the nonlinear optimization problem, convergence heavily depends on the conditioning of the Jacobian and the proximity of the initial guess to the true solution. When the system is stressed or presents with topology errors [14, 15]:

The Jacobian matrix loses rank, causing the linear system at each iteration to become unstable or unreliable.
The solution trajectory can oscillate between points without making meaningful progress.
The algorithm may take tiny step sizes, resulting in premature termination before reaching a true minimum.
In severe cases, the iteration process may diverge entirely, failing to produce any valid estimate.

These convergence failures are especially problematic because they tend to occur precisely when accurate state information is most critical—during emergencies or system anomalies. A 2020 report [16] from the MIT Energy Initiative observed that traditional state estimators “have trouble estimating a system state during unusual or emergency conditions—unfortunately, when they are most needed.”

The convergence issue is not merely a theoretical inconvenience; it poses practical risks. An estimator that fails to converge can leave operators blind to critical system changes, delaying emergency responses and corrective actions. In some utilities, this has led to operators disabling bad data detection routines or skipping iterations to force convergence, further compromising the reliability of situational awareness.

These limitations emphasize the need for robust, non-divergent state estimation algorithms that can maintain stability and accuracy even under stressed or degraded operating conditions. Such methods must be capable of handling nonlinearities more gracefully, avoiding dependency on favorable initial conditions, and tolerating model inaccuracies without collapse.

2.4 The issue of leverage points

In statistical estimation and regression theory, leverage points are data points that have a disproportionately large impact on the estimated solution. In the context of power system state estimation, leverage points correspond to measurements whose influence on the solution is significantly higher than that of others, due to either network structure, measurement type, or weight assignment. These points can unduly shape the outcome of the state estimation process, especially when they are erroneous, making them particularly dangerous if undetected.

Common examples of leverage points in power systems include [17]:

Injection measurements at highly connected buses: Buses with a large number of incident branches have a wide influence on the network. Errors in injection measurements at these buses can distort the voltage estimates across large portions of the system.
Measurements on very low-impedance lines: Because low-impedance lines exhibit minimal voltage drop, any flow or current measurement across such lines tightly constrains the voltage difference between connected buses. As a result, an error in such a measurement can propagate strongly through the Jacobian matrix, distorting the entire state vector.
Measurements assigned excessive weights: Perhaps the most notable example is zero-injection buses, which are often treated as exact measurements. These indicate that a bus has neither generation nor load, leading operators to assign extremely high weights (or effectively infinite precision) to enforce the zero-injection constraint.

While this might seem reasonable under ideal modeling assumptions, zero-injection buses are highly susceptible to becoming hidden sources of error. If any branch connected to a zero-injection bus contains a topology error (e.g., incorrect breaker status) or parameter error (e.g., wrong line impedance), then the actual power injection will no longer be zero. However, because the zero-injection constraint is treated as exact, the estimator forcefully tries to satisfy it, adjusting nearby voltage estimates in a biased manner.

This creates a hidden bad data condition−the measurement is incorrect, but residual-based detection fails because the weight is so high that the residual is suppressed. Standard bad data detection methods, such as the largest normalized residual tests, become ineffective at identifying errors at leverage points, especially when the measurement dominates the estimation process.

Moreover, the presence of such high-leverage bad data can mask other errors, compromise convergence, and result in a false sense of accuracy in the estimated state. Leverage points can be especially problematic in sparse or weakly observable networks, where each measurement carries more influence.

To mitigate these risks, alternative approaches are needed, such as:

Robust estimators that reduce sensitivity to outliers and high-leverage data,
Influence analysis to identify and monitor measurements with excessive impact on the solution,
Adaptive weighting that accounts for the possibility of modeling errors, even in traditionally trusted measurements like zero-injections.

In summary, leverage points represent a critical blind spot in traditional state estimators whose objective functions are based on the residual magnitudes, including WLS and LAV-based estimation, one that can severely undermine both the accuracy and reliability of state estimates—particularly in the presence of hidden model errors.

2.5 Inability to accurately recover true state values

A fundamental limitation of the Weighted Least Squares estimator is its inherent tendency to fit all measurements simultaneously, including those that are flawed. Because WLS minimizes the total weighted sum of squared residuals, it does not inherently distinguish between correct and incorrect data unless additional mechanisms, such as bad data detection and removal, are explicitly implemented. As a result, the estimated state variables often reflect a compromise between accurate and inaccurate information.

This compromise leads to several undesirable consequences [9]:

True system states become biased: When even a few bad measurements are present, they can pull the solution away from the true values. The estimated voltage magnitudes and angles may no longer represent the actual operating condition of the system.
Bad data are absorbed rather than rejected: Since WLS gives all measurements a role in shaping the estimate, erroneous values are not excluded by default. Instead, their errors are redistributed across the network, contaminating the state estimate in subtle but significant ways.
Recovered measurement values are distorted: The measurement values recovered by projecting the estimated state back through the measurement functions hx may no longer resemble the true system behavior. This includes power injections, flows, and bus voltages, which can deviate from actual values even when the residuals appear small.

For example, consider a case where a power injection at a bus is erroneously recorded as 500 MW due to a sensor malfunction, while the actual value is 200 MW. WLS will attempt to adjust the surrounding voltage angles and magnitudes to minimize the mismatch, rather than reject the measurement. This typically results in:

A distorted voltage profile around the erroneous bus,
Artificial shifts in power flows in adjacent lines,
And propagation of estimation error to other buses—even those with no measurement error themselves.

Critically, the flawed measurement is never truly corrected. Instead, the estimator masks the problem by diffusing it, potentially lowering the residual to within acceptable limits but corrupting the overall solution. The resulting estimate may still appear to “fit the data” in a statistical sense, but it no longer reflects the physical reality of the system.

This inability to recover the true state is particularly dangerous in scenarios where precise estimates are required, such as:

Contingency analysis,
Voltage stability assessment,
Real-time control decisions, and
Market settlement calculations.

Moreover, in large-scale systems with thousands of measurements and states, these subtle biases can accumulate and lead operators to misinterpret system conditions, such as underestimating load, overlooking overloads, or misidentifying the location of a disturbance.

To address these limitations, modern estimation methods must move beyond WLS’s “fit-all” approach. Instead, they should be capable of identifying and isolating erroneous data, or better yet, inherently rejecting the influence of flawed measurements during the estimation process. Only then can the state estimator reliably recover true system values and provide accurate situational awareness.

2.6 Negative loads in state estimation

In WLS-based state estimation widely used in power systems, it is common to encounter the phenomenon of negative loads, where the estimated power injection at load buses appears positive, suggesting that the bus is generating power rather than consuming it. These physically unrealistic results arise because WLS does not impose explicit constraints on the direction or sign of power injections during the estimation process. Instead, the algorithm simply adjusts the state variables to minimize the residuals, regardless of whether the resulting estimates conform to known operational characteristics.

As a consequence, WLS may produce dozens or even hundreds of negative loads in a moderately sized power system, especially under noisy measurement conditions or in the presence of bad data. These outcomes conflict with the basic understanding of system operation, where load buses are expected to consume power. Furthermore, negative loads can:

Mislead operators by distorting load distribution across the network,
Affect contingency and reliability analysis,
Complicate post-processing and validation procedures.

In contrast, Static State Estimation Plus (SSE+) addresses this issue by significantly reducing the occurrence of negative loads, often limiting them to fewer than 10 in practical implementations. This improvement stems from SSE+’s strategic selection of a clean basis set, composed of measurements that are consistent, well-behaved, and physically trustworthy. By focusing the estimation process on this reliable subset of data, SSE+ avoids overfitting to flawed or noisy measurements that could otherwise distort the results.

Rather than attempting to fit all data equally, SSE+ leverages the structure and redundancy in the measurement set to solve the estimation problem using only the most reliable information. This inherently suppresses anomalous artifacts like negative loads and leads to state estimates that are not only mathematically consistent but also physically meaningful and operationally realistic.

The ability to minimize the occurrence of negative loads underscores the robustness of SSE+ and its superiority over traditional WLS in producing state estimates that truly reflect the actual conditions of the power system. This capability is especially important in real-time applications, where credible and interpretable estimates are essential for safe and reliable system operation.

3. Illustrative examples of WLS and LAV limitations

To demonstrate the performance limitations of traditional estimators, we present a small but representative test case. This example uses a simplified 4-bus DC model with various measurement scenarios. It reveals how bad data corrupt the solution under the WLS method and how such estimators struggle with identification and correction.

3.1 Case study: Bad data impact on state estimation

3.1.1 WLS state estimator as average estimator

We want to use this small example to show that the WLS state estimator is essentially an average estimator.

A scalar variable x is defined, and we have three noisy linear measurements:

zi=x+ei,i=1,2,3

Z=1.02.01.5

Noise variances: σ12=1,σ22=4, σ32=1.

WLS state estimator reached an estimate of x:

xWLS=1.3.

Let us calculate the weighted average of all the measurements:

x̂=z1σ12+z2σ22+z3σ321σ12+1σ22+1σ22=1.0+0.5+1.51+0.25+1=3.02.25=1.3

It is obvious that the WLS state estimator is basically an average estimator, which calculates the weighted average of all the measurements, including the bad ones.

The next example shows how the bad measurements affect the solution of the WLS state estimator.

3.1.2 4-bus DC system

For simplicity, we use a 4-bus power system to show the performance of WLS, LAV, and SSE+ with bad data. The system information is given below:

A 4-bus system with buses labeled 1 through 4.
All transmission line reactance x=1.0 p.u.
Bus 1 is the reference bus with θ1=0.0o.
The state variables to estimate are the voltage phase angles at buses 2, 3, and 4.
Measurements are given as real power flows along lines or at buses.

The single-phase graph of this system is given as follows (Figure 1):

For the given set of measurements, we have the Jacobian matrix H as follows:

H=1−100−1100010−1−10012−10−1−13−1−10−12−1−1−1−131000

The real power measurements are given in per-unit values:

Z=−0.10.1−0.20.3−0.4−0.20.00.60.0

θtrue=0.00.10.20.3

3.1.2.1 Case 1: No bad data

When the measurements contain only noise, WLS performs very well, and it can reach the true state values:

θestimated=0.00.10.20.3

θtrue=0.00.10.20.3

The results show that WLS state estimator is able to achieve accurate state variables when only noise is present in the measurements.

3.1.2.2 Case 2: Single bad measurement

We now inject a gross error into the measurement P2 by forcing (Figure 2) P2=0.5:

Figure 2.
4-bus system with a single bad measurement.

3.1.2.2.1 WLS state estimator

WLS state estimator reached the following values for the state variables:

θWLSestimated=0.00.21810.14670.2770

Using the obtained estimate of state variables, we calculated the measurement recovered values and the residuals (Table 1).

	P12	P21	P24	P41	P1	P2	P3	P4
Measurements	−0.1	0.1	−0.2	0.3	−0.4	0.5	0	0.6
Recovered measurements	−0.2181	0.2181	−0.0588	0.277	−0.4951	0.2307	−0.2016	0.466
Residuals	0.1181	−0.1181	−0.1412	0.023	0.0951	0.2693	0.2016	0.134

Table 1.

WLS state estimator recovered values and obtained residuals of the measurements.

It is obvious to see that the residuals of all measurements are affected because all the state variables are biased due to this single bad measurement.

3.1.2.2.2 SSE+ (static state estimator plus)

SSE+ was introduced in Ref. [9], which optimizes the selected basis set of measurements such that all measurements in the basis set are error-free. The basis set was formed by measurements P12,P24,P3, highlighted in yellow (Figure 3).

Figure 3.
4-bus system with SEE+ basis set.

SSE+ used the basis set measurements to solve the state variables. It exactly solved the state variables because the basis set is error free.

θSSE+estimated=0.00.10.20.3

Using the obtained estimate of state variables, we calculated the measurement recovered values and the residuals (Table 2).

	P12	P21	P24	P41	P1	P2	P4
Measurements	−0.1	0.1	−0.2	0.3	−0.4	0.5	0.6
Recovered measurements	−0.1	0.1	−0.2	0.3	−0.4	−0.2	0.6
Residuals	0	0	0	0	0	0.7	0

Table 2.

SSE+ recovered measurement values and obtained measurement residuals.

We can see that all measurements except the bad measurement P2 are exactly recovered because the estimate of the state variables is accurate. Even for the bad measurement P2, its true value −0.2 is exactly recovered.

3.1.2.3 Case 3: Multiple bad measurements

In this scenario, we investigate the impact of multiple bad measurements on the state estimation results. Specifically, we intentionally corrupt three measurements in the system:

z₂ = 0.5
z₃ = 0.0
z₄ = −0.3

The simultaneous presence of multiple bad measurements makes it more difficult for traditional state estimators to isolate and reject errors. WLS, in particular, attempts to fit all data points, including the bad ones, which often leads to a distorted estimate. The influence of these errors may spread across the entire state vector, and residual-based detection becomes less effective (Figure 4).

Figure 4.
4-bus power system with three bad measurements.

In contrast, SSE+ identifies these measurements as inconsistent by focusing on minimizing the number of abnormal residuals rather than minimizing the objective function formed by their magnitudes. This allows SSE+ to reject the bad data with the highest possibility and recover a more accurate and physically meaningful estimate.

3.1.2.3.1 SSE+ (static state estimator plus)

Table 3 shows that SSE+ can minimize the influence of bad measurements because only one state variable is biased and two measurements received non-zero residuals. However, if all three bad measurements are rejected, the entire system becomes unsolvable.

Variable	True values	Estimate values
θ1	0.0	0.0
θ2	0.1	0.1
θ3	0.2	0.7
θ4	0.3	0.3

Measurements	P12	P21	P24	P41	P1	P2	P3	P4
Measurements values	−0.1	0.1	−0.2	0.3	−0.4	0.5	1	1.6
Recovered measurements	−0.1	0.1	−0.2	0.3	−0.4	−0.7	1	0.1
Residuals	0	0	0	0	0	1.2	0	1.5

Table 3.

SSE+ recovered measurement values and obtained measurement residuals for multiple bad measurements.

3.1.2.3.2 WLS state estimator

In this situation, the solution becomes even more distorted. The voltage estimates are now significantly removed from their true values. The WLS estimator spreads the impact of errors across the solution space to satisfy the optimization goal (Table 4).

Variable	True values	Estimate values
θ1	0.0	0.0
θ2	0.1	0.2840
θ3	0.2	0.5438
θ4	0.3	0.5568

Measurements	P12	P21	P24	P41	P1	P2	P3	P4
Measurements values	−0.1	0.1	−0.2	0.3	−0.4	0.5	1	1.6
Recovered measurements	−0.284	0.284	−0.2728	0.5568	−0.8407	−0.2488	0.2469	0.8426
Residuals	0.184	−0.184	0.0728	−0.2568	0.4407	0.7488	0.7531	0.7574

Table 4.

WLS state estimator recovered measurement values and obtained measurement residuals for multiple bad measurements.

This demonstrates a critical vulnerability: WLS cannot isolate or recover the true state values once multiple errors are present.

3.2 Analysis of leverage points and biased solutions

If a leverage point measurement is corrupted, WLS and LAV state estimators are forced to “fit” it, regardless of its validity.

This behavior makes WLS and LAV particularly vulnerable in practice because:

Residuals for leverage points are suppressed,
Bad data in such locations go undetected,
Bias accumulates silently in the estimated state.

The following example gives the responses of the LAV state estimator and SSE+ to leverage point measurements.

Consider a simple regression model given below [2]:

zi=Ai1x1+Ai2x2+eii=1,•••,5

Where Aij, represents the j^th entry in vector Ai as defined above. The associated observation data are given in the table below. SD=0.001.

3.2.1 Test 1: Add a bad data on a normal measurement

The measurement configuration (a bad data is added on the third measurement) (Table 5).

i	zi	Ai1	Ai2
1	14.9954	1.0	1.0
2	18.0053	2.0	1.0
3	3.9969	3.0	1.0
4	24.0213	4.0	1.0
5	161.8142	50.0	1.0

Table 5.

Two variables example for a regular bad measurement.

XTrue=312T

It is clear to see that z5 is a leverage point because of big 50.0.

3.2.1.1 WLS estimate

X̂T=3.08057.6000∆XT=−0.0805,4.4

See Table 6.

i	zi	ẑi	ri
1	14.9954	10.6806	4.3149
2	18.0053	13.7611	4.2442
3	3.9969	16.8417	−12.8448
4	24.0213	19.9222	4.0991
5	161.8142	161.6275	0.1867

Table 6.

Results of WLS state estimator for two variables example for a regular bad measurement.

3.2.1.2 LAV estimate (selects z1, z5)

X̂T=2.996311.9991∆XT=0.00370.0009

See Table 7.

i	zi	ẑi	ri
1	14.9954	14.9954	0.0
2	18.0053	17.9917	0.0135
3	3.9969	20.9880	−16.9912
4	24.0213	23.9843	0.0370
5	161.8142	161.8142	0.0

Table 7.

Results of LAV state estimator for two variables example for a regular bad measurement.

x1true=3,x2true=12

The bad data has a significant negative influence on the WLS solution.
The bad measurement was successfully rejected by LAV.

3.2.2 Test 2: Response to the leverage points

The above example shows that WLS state estimator is not able to reject normal bad measurements, while LAV state estimator is more robust than WLS state estimator because of being capable of rejecting bad measurements.

However, LAV is still suffering from the problem of leverage points. The definition of leverage points is given below:

A leverage point is a measurement that affects (dominates) the state estimator in such a way that the state estimate residuals will closely satisfy that particular measurement.

Influence of bad leverage point leads state estimators to a solution that is significantly biased due to the bad leverage point, and it is hard to detect those bad leverage points because their residuals are close to zero (Table 8).

i	zi	Ai1	Ai2
1	14.9954	1.0	1.0
2	18.0053	2.0	1.0
3	20.9382	3.0	1.0
4	24.0213	4.0	1.0
5	20.0065	50.0	1.0

Table 8.

Two variables example for bad leverage point.

z5 is a bad leverage point. Standard deviation is 0.001. x1true=3,x2true=12.

3.2.2.1 LAV estimate (selects z1,z5)

X̂T=0.102314.8931

∆XT=2.8977−2.8931

See Table 9.

i	zi	ẑi	ri
1	14.9954	14.9954	0.0
2	18.0053	15.0977	2.9076
3	20.9382	15.1999	5.7383
4	24.0213	15.3022	8.7191
5	20.0065	20.065	0.0

Table 9.

Results of LAV state estimator for two variables example for bad leverage point.

3.2.2.2 SSE+ estimate (selects z1,z2)

X̂T=3.009911.9855

∆XT=−0.00990.0145

See Table 10.

i	z_i	residual (z₁, z₂)	residual (z₁, z₅)	residual (z₂, z₄)	residual (z₂, z₅)
1	14.9954	0.0	0.0	−0.0019	−2.9682
2	18.0053	0.0	2.9076	0.0	0.0
3	20.9382	−0.077	5.7383	−0.0751	2.8912
4	24.0213	−0.0038	8.7191	0.0	5.9326
5	20.0065	−142.474	0.0	−142.3828	0.0
# of abnormal residuals		3	3	1	3

Table 10.

Results of SSE+ state estimator for two variables example for bad leverage point.

Any combination of two measurements except z5 is a solution of SSE+, which rejects the bad leverage point effectively.

4. The root causes of the vulnerability of traditional state estimators

This section explores the reasons why the performance of WLS and LAV estimators cannot meet the standards in real-world power system applications. These issues are not just implementation artifacts—they stem from fundamental assumptions baked into their mathematical formulations.

4.1 Including all measurements in the objective function vs. applying a rejection mechanism

In most traditional state estimation frameworks, like WLS state estimator, the optimization objective includes all available measurements, regardless of quality:t

Jx=∑i=1mwizi−hix2

This approach implicitly assumes that the majority of measurements are correct and that any errors will “average out.” However, even a few bad data points, especially if they are strategically located or highly weighted, can distort the entire solution.

A more robust alternative would apply a measurement selection or rejection mechanism prior to or during optimization. Rather than treating all data equally, the algorithm would aim to:

Identify a subset of consistent measurements
Exclude or downweight measurements likely to be erroneous.

This change in philosophy transforms state estimation from a global averaging problem into a local consistency evaluation, more aligned with practical grid behavior.

4.2 Evaluating residual magnitudes vs. counting abnormal residuals

Traditional state estimators, such as WLS and LAV, are fundamentally designed to minimize objective functions formed by residual magnitudes, where residuals are defined as the difference between measured and estimated values. Their objective functions, whether based on squared residuals (in WLS) or absolute residuals (in LAV), are built on the assumption that reducing the total residual magnitude leads to a more accurate estimate of the system state.

However, this reliance on residual magnitudes introduces a fundamental structural vulnerability. When a measurement is assigned a large weight, any deviation from its expected value contributes significantly to the objective function. To avoid a large penalty, the estimator is forced to reduce the residual of that measurement to near zero, even if the measurement is corrupted. This mechanism causes such measurements to become leverage points, exerting undue influence on the solution. If the leveraged measurement contains bad data, its residual still remains artificially small, effectively masking the error and allowing it to corrupt the overall state estimate.

Moreover, estimators that rely solely on residual magnitudes face additional limitations:

Bad data at leverage points evade detection because their residuals are minimized by design.
Model errors may distort residuals in misleading ways, leading to incorrect interpretations of measurement quality.

In contrast, SSE+ adopts a fundamentally different approach by shifting the objective from minimizing residual magnitude-based objective functions to minimizing the number of abnormal residuals. This reflects a shift in philosophy: rather than forcing every residual to be small, SSE+ allows some measurements to deviate if doing so improves the global consistency of the state estimate. By identifying and minimizing the count of residuals that exceed a consistency threshold, SSE+ is less vulnerable to leverage effects, better able to isolate bad data, and more structurally aware of how measurements interact. This leads to estimates that are not only numerically stable but also physically meaningful, even under stressed system conditions or when the model includes parameter or topology errors.

This transition from measuring “how much” a residual deviates to “how many” residuals are inconsistent represents a key innovation in state estimation and reflects the kind of structural robustness needed for future power system operations [9, 18].

4.3 Linearizing the objective function vs. preserving nonlinear characteristics

Traditional state estimators, including WLS and LAV methods, typically rely on linearizing the inherently nonlinear power flow equations. This is done to construct the Jacobian matrix required by iterative solvers such as Gauss-Newton. These nonlinear equations define the relationship between power injections, line flows, and bus voltages, but are approximated locally to enable tractable solution steps.

While linearization provides computational efficiency, particularly for large-scale systems, it introduces fundamental weaknesses that degrade both robustness and estimation quality:

Ill-conditioned Jacobian matrix under stressed conditions: When the system is heavily loaded, near voltage collapse, or contains topology/parameter errors, the Jacobian matrix may become nearly singular or lose rank. This often leads to divergence or instability in Newton-type iterations.
Strong dependence on the initial guess: Since linearization is only valid near the expansion point, a poor initial estimate can cause the algorithm to follow a misleading path, fail to converge, or settle on a physically implausible solution.
Loss of critical nonlinear behaviors: Features like transformer tap changers, reactive power limits, and the behavior of lines with zero impedance are nonlinear in nature and cannot be captured accurately using first-order approximations. Linearized models tend to distort or suppress these effects.
Propagation of bad data: The structure of the linearized system tends to diffuse the impact of bad measurements throughout the entire solution, making it difficult to isolate erroneous data without intensive post-processing.

In contrast, SSE+ avoids the linearization process entirely by solving the original nonlinear equations directly. It does not construct or depend on a Jacobian matrix, nor does it rely on step-by-step approximation techniques like Gauss-Newton. Instead, SSE+ retains the full nonlinear structure of the measurement model throughout the estimation process, enabling it to operate with greater numerical stability and higher physical fidelity.

Moreover, SSE+ adopts a sparsity-aware formulation that minimizes the number of abnormal residuals—rather than minimizing their magnitudes—while preserving nonlinear interactions between state variables and measurements. This provides several key advantages:

Robust convergence even under extreme or misconfigured system conditions, since no fragile Jacobian matrix needs to be inverted or updated.
Greater accuracy near the operating limits of the system, where linear approximations fail to capture system dynamics correctly.
Natural treatment of non-standard components, such as tap transformers or partially observable buses, without artificial simplification.
Reduced dependence on the initial state guess, as the estimation is guided by structural consistency in the measurement set, not by iterative local updates.

By working with the full nonlinear model from the outset, SSE+ shifts the trade-off between computational efficiency and physical accuracy. Rather than sacrificing realism for speed, SSE+ achieves both, using a structurally grounded formulation that scales efficiently and converges reliably—even under stress or uncertainty.

This highlights a crucial shift in the philosophy of state estimation: preserving the nonlinear nature of the problem is not a computational burden to be avoided, but a robustness-enhancing strategy essential for modern power systems [18].

4.4 Achieving a converged solution by adjusting measurement weights vs. obtaining voltage estimates best reflecting the operating state

Traditional WLS-based state estimators often achieve convergence by adjusting the weights of measurements during the iterative solution process. When faced with large residuals caused by bad data, topological or parameter errors, or stressed operating conditions, these estimators reduce the influence of problematic measurements—sometimes artificially—to force convergence. However, this practice can yield voltage estimates that do not accurately reflect the true operating state of the system. The convergence, in such cases, is numerical rather than physical, and the final state estimate may be significantly biased.

In contrast, SSE+ takes a fundamentally different approach. It focuses on identifying a clean basis set—a subset of measurements that is both sufficient for observability and minimally contaminated by bad data. Through compressive sensing and sparse recovery techniques, SSE+ effectively pushes bad or inconsistent measurements out of the basis set and into the redundant set. By solving the state estimation problem using only the clean basis set, SSE+ ensures that the resulting voltage estimates closely reflect the actual system conditions. This approach prioritizes physical accuracy over artificial convergence, yielding robust and reliable estimates even in the presence of significant measurement errors or modeling uncertainties [18].

5. Simulation results and performance evaluation

This section highlights the numerical results of state estimation under different conditions—ranging from ideal to highly corrupted data environments—focusing on convergence rate, bad data rejection, recovery accuracy, and computational efficiency [9].

5.1 Tests on 13659 bus system

The system data and measurement configuration for this test are given in Table 11.

	Configuration
Total # of buses	13659
Total # of branches	20467
Total # of line flow measurements	35510
Total # of bus injection measurements	27318
Total # of bus voltage magnitude measurements	5772
Measurement redundancy ratio	2.51

Table 11.

Measurement configuration of 13659 bus system.

In this study, three categories of bad data—topology errors, parameter errors, and measurement value errors—were systematically introduced into the system to evaluate estimator performance under realistic conditions. The measurement redundancy ratio, defined as the ratio of the total number of measurements to the number of state variables, was set at 2.51. This level reflects a moderate redundancy typical in practical applications. For each type of bad data, the number of erroneous entries varied from 10 to 500, and 100 randomized trials were conducted for each case. To further assess the impact of significant errors, topology and parameter faults were injected specifically on branches carrying at least 80 MW of real power flow. Across all test scenarios, the SSE+ algorithm demonstrated a 100% convergence rate. The results for all three types of bad data are summarized in Figures 5–7.

Figure 5.
Rejection testing on the typical three bad data.

Figure 6.
Calculation time testing on typical three bad data. The PC computer has 4 cores with 1.80GHz.

Figure 7.
Average voltage magnitudes for the testing of typical three bad data.

5.2 Test on a real power system case

SSE+ was tested on about 100 real-world cases provided by four major utility companies in North America. This subsection presents the testing results for one representative case. Detailed system information and the corresponding measurement configuration are summarized in Table 12. Notably, the measurement redundancy ratio in this case is relatively low, reflecting the challenges typically encountered in practical utility environments.

# of buses	3409
# of transmission branches	4337
# of line flow measurements	3571
# of injection measurements	3022
# of voltage measurements	1457
Measurement Redundancy ratio	2.15

Table 12.

System data and measurement configuration of the utility real case.

We tested the solution quality and capability of bad data recovery for SE+ in the following three aspects.

5.2.1 Comparison of voltage solution quality with a standard commercial SSE+

For this real-world system case, the utility also provided the voltage solution obtained using their standard commercial state estimator. To evaluate and compare the quality of voltage estimates produced by SSE+ and the commercial SSE+, we computed the residuals of the voltage measurements for both solutions. The comparative results are presented in Table 13.

	Standard commercial SSE+	SSE+
Sum of voltage measurement residuals	15.45 p.u.	12.97 p.u.
Average of voltage measurement residuals	0.0108 p.u.	0.0091 p.u.
Standard deviation	0.0163	0.0158
Mean voltage solution value in p.u.	1.0259	1.0257

Table 13.

Comparison of voltage solution quality.

Voltage measurement residuals are defined as the differences between the measured bus voltage magnitudes and the corresponding solved voltage magnitudes. Smaller residuals indicate that the estimated voltages closely match the actual measurements. To facilitate an overall comparison, we sorted the voltage residuals from both solutions and plotted them as two curves on the same graph. As shown in Figure 8, the voltage estimates produced by SSE+ consistently exhibit smaller residuals, demonstrating that SSE+ provides a solution more closely aligned with the measured data than the standard commercial SSE+.

Figure 8.
The comparison of voltage residuals between solutions of commercial SSE+ and SSE+.

5.2.2 Recovery of accurate values lost due to measurement errors

It is well known that the WLS state estimator is generally unable to recover the true values lost due to bad measurements, as it lacks the inherent capability to reject erroneous data. In contrast, the SSE+ approach effectively mitigates the impact of bad measurements by constructing a base set composed of the minimum number of most consistent measurements. This process naturally excludes inconsistent—or bad—measurements from the estimation, thereby enhancing the reliability of the solution. As a result, the voltage estimates produced by SSE+ are not affected by rejected bad data and are expected to be more accurate. To evaluate the bad measurement recovery capability of SSE+, we conducted the following two tests.

5.2.2.1 Recovery of bad data on injection measurements

In this test, large errors of 1000 MW were intentionally introduced to 10 real power injection measurements. Due to the low measurement redundancy ratio in the system, SSE+ selected five of these ten measurements to form part of the base set, while the remaining five were placed in the redundant set. Remarkably, four of the bad measurements were accurately recovered by SSE+, as illustrated in Figure 9.

Figure 9.
Test results for bad data recovery for bus injection measurements.

One of the bad injection measurements could not be accurately recovered: the original value was 0.0 MW, while the recovered value was 3424.0 MW. This large deviation occurred because the voltage at the associated bus was biased, caused by a neighboring bad injection measurement that was included in the base set. However, the fact that the other four bad measurements were precisely recovered indicates that the negative impact of bad measurements within the base set is limited and localized. SSE+ effectively isolates and minimizes the influence of such errors, enabling accurate recovery in most cases even under challenging conditions.

5.2.2.2 Recovery of bad data on powerline flow measurements

To evaluate the recovery capability of SSE+ for contaminated line flow measurements, we introduced large errors of 500 MW to 10 real power line flow measurements. Among these, SSE+ selected one bad measurement to enter the base set, while the remaining nine were rejected and placed in the redundant set. To illustrate the recovery accuracy, Figure 10 presents the differences between the original values and their recovered counterparts. The small differences shown in the figure indicate that SSE+ accurately recovered the contaminated line flow measurements, even when some erroneous data were present in the base set (Figure 11).

Figure 10.
Testing of bad data recovery for the line flow measurements: big errors on line flow measurements.

Figure 11.
Testing of bad data recovery for the line flow measurements: recovered values of line flow measurements with big errors.

5.3 Computational efficiency and scalability

The computational performance of SSE+ was evaluated using a diverse set of real-world power systems of varying sizes, as illustrated in Figure 12. The number of buses in these systems ranges from approximately 2500 to over 13,000. Despite the significant variation in system size and complexity, SSE+ consistently demonstrates high computational efficiency. For all systems containing fewer than 7000 buses, the state estimation process is completed in under 0.6 seconds. Even in the largest case—comprising 13,659 buses—SSE+ required only about 1.2 seconds of CPU time. These results underscore the scalability and real-time capability of the SSE+ algorithm. Its ability to deliver robust and accurate state estimates with minimal computational burden confirms its suitability for deployment in modern energy management systems.

Figure 12.
The CPU time of SSE+ for power systems with different sizes.

6. Conclusions and future work

This chapter aimed to revisit the fundamentals of power system state estimation, focusing on basic concepts, practical challenges, and small illustrative examples, rather than on advanced mathematical theories or abstract generalizations. The intent was to provide a clear and accessible understanding of how traditional estimators operate, why they often fail under real-world conditions, and what conceptual shifts are needed to develop more robust and reliable estimation methods for modern power systems.

The analysis centered primarily on the two most widely used estimation techniques in power systems: Weighted Least Squares (WLS) and Least Absolute Value (LAV) estimators. These methods are grounded in general-purpose optimization frameworks that assume measurement noise is Gaussian, the system model is accurate, and the estimation problem is well-conditioned. However, this chapter demonstrated that these assumptions break down frequently in actual grid operations—where measurement errors, topology inaccuracies, parameter uncertainties, and stressed system conditions are common.

Through simplified examples and structural analysis, the chapter revealed the deep underlying causes of the limitations inherent in these estimators. WLS, in particular, distributes the influence of all measurements—accurate or flawed—across the entire system state vector. When even a single bad data point exists, especially at a leverage point, it can bias the solution without being detected by standard residual analysis. In stressed conditions, such as after disturbances or during high loading scenarios, the linearized Jacobian matrix used in Newton-based solvers may become ill-conditioned, leading to poor convergence or complete divergence. These failures are not anomalies but rather consequences of core design assumptions that treat all data equally and depend on residual magnitudes as the primary signal of inconsistency.

To illustrate these issues, a simple 4-bus example was used to show how traditional estimators behave in the presence of bad data. Even in this minimal setting, WLS SE failed to produce reliable results when incorrect measurements were introduced. These examples were deliberately kept small in order to isolate and visualize the behavior of the algorithms, making it easier to trace the source of failure and to expose the conceptual weaknesses of the traditional estimation framework.

In contrast, the chapter introduced SSE+, a more robust estimator designed with the unique characteristics of power systems in mind. Unlike general-purpose estimators, SSE+ does not attempt to fit all data equally. Instead, it operates on the principle of selectivity and consistency, identifying a minimal subset of measurements that are physically consistent and trustworthy, and solving the estimation problem primarily from this clean basis set. This shift in philosophy—away from full-data optimization and toward structure-aware selection—represents a fundamental departure from traditional methods. SSE+ is not designed for arbitrary systems, but rather for the specific challenges posed by real-world power grids, including limited observability, network nonlinearities, and error-prone measurements.

The examples presented in the chapter demonstrate this philosophical difference. While WLS aims to minimize global error through averaging, SSE+ seeks to recover the most physically meaningful and internally consistent solution, even if that means discarding a portion of the available data. This makes SSE+ inherently more robust to bad data, less dependent on ideal assumptions, and more reflective of actual system behavior.

These insights have important implications for power system operations. As power grids become more complex, with increasing integration of renewables, growing measurement diversity, more frequent extreme weather events, and heightened cybersecurity risks, the need for reliable and adaptable estimation grows. Traditional residual-based bad data detection techniques are inadequate in the presence of topological uncertainty, correlated errors, or stealthy data manipulation. Estimators must evolve to incorporate mechanisms that can detect inconsistency patterns, adapt to modeling inaccuracies, and respond intelligently to conflicting information.

Looking ahead, the future of state estimation lies in developing algorithms that are selective, resilient, and system-aware. One promising direction is the advancement of data-selective estimators—methods that focus on identifying a self-consistent core of data and solving the estimation problem from that subset, rather than attempting to reconcile all measurements. This approach directly addresses the limitations exposed in this chapter and aligns more closely with how real operators evaluate system reliability: not by assuming everything is correct, but by identifying what can be trusted.

Another critical area is the integration of machine learning and artificial intelligence into the estimation process. These tools can help identify suspicious measurements, adjust weights dynamically, or learn system behavior patterns that may indicate model errors or cyberattacks. When combined with physical modeling, AI-driven insights can enhance both accuracy and robustness.

Additionally, hybrid estimation frameworks that fuse data from SCADA and PMUs can increase observability and redundancy, offering more reliable estimates even under stressed or abnormal operating conditions. As measurement infrastructure evolves, estimators must be capable of seamlessly integrating diverse data sources with different resolutions, time scales, and error characteristics.

Moreover, in a world where cyber-physical threats are increasingly sophisticated, the development of cybersecurity-aware estimation techniques is essential. Estimators must be designed not only to tolerate noise but also to identify and reject intentional data manipulation, stealth topology alterations, and coordinated attacks that traditional techniques cannot detect.

Finally, there is a growing need for formally robust estimation algorithms that offer performance guarantees under specified conditions. Estimators with bounded error characteristics and provable resilience to data corruption will become essential tools in mission-critical operations and regulatory compliance.

Hanbing Ke

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