 Research
 Open Access
An efficient method to include equality constraints in branch current distribution system state estimation
 Carlo Muscas^{1}Email author,
 Marco Pau^{1},
 Paolo Attilio Pegoraro^{1} and
 Sara Sulis^{1}
https://doi.org/10.1186/s1363401502061
© Muscas et al.; licensee Springer. 2015
Received: 30 October 2014
Accepted: 9 February 2015
Published: 3 March 2015
Abstract
Distribution system state estimation is a fundamental tool for the management and control functions envisaged for future distribution grids. The design of accurate and efficient algorithms is essential to provide estimates compliant with the needed accuracy requirements and to allow the realtime operation of the different applications. To achieve such requirements, peculiarities of the distribution systems have to be duly taken into account. Branch currentbased estimators are an efficient solution for performing state estimation in radial or weakly meshed networks. In this paper, a simple technique, which exploits the particular formulation of the branch current estimators, is proposed to deal with zero injection and mesh constraints. Tests performed on an unbalanced IEEE 123bus network show the capability of the proposed method to further improve efficiency performance of branch current estimators.
Keywords
 Distribution systems
 State estimation
 Branch current estimator
 Weighted least squares
 Equality constraints
1 Introduction
In the smart grid (SG) scenario, where control and management activities of the electric distribution network are expected to play a relevant and increasing role, distribution system state estimation (DSSE) is conceived as a fundamental monitoring tool. In fact, control systems, such as distribution management systems (DMSs), must rely on a possibly complete and accurate knowledge of the state of the network given by DSSE [1].
In the current evolving scenario, the increasing presence of distributed generation (DG), storage devices and flexible loads to be controlled leads to unforeseen dynamics, which require suitable measurement responsiveness. In this context, DSSE techniques able to work at high reporting rates, while safeguarding the estimation accuracies required by specific applications, are needed. DSSE has to be able to include all the different measurement types, provided by traditional and modern measurement devices, which can be available with different frequencies and accuracies. In particular, the phasor measurement units (PMUs) [2,3], which give phasor measurements of both voltages and currents synchronized with respect to a common time reference (the socalled synchrophasors) are becoming increasingly widespread in transmission systems [46] and are expected to be widely used also in distribution systems, along with newgeneration power meters.
For these reasons, innovative and dedicated solutions able to estimate the operating point of the future distribution network are increasingly needed.
DSSE techniques are based on measurement models that link measurements with the state variables to be estimated. They are intended to elaborate the measurements acquired from the field, by means of distributed measurement systems, along with all the a priori information that can be collected about load and generator activity.
In fact, it is impractical and economically infeasible to have a fully monitored distribution network, where each node is equipped with a measurement device connected to the monitoring infrastructure. The DSSE thus reaches observability by relying on the socalled pseudomeasurements [7], which include historical or forecast data on generator production and load consumption.
DSSE approaches proposed in the literature are mainly based on weighted least squares (WLS) algorithms [712] and mainly differ between each other in the chosen state variables and in the way the different measurement types are included. In particular, two main classes of DSSE exist, based on two different choices of the underlying state vector: node voltage state estimators, NVDSSEs (as in [7,13]) and branch current estimators, BCDSSEs (as [812]). BCDSSE is suitably designed to better keep into account the peculiar characteristics of distribution systems, as the radial or weakly meshed topology and the high r/x ratios, and it is usually faster with respect to those based on voltage state (see [14]).
An important topic, which can influence both accuracy and speed of DSSE based on WLS methods, concerns the constraints given by a priori knowledge on network operation and topology. For instance, a priori information exploitable by DSSE includes also the identification of the socalled ‘zero injections’, that is of the nodes that are surely known to have no power consumption or generation. Zero injections are frequent in a distribution grid, in particular, because in a threephase unbalanced context, some nodes may have no loads or generators connected to some of the phases. Besides, the inactivity of a load or generator, if it represents an absolutely sure information, could be also translated in a zero injection constraint. Additional constraints can be also present. As an example, in a branch current formulation, possible meshes have to be duly considered.
Such constraints can be treated in different ways. The simplest method is to consider them as virtual pseudomeasurements and to use a large weight to enforce them. This choice can lead to numerical conditioning problems [15], and thus other options have been advanced in the literature. In [16], for instance, it is proposed to include them using constraints expressed through Lagrange multipliers, while in [17], they are treated as normal measurements with a low weight and the constraints are reimposed between the subsequent iterations of the WLS.
In this paper, a simple way to deal with the equality constraints, wellsuited to BCDSSE (and in particular to the efficient formulation presented in [12]) and based on state vector reduction, is proposed. This approach is compared with other traditional and commonly used techniques to underline the advantages by means of simulation results obtained on a IEEE 123buses threephase test network.
2 Branch current state estimation
State estimation techniques are based on mathematical relations between system state variables and measurements collected from the distributed measurement system. The measurements in a distribution grid can be the traditional ones, as voltage and current magnitudes, real and reactive power flows, and power injections at buses, or the current and voltage synchrophasors provided by PMUs. Usually, distribution networks are only partially monitored. As a consequence, prior information on the loads (the socalled pseudomeasurements) are necessary to perform state estimation. Thus, the forecasts of power injections usually constitute the majority of the measurements available for DSSE.
As aforementioned, different state variables can be considered for the estimation algorithm and in particular node voltages or branch currents. Such variables can be represented either in polar or rectangular coordinates. In this paper, the enhanced branch currentbased estimator proposed in [12] is adopted, as it was shown to be as accurate as those traditionally based on node voltages and more efficient in the practical application of distribution networks. This algorithm will be referred to in the following as BCDSSE.
where z=[z _{1}…z _{ M }]^{T} is the vector of the measurements obtained from real instrumentation in the network and of the chosen pseudomeasurements, h=[h _{1}…h _{ M }]^{T} is the vector of measurement functions, x=[x _{1}…x _{ N }]^{T} is the vector of the chosen state variables and e is the measurement error vector, which is a zeromean random vector with covariance matrix Σ _{ z }. Measurement functions in h can be nonlinear, depending on the type of considered measurements, and are strictly influenced by the topology and the parameters (impedances) of the network.
The state vector x, in the BCDSSE, is given by the branch currents of all the N _{br} network branches, in rectangular coordinates, and the voltage v _{ s } at a reference node, for instance the slack bus. In a threephase framework, x is \(\left [\mathbf {x}_{A}^{\mathrm {T}},\mathbf {x}_{B}^{\mathrm {T}},\mathbf {x}_{C}^{\mathrm {T}}\right ]^{\mathrm {T}}\), with x _{ ϕ } (ϕ=A,B,C) equal to \(\left [v^{r}_{\textit {s}\phi }, v^{x}_{\textit {s}\phi }, i^{r}_{1\phi }\ldots i^{r}_{N_{\text {br}\phi }}, i^{x}_{1\phi }\ldots i^{x}_{N_{\text {br}\phi }}\right ]^{\mathrm {T}}\), under the hypothesis that synchronized measurements are present. This formulation exploits the absolute phase angles provided by PMU measurements, which use the common time reference of the coordinated universal time (UTC). Such time synchronization can be obtained by means of Global Positioning system (GPS) or other synchronization sources (see for instance [18,19]).
Pseudomeasurements, in the model (1), are handled as measurements that are assigned with a higher standard deviation σ to highlight the lower accuracy due to the fact they are not based on real measurements but rather on historical and forecast data.

definition/update of measurements and residuals;

branch current estimation applying a WLS method;

network voltage state computation through a forward sweep calculation.
For each iteration k, in the first step of BCDSSE, power measurements are translated into equivalent phasor current measurements using the node voltages estimated in the previous iteration. This approach allows including power measurements (and above all pseudomeasurements) easily in the estimator, since equivalent current injections are linearly linked to the branch current variables. Using the updated vector z _{ k } of the measurements and the previously estimated state \(\hat {\mathbf {x}}_{k1}\), the measurement residuals \(\mathbf {r}_{k}=\mathbf {z}_{k}\mathbf {h}(\hat {\mathbf {x}}_{k1})\) are computed. In the first iteration, when estimates are not still available, an initialization of the state variables is needed.
where \(\mathbf {H}_{k}=\mathbf {H}(\hat {\mathbf {x}}_{k1})\) is the Jacobian of the measurement functions at iteration k and W is the weighting matrix, equal to the inverse of Σ _{ z }.
Matrix G=H ^{T} W H (subscript k will be dropped in the following, for the sake of simplicity) represents the socalled gain matrix, which has to be inverted or factorized to find the solution of (2). Such matrix and its characteristics play thus a key role in the estimation process.
where Z _{paths} is the matrix that contains, for each row i, the branch impedances z _{ j } that belong to the path that links v _{ i } to the reference bus v _{ s }.
It is worth noting that, in the case of meshed networks, a radial tree of the network can be considered in order to identify the paths linking the slack bus to each node of the grid and to make the forward sweep step possible. The chosen tree can be whatever, since the inclusion of the mesh constraints in the preceding WLS step ensures the final achievement of the same voltage results independently from the particular choice of the path.
The procedure is repeated until a given threshold in estimated state variation is reached.
3 Formulation of the equality constraints
where c(·) is a N _{ c } size vector that represents the constraints to be kept into account in state estimation. It is worth noting that these constraints can be nonlinear, depending on the chosen state variables of the system, but they are linear in the case of rectangular BCDSSE. In a similar way, possible presence of meshes also leads to equality constraints that, in the branch current formulation, have to be suitably considered. The way to handle such constraints in the DSSE can affect the performance of the estimator, in terms of both accuracy and speed. Different methods have been used in the literature. In the following, virtual measurements and Lagrange multipliers are first described, in order to present the equality constraints issue in a selfcontained discussion. Then, a new method, based on state vector reduction, is proposed.
3.1 Virtual measurement method
The zero injection weighting matrix is \(\mathbf {W}_{\textit {zi}}=\sigma _{\textit {zi}}^{2}\mathbf {I}_{N_{c}}\), where \(\mathbf {I}_{N_{c}}\) is an identity matrix, whose size is equal to the number of virtual measurements. The residual vectors are \(\mathbf {r}_{m} = \mathbf {z}_{m}\mathbf {h}_{m}(\hat {\mathbf {x}}) \) and \(\mathbf {r}_{\textit {zi}} = \mathbf {c}(\hat {\mathbf {x}})\).
where the only nonzero elements are those corresponding to the real and imaginary parts of the branch currents to and from the considered node (the sign depends on the direction assumed for the currents).
Possible meshes in the network can be expressed as additional constraints among the branch current state variables, using the Kirchoff voltage law, as m(x)=M x=0, where each couple of rows j and j+1 refers to the Kirchoff voltage law of a mesh, expressed in real and imaginary parts, respectively. Matrix M thus contains, for each mesh, the resistances or inductances of the branches involved in the mesh itself.
where M and W _{ zm } are the Jacobian and the weighting submatrix (with large weights) of the mesh constraints, respectively.
3.2 Lagrangian method
where C is the Jacobian of the zero injection constraints, λ _{ zi } is the N _{ c } vector of the associated Lagrange multipliers and, using the same notation as in (5), \(\mathbf {G}_{m}=\mathbf {H}_{m}^{\mathrm {T}}\mathbf {W}_{m}\mathbf {H}_{m}\) represents the gain matrix referred only to the set of measurements and pseudomeasurements. With this approach, the number of the unknowns increases because of the multipliers, but the sparsity of system (11) is higher with respect to the case of virtual measurements.
where λ _{ zm } is the vector of Lagrange multipliers included to deal with the mesh constraints m(x).
3.3 Proposed method: state vector reduction
where \(\mathbf {I}_{\tilde {N}}\) is a \(\tilde {N} \times \tilde {N}\) identity matrix and Γ _{ zi } is a \(N_{c} \times \tilde {N}\) sparse matrix of ±1 elements linking the eliminated variables x _{ zi } to the remaining ones.
and the residual vector is computed as \(\mathbf {r}_{m}=\mathbf {z}_{m}\mathbf {h}\left (\left [ \tilde {\mathbf {x}}^{\mathrm {T}}, \tilde {\mathbf {x}}^{\mathrm {T}} \boldsymbol {\Gamma }_{\textit {zi}}^{\mathrm {T}} \right ]^{\mathrm {T}}\right)\).
where Ω identifies the set of branches in the mesh, and Z _{ j } and i _{ j } are the impedance and the current of the jth branch, respectively. It is worth noting that, in a threephase framework, Z is a 3 × 3 impedance matrix, which includes also the mutual impedance terms, while i is the vector of the phase currents.
where k is the index of the branch whose currents will be eliminated from the state variables.
where Γ _{ zm } is a \(N_{m} \times \tilde {N}\) matrix linking the N _{ m } removed current variables x _{ zm } to the remaining ones. Thus, in this case, the resulting state vector has a reduced length equal to \(\tilde {N} = N  N_{c}  N_{m}\). Once the transformation matrix linking the starting state vector to the reduced one is defined, the same considerations involved in (16) and (17) hold for the execution of the estimation algorithm.
The performed transformation leads to a lower sparsity of the system, reflecting the fact that each eliminated variable is expressed in terms of more remaining variables. However, since distribution grids usually have a large number of zero injections, the transformation also allows a significant reduction of the dimensions of the equation system to be solved. It is worth underlining that Γ _{ zi } and Γ _{ zm } are constant matrices that can be built a priori knowing the operation and topology of the network; thus, there is no need to compute them at each run of the DSSE.
It is important highlighting that such approach can be applied in an efficient way only because of the chosen state variables: a similar logic does not apply, with the same simplicity, to traditional estimators based on polar node voltages.
The proposed approach allows an efficient implementation even for the bad data detection and identification functions. In fact, the same techniques traditionally adopted in WLS estimators, based on the computation of the normalized residuals, can be conveniently implemented here, thanks to the reduced sizes of the Jacobian and gain matrices that are involved in the computation of the residual covariance matrix (see [15] for further details). Identified bad measurement data are removed, and the BCDSSE estimation steps previously described are performed again on the reduced measurement set. It is worth noting that, in order to avoid the computation of the residual covariance matrix when bad data are not present, the bad data detection can be also implemented by using the wellknown χ ^{2} test [15].
4 Tests and results
4.1 Test assumptions and metrics

number of Monte Carlo trials: N _{MC}=25,000;

pseudomeasurements available for all the loads of the network and characterized by normally distributed uncertainty with maximum deviation 3σ=50% of the nominal value;

PMU measurements characterized by uncertainty with uniform distribution and variance σ ^{2} equal to one third of the squared accuracy value; in particular, an accuracy equal to 0.7% and 0.7 crad (i.e. 0.7×10^{−2} rad) is used for magnitude and phaseangle measurements, respectively, in order to simulate the accuracy limits specified in the synchrophasor standard [3].
Results of the tests have been analysed to assess the performance of the proposed approach in comparison to virtual measurements and Lagrange multiplier methods. In particular, accuracy, computational properties and efficiency of the different approaches have been analysed in order to have an overall performance evaluation.
4.1.1 Estimation accuracy
where \(\hat {y}_{i}\) is the estimation of y at the ith Monte Carlo trial and y _{true} is the true value of the quantity. In this paper, as overall index for the whole network, the mean RMSE, obtained averaging the RMSEs of all the nodes or branches (depending on the considered electrical quantity), has been used.
4.1.2 Numerical properties
An important issue for the estimator design is the numerical conditioning of the equation system. In fact, as described in [15], due to illconditioning, small errors in the different entries of the equation system may be translated in large errors for the solution vector. As a consequence, the accuracy and the convergence properties of the algorithm can be significantly affected and numerical instabilities can appear.
where G _{ tot } is a coefficient matrix, Δ x _{ tot } is the total vector of the unknowns (note that for the Lagrangian method also the Lagrange multipliers are included in this vector) and u is a vector resulting from the measurement residuals.
where ∥·∥ is the 2norm of the matrix.
Other interesting properties from the computational point of view are the density and the size of the coefficient matrix. A low density (defined as the ratio between nonzero terms and total number of elements in the matrix) implies a large number of zero elements in the matrix and thus the possibility to use sparse matrix techniques for the calculations. The size, instead, is obviously associated to the number of unknowns and represents the dimension of the equation system to be solved. In the following, all these parameters will be taken into account to discuss the obtained results.
4.1.3 Computational efficiency
The efficiency of the estimator is a crucial factor for the realtime management and control of future distribution systems. For this reason, in the following discussion, the average execution times of the different methods (obtained averaging among the N _{MC} Monte Carlo trials) will be compared. Moreover, since the execution times strictly depend on the number of iterations needed for the algorithm convergence, also the average number of iterations of the different approaches will be evaluated. Tests have been performed under Matlab environment and run on a 2.4GHz quadcore processor with 8GB RAM.
4.2 Test results
In this section, the performance of the approaches virtual measurement method (VM), Lagrangian method (LM) and state vector reduction (SVR) will be analysed and discussed.
First, a test has been carried out considering a possible realistic measurement configuration for the network. In particular, three measurement points have been supposed to be available in nodes 150 (primary station), 18 and 67. Each measurement point is composed of a voltage synchrophasor measurement on the node and of current synchrophasor measurements on all the branches converging to that node.
Variation of P _{0inj} [kW] and Q _{0inj} [kvar] in VM
Virtual measurement weight  

10 ^{9}  10 ^{10}  10 ^{11}  10 ^{12}  
P _{0inj}  1.9·10^{−1}  1.9·10^{−2}  1.9·10^{−3}  1.9·10^{−4} 
Q _{0inj}  3.0·10^{−1}  3.0·10^{−2}  3.0·10^{−3}  3.0·10^{−4} 
Average iteration numbers and execution times
Method  Iteration number  Execution time [ms] 

VM  3.75  18.2 
LM  3.75  21.5 
SVR  3.64  12.8 
Numerical properties of the coefficient matrix
Method  Coefficient matrix  Coefficient matrix  Condition 

density (%)  size  number  
VM ^{a}  3.26  454 × 454  3.02 · 10^{6} 
LM  1.48  692 × 692  4.63 · 10^{4} 
SVR  21.17  216 × 216  1.02 · 10^{4} 
An additional advantage guaranteed by the use of SVR concerns the condition number of the coefficient matrix. As it can be observed in Table 3, SVR gives the lowest conditioning. It is worth recalling that, in general, VM suffers possible illconditioning problems because of the use of very large weights to enforce the constrained measurements. As a confirmation of such an impact, the condition number obtained changing the weights to 10^{8} and 10^{12} has been checked: in both the cases, a consequent variation of the condition number (3.74·10^{4} and 3.02·10^{8}, respectively) has been found.
Several tests have been performed also to verify the proper operation of the aforementioned bad data detection and identification function when using the proposed approach. For instance, tests have been performed by adding intentional errors (of 5%, 10% and 20%) to the voltage magnitude measurement at node 18 (on the first phase). In such a scenario, all the analysed methods allow the detection of the bad data, through the χ ^{2} test, and the proper identification of the erroneous voltage, by means of the largest normalized residual technique. It is important to note that, since the presence of the bad data implies the computation of the residual covariances and the need to repeat the estimation process, the aforementioned improvements brought by the SVR method on the execution times further increase. It is also worth noting that, in some cases, depending on measurement configuration, when considering the bad data on other measurements (for example, on PMU currents), the bad data identification function could be unable to properly identify the erroneous measurement. However, as known from the literature, this is a general issue that can occur due to the low redundancy of the measurements, and it does not depend on the particular method used to handle the equality constraints. In fact, several tests (not reported here for the sake of brevity) have been performed changing the corrupted measurement, and as expected, test results prove that all the approaches exhibit exactly the same behaviour, with the same identification results, in all the different scenarios. Since possible identification problems are generally related to the measurement system deployed in the distribution grid, a deeper analysis on this issue is out of the scope of this paper.
4.2.1 Impact of the measurement configuration
Further tests have been performed to assess the performance of the methods with different measurement types and configurations.
First of all, the general validity of the previous considerations has been tested using different measurement devices. To this purpose, voltage and current phasor measurements have been replaced by voltage magnitude and active and reactive power measurements (with accuracy equal to 1% and 3%, respectively). It is worth noting that in this case, since synchronized measurements are not available, the estimator model has to be suitably adapted to take into account the absence of an absolute phase angle reference (see [12] for more details).
Average iteration numbers and execution times with traditional measurements
Method  Iteration number  Execution time [ms] 

VM  5.30  23.3 
LM  5.30  28.4 
SVR  5.28  17.8 
Numerical properties of the coefficient matrix with traditional measurements
Method  Coefficient matrix  Coefficient matrix  Condition 

density (%)  size  number  
VM ^{a}  3.09  451 × 451  5.75 · 10^{6} 
LM  1.40  689 × 689  2.26 · 10^{4} 
SVR  20.56  213 × 213  5.41 · 10^{3} 
As for the measurement configuration, the attention has been mainly focused on the impact of additional voltage measurements, since they can significantly affect numerical properties and efficiency of the branch currentbased formulation of DSSE. In fact, voltage measurements lead to nonzero terms in the Jacobian H _{ m } corresponding to all the derivatives with respect to the branch currents included in the path between the bus used as reference in the state vector and the measured node (for details, see [12]). This, in turn, causes a lower sparsity of the coefficient matrix in (23), thus affecting the efficiency of the equation system solution.
Numerical properties of the coefficient matrix with two additional voltage measurements
Method  Coefficient matrix  Coefficient matrix 

density (%)  size  
VM  5.95  454 × 454 
LM  2.68  692 × 692 
SVR  24.52  216 × 216 
Average iteration numbers and execution times with two additional voltage measurements
Method  Iteration number  Execution time [ms] 

VM  3.62  22.3 
LM  3.62  27.7 
SVR  3.54  14.0 
As expected, the presence of additional voltage measurements affects the coefficient matrix leading to higher densities: this is the main reason for the increased execution times shown in Table 7 (with respect to the results in Table 2). Moreover, it is worth highlighting that the increased density brings different impacts on the different methods. In fact, since the solution of the equation system in SVR is managed without using sparse matrix techniques, the impact on this approach is smaller with respect to the other methods. This is confirmed by the enhancements obtained on the execution times that, in this scenario, rise up to almost 37% and 50% with respect to VM and LM, respectively.
Coefficient matrix density and execution times with six additional voltage measurements
Method  Coefficient matrix  Execution 

density (%)  time [ms]  
VM  10.36  28.1 
LM  4.64  39.0 
SVR  31.49  15.2 
As for the accuracy and the numerical conditioning of the analysed methods, considerations similar to those made for the first test can be derived also in these cases: all the methods provide very similar accuracy results, and SVR shows the best conditioning properties.
4.2.2 Impact of the size of the network
One of the main issues involved in the handling of distribution systems is the large size of these networks. This aspect is particularly critical from the standpoint of the execution times, since it implies a significant increase of the size of the equation system to be solved. In such a situation, possible drawbacks can arise for the SVR method due to the fillins resulting from the elimination of the state variables.
To assess the impact of such issue, additional tests have been performed with networks having a larger number of nodes. In particular, in order to simulate different sizes of the grid, new networks with an increasing number of feeders have been built, where each feeder replicates the topology of the previously considered 123bus network. For all the tests, the measurement configuration is supposed to be composed of a measurement point in the substation and, for each feeder, of two measurement points in the buses corresponding to the nodes 18 and 67 of the original 123bus network (see Figure 1). In order to obtain different loading conditions in the feeders, power consumptions of the loads have been modified adding random variations.
Average iteration numbers and execution times with tenfeeder network
Method  Iteration number  Execution time [ms] 

VM  4.00  166.7 
LM  4.00  206.0 
SVR  4.00  129.2 
4.2.3 Impact of the mesh constraints
In this section, the performance of the proposed method are tested for the case of weakly meshed networks, referring to the original 123bus grid. To this purpose, the presence of a branch between nodes 151 and 300 and between nodes 54 and 94 of the benchmark network has been supposed in order to create two meshes. Simulations have been carried out considering the base monitoring configuration composed of the three measurement points in nodes 150, 18 and 67.
Numerical properties of the coefficient matrix with weakly meshed network
Method  Coefficient matrix  Coefficient matrix 

density (%)  size  
VM  11.42  466 × 466 
LM  1.78  716 × 716 
SVR  34.43  216 × 216 
It is possible to observe that the density of the coefficient matrix in VM and SVR is significantly affected by the presence of the meshes. In fact, as clear from (18), the mesh constraint involves all the threephase currents of the branches belonging to the mesh. Thus, the matrix multiplications needed to create the coefficient matrix G _{ tot } and involving the Jacobian matrix M for VM (see Equation 10) and the transformation matrix Γ _{ z m } for SVR (see Equations 20, 16 and 17) lead to a significant increase of the nonzero elements. At the same time, the presence of the additional branch currents (due to the meshes) lead to an increase of the dimension of the equation system for VM, while no change appears for SVR since such currents are expressed in terms of the reduced state vector. In the case of LM, instead, the explicit expression of the mesh constraints, together with the growth of the dimensions of the equation system (due to both the additional branch currents and the additional constraints) allows to keep the coefficient matrix very sparse.
Average iteration numbers and execution times with weakly meshed network
Method  Iteration  Execution 

number  time [ms]  
VM  3.89  33.8 
LM  3.89  32.8 
SVR  3.82  18.2 
The same considerations already made for the tests with the radial version of the network hold for the accuracy performance and the conditioning properties.
5 Conclusions
In this paper, an efficient way to include the equality constraints in a branch currentbased state estimator is presented. The method exploits the use of rectangular branch currents as state variables of the system to perform a simple elimination of one of the currents involved in a constraint, expressing it as a linear function of the remaining ones. The method not only is particularly efficient in the management of zero injections but also allows the treatment of mesh constraints. Performed tests prove the goodness of the proposed technique and in particular its capability to significantly improve the computational efficiency of the estimator (with respect to other traditionally used methods). Moreover, full fulfilment of the constraints is guaranteed, and additional benefits can be achieved for the numerical conditioning of the system.
Declarations
Authors’ Affiliations
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