WO2016050309A1

WO2016050309A1 - Method of estimating a system value

Info

Publication number: WO2016050309A1
Application number: PCT/EP2014/071161
Authority: WO
Inventors: Izudin Dzafic
Original assignee: Siemens Aktiengesellschaft
Priority date: 2014-10-02
Filing date: 2014-10-02
Publication date: 2016-04-07
Also published as: US20180231596A1; US10627437B2; EP3186681A1

Abstract

An embodiment of the invention relates to a method of estimating a system value which indicates a state of an electrical distribution system, the method comprising the steps of: - measuring one or more electrical quantities at a plurality of system nodes and providing measurement values, and - generating the system value based on the measurement values as well as on weighting coefficients that define or reflect the timely change rate of the measurement values and/or the transmission rate of the measurement values.

Description

Method of estimating a system value The invention relates to a method of estimating a system value which indicates a state of an electrical distribution system, and a substation comprising a calculating unit configured to estimate a system value. Background

The notion of state estimation (SE) for transmission systems can be traced back to the seventies [1] . Some twenty years later, SE algorithms specifically tailored to distribution systems were introduced [2], [3] . In practice, however, it has not been until very recently that SE tools for distribu^¬ tion feeders have been comprehensively considered [4]-[6]. Smart grid developments are progressively bringing more and more information to Distribution Management Systems (DMS) , allowing applications that were long ago conceptually mature but still waiting for the required infrastructure to be de^¬ ployed at the distribution level [7], [8] . Eventually, the massively distributed nature of medium-voltage and low- voltage subsystems, and the resulting communication bottle^¬ necks, will force utilities to consider some kind of hierar- chical organization in today's fully centralized DMS [9] . In^¬ deed, only if raw data are processed in a local manner [10] will it be possible for new and ubiquitous sources of infor^¬ mation, such as smart meters and the associated concentra^¬ tors, to be scanned at rates which are fast enough for real- time network operation. Until this partly decentralized envi^¬ ronment arrives, DMS operators can only expect to have once- a-day or few-times-a-day values of energy consumed by custom^¬ ers connected to the distribution system [11] . This has moti- vated the development of heuristic methods combining load flow calculations [13], [14], machine learning functions [12] or pattern-based load allocation [15] with ad hoc SE tech^¬ niques .

What these hybrid schemes generally have in common is a pre^¬ processing phase in which delayed smart meter data or daily load patterns are somehow exploited to generate pseudomeas- urements for the SE phase. In the foreseeable future, if not in the near term, smart meter data will be collected and pre- processed by substation-level management systems, at much faster scan rates than those achievable if every piece of in^¬ formation had to be gathered at the centralized DMS. Whereas a DMS is in charge of an entire system, typically serving several million customers, a 60-MW primary substation may serve three orders of magnitude less customers, whose smart meter data are in turn concentrated at less than a hundred intermediate points (generally secondary substations serving the LV subsystem) . Having these data collected at the primary substation, at rates ranging from 5 to 20 times an hour, is a feasible choice even with today's bandwidths and technology.

In this context, the substation-level SE tool will have to deal with two heterogeneous types of information, as ex- plained in more detail further below:

1) regular SCADA measurements, and eventually those coming from new smart grid sensors, captured every few seconds;

2) smart meter (or smart meter concentrator) readings and distributed generation production, updated every few minutes.

This naturally leads to an information processing model in two time scales. Even though two-time-scale problems have long been known and exploited in several engineering fields (see for instance [16] -[19]), including SE of chemical or biological processes [20], to the inventor's knowledge such a notion has not been explored so far in power system SE . Objective of the present invention

An objective of the present invention is to provide a method of estimating a system value which indicates a state of an electrical distribution system, where the estimation needs to be based on measurement values with differing properties.

A further objective of the present invention is to provide a substation capable of estimating a system value which indicates a state of an electrical distribution system, where the estimation needs to be based on measurement values with dif- fering properties.

Brief summary of the invention

An embodiment of the invention relates to a method of esti^¬ mating a system value which indicates a state of an electri- cal distribution system, the method comprising the steps of: measuring one or more electrical quantities at a plural^¬ ity of system nodes and providing measurement values, and generating the system value based on the measurement val^¬ ues as well as on weighting coefficients that define or reflect the timely change rate of the measurement values and/or the transmission rate of the measurement values.

An advantage of this embodiment of the invention is that a fast and reliable state estimation can be carried out even if the measurement values belong to different time scales. For instance, the state estimation can be based on a combination of measurement values that include pseudomeasurement values as well as redundant measurement values, which are far more accurate .

The method preferably further comprises the steps of dis- criminating the measurement nodes into groups by the timely change rate of their measurement values and/or by their transmission rate of the measurement values. A weighting co^¬ efficient is preferably assigned to each group of said groups of measurement nodes depending on the timely change rate and/or the transmission rate of the measurement values of the respective group. The step of estimating the system value is preferably based on the weighted measurement values and the weighting coefficients. Moreover the method may be used in distribution systems com^¬ prising a distribution feeder. The system value may then be estimated to define the electrical load connected to said distribution feeder. Further, the system value may be estimated using a state es^¬ timation algorithm.

Moreover the system nodes may be discriminated into two groups, namely a first group comprising frequently available measurement values and a second group comprising less fre^¬ quently available measurement values.

Furthermore, the system value may be estimated through itera- tively solving the normal equation

(H_p ^TW_pH_p+H_m ^TW_mH Ax = H_p ^TW_p[z_p-h_p(x)]+H_m ^TW_m[z_m-h_m(x)] wherein H_m designates a measurement matrix comprising the measure^¬ ment values of the first group,

H_m ^T designates the transposed matrix of H_m ,

H_p designates a measurement matrix comprising the measure- ment values of the second group,

H_p designates the transposed matrix of H _p ,

W_m designates a weighting matrix comprising the weighting coefficient or coefficients assigned to the first group, W designates a weighting matrix comprising the weighting coefficient or coefficients assigned to the second group,

Δχ designates an increment vector of calculated values, z_p designates a vector of measurement values assigned to the second group,

(x) designates a vector of functions that defines measure^¬ ments of the second group,

z_m designates a vector of measurement values assigned to the first group, and

h_m(x) designates a vector of functions that defines measure- ments of the first group.

The Cholesky factorization may be applied to the normal equa^¬ tions . The weighting coefficients preferably define or at least re^¬ flect the information uncertainty of the corresponding meas^¬ urement values.

Moreover the weighting coefficients preferably define or at least reflect the hierarchical position of the system nodes providing the measurement values. The method as explained in an exemplary fashion above may be used in a distribution system comprising a medium voltage grid and a low voltage grid.

In the latter case, the measurement nodes may be discrimi^¬ nated into a first group and a second group. The first group may comprise the measurement values of the low voltage grid and the second group may comprise the measurement values of the medium voltage grid. One or more weighting coefficients may be assigned to the first group of measurement nodes, and one or more weighting coefficients may be assigned to the second group of measurement nodes. Then, the system value may be estimated based on the measurement values of the first and second group and the weighting coefficients of the first and second group.

The medium voltage grid and the low voltage grid are prefera^¬ bly separated by substation. In this case, said step of esti- mating the system value is preferably carried out in the sub^¬ station .

A further embodiment of the present invention relates to a substation having a calculating unit configured to estimate a system value which indicates a state of an electrical distri^¬ bution system, the calculating unit further being configured to carry out the steps of:

- receiving measurement values indicating one or more elec^¬ trical quantities from a plurality of system nodes, and - generating the system value based on the measurement val^¬ ues as well as on weighting coefficients that define or reflect the timely change rate of the measurement values and/or the transmission rate of the measurement values. An advantage of this substation is that it may carry out a fast and reliable state estimation even if the measurement values belong to different time scales. For instance, the substation can handle a combination of measurement values that include pseudomeasurement values as well as redundant measurement values.

Brief description of the drawings

In order that the manner in which the above-recited and other advantages of the invention are obtained will be readily un^¬ derstood, a more particular description of the invention briefly described above will be rendered by reference to spe^¬ cific embodiments thereof which are illustrated in the ap- pended drawings. Understanding that these drawings depict only typical embodiments of the invention and are therefore not to be considered to be limiting of its scope, the inven^¬ tion will be described and explained with additional speci^¬ ficity and detail by the use of the accompanying drawings in which in an exemplary fashion

Figure 1 shows fast measurement snapshots (designated by z_m) and slow pseudomeasurement snapshots (desig^¬ nated by z_p) of two electrical quantities, wherein the graphs designate the actual evolu^¬ tion and t designates the time,

Figure 2 shows generic scenarios illustrating an Ampere- measured branch with two pseudo-measured buses downstream, wherein in case a) each bus may represent the aggregated load of two main laterals downstream bifurcation, both of similar size in case b) bus 1 can be a single load while bus 2 may represent the aggregated load of the re^¬ maining buses downstream, shows an example with homogeneous load trends, wherein P designates the active power injection (p.u.), t designates the time in minutes, the upper set of lines designate bus 1, the lower set of lines designate bus 2, reference numeral 31 designates exact measurements, reference nu^¬ meral 32 designates pseudomeasurements, refer^¬ ence numeral 33 designates 8 I measurements, shows an example with opposite load trends, wherein P designates the active power injection (p.u.), t designates the time in minutes, the upper set of lines designate bus 1, the lower set of lines designate bus 2, reference numeral 41 designates exact measurements, reference nu^¬ meral 42 designates pseudomeasurements, refer^¬ ence numeral 43 designates 8 I measurements, shows the generation of pseudomeasurement inter^¬ mediate values in a stepwise fashion (diagram at the top) , in an extrapolation fashion (diagram in the middle) and in an interpolation fashion (diagram at the bottom) , shows a 100-bus, 15-kV test distribution network made up of two feeders and 38 secondary trans^¬ formers, delivering energy to a mix of residential, industrial and commercial loads, compris^¬ ing heterogeneous load patterns, Figure 7 shows examples of 24-hour heterogeneous load patterns, wherein P designates the active power in kW, t designates the time in hours, Figure 8 shows branch active power flows in decreasing order, wherein P designates the active power in kW, t designates the time in hours,

Fig. 9a) -c) shows an evolution of active power estimates at a typical load bus in scenario A, when z_p is kept constant between snapshots (diagrams at the top) , extrapolation is used (diagrams in the middle) and interpolation is used (diagrams at the bottom) , wherein P designates the active power in kW, t designates the time in hours, reference numeral 91 designates exact measure^¬ ments, reference numeral 92 designates pseu- domeasurements , reference numeral 93 designates 8 I measurements,

Figure 10 shows an evolution of active power flow errors for the 5 branches with the largest load, wherein P designates the active power error in kW, t designates the time in hours, reference numeral 101 designates pseudomeasurements , ref^¬ erence numeral 102 designates 8 I measurements, and

Figure 11 shows an exemplary embodiment of a substation according to the present invention.

Detailed description of the preferred embodiment The preferred embodiment of the present invention will be best understood by reference to the drawings, wherein identi^¬ cal or comparable parts are designated by the same reference signs throughout.

It will be readily understood that the present invention, as generally described and illustrated in the figures herein, could vary in a wide range. Thus, the following more detailed description of the exemplary embodiments of the present in- vention, as represented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of presently preferred embodiments of the in^¬ vention . Sources of information in smart distribution systems

Unlike transmission and subtransmission systems, where real^¬ time telemetry provides sufficient redundancy to assure net^¬ work observability, Medium Voltage (MV) distribution feeders have so far lacked the required infrastructure (sensors and telecommunication) allowing the operating point to be accurately determined. In the upcoming smart grid paradigm, though, distribution systems will have to cope with a heterogeneous set of information sources, most of them not yet available at the DMS, which can be roughly classified into the following categories:

1. RTU measurements captured at High Voltage (HV) -MV sub^¬ stations, collected by the SCADA system of the DMS at rates ranging from few seconds to about a minute (in general, much lower refreshing rates than those employed at transmission-level SCADAs [22]) . As far as radial feeders are concerned, such measurements typi^¬ cally reduce to the MV busbar voltage magnitude and head line currents (assuming passive loads with an av^¬ erage power factor, this allows the total P&{Q} deliv^¬ ered by the feeder to be computed) . So far, this is es^¬ sentially the only telemetered information at the MV level for a majority of utilities which, unless a fault occurs, can only have a very crude idea of what is go^¬ ing on downstream with the help of load allocation techniques (item 4) .

Higher reliability standards (i.e. lower SAIFI/SAIDI indexes) are forcing distribution utilities to deploy more and more feeder automation devices, including remotely-operated intermediate switching points for fault management. Once the required communication channel is available, such points can be converted into true RTUs with very little extra investment. In fact, most ven^¬ dors currently offer this product in their catalogues. The information provided by these additional RTUs can be useful for both fault location and state estimation purposes .

Distributed generation is already a reality and will increasingly spread in many radial feeders worldwide. Depending on the specific regulation and rated power, the production of DGs is required to be monitored at different rates, ranging from day-ahead hourly fore^¬ casting to real telemetry periodically submitted to the DMS .

Distribution utilities have customarily kept a more or less elaborated data base of historic load pat^¬ terns/profiles. This information originates in several sources, including load forecasting, load allocation techniques in combination with feeder head measurements, characteristic power factor values of aggregated loads and systematic metering campaigns performed at specific points. The feeder-level state estimator can benefit from these not very precise values of P and Q, which can be used as pseudo-measurements to extend the observable area.

The latest and eventually most important addition to the list of information sources at the feeder level comes from the AMR/AMI infrastructure (typically smart meter concentrators) , provided the right communication ^xbridge' is built between AMI and DMS subsystems. Nowadays this information is col^¬ lected once a day in many systems but, depending on bandwidth availability, snapshot latencies of up to 15 minutes have been reported. Notice that not all of the above data will necessarily reach the DMS, but may remain at an intermediate place much closer to the points where they are captured from the field. In the hierarchical control system architecture envisioned elsewhere [procieee, smartsub, Bose] , the right place where the raw in- formation should be collected and processed is the distribu^¬ tion substation, since there are currently no technical bar^¬ riers for a state estimator to be implemented in this envi^¬ ronment. For the purposes of this work, all sources of infor^¬ mation summarized above, to which the feeder-level state es- timator can resort, will be grouped in two broad classes of different nature, each with different accuracy and latency:

Telemetered data provided by RTUs (items 1, 2 and, in some cases, 3) . This comprises quite accurate snapshots captured with latencies ranging from few seconds to about a minute. The set of measurements is insufficient in any case to assure network observability. Pseudomeasurements (items 3, 4 and 5) . Updated at inter^¬ vals ranging from 15 minutes to 24 hours, these bus-level data are barely critical for observability purposes.

Clearly, in order to achieve a minimum redundancy level, both information types should be properly combined, which leads to the particular SE model described in the sequel.

State estimation with two time scales

Let z_m and z_p denote the fast-rate measurement and slow-rate pseudomeasurement vectors, respectively. As suggested by Fig^¬ ure 1, z_m snapshots are updated at regular intervals of width T_m, while Zp is refreshed at much wider intervals of period T_p= n T_m. Between two consecutive snapshots of z_p, n snapshots of z_m are captured (n= 4 in the Figure 1) .

At a given time instant, t_k, the available information is composed of the current snapshot z_mik and the past pseudomeas^¬ urement value z_Pfj . Therefore, the faster the load increases or decreases the quicker and more obsolete z_{Pi j} becomes. When the sign of the slope does not change between tj and tj_+n, the worst condition in terms of pseudomeasurement obsolescence arises for tj_+n-i, just before z_p is updated again. Dropping for simplicity the discrete-time indices, the resulting meas^¬ urement model is:

(1)

where h (·) and h_m(») represent the respective measurement functions and ε_ρ and s_m the associated errors. Notice that the variance of ε_ρ is generally much higher than that of s_m . The

WLS SE solution is obtained by iteratively solving the normal equations : (H_p ^TW_pH_p+H_m ^TW_mH Ax = H_p ^TW_p[z_p-h_p(x)]+H_m ^TW_m[z_m-h_m(x)] (2) where the weighting coefficients should reflect whenever pos- sible the information uncertainty:

W_m ^l=cov(s

The special structure of the normal equations (2) can be ex- ploited to save computational effort. Considering the rela^¬ tively few number of measurements in vector z_m, a major source of computational saving arises when the Cholesky fac^¬ torization of the gain matrix is not repeated at each SE run, but only when the set z_p is updated. Approximating the gain matrix in this fashion may slightly increase the number of iterations, particularly when loads evolve quickly, but will not affect the solution as long as the right-hand side of (2) is exactly computed. Needless to say, using the solution of the previous run as starting point, rather than the customary flat start profile, is a convenient strategy to save itera^¬ tions .

Limitations arising from the use of reduced redundancy levels This section is devoted to qualitatively analyzing the limi- tations of the two-scale state estimator (TSSE) in a context characterized by extremely low redundancy levels. Intui^¬ tively, one expects that adding a few branch Ampere measure^¬ ments, scattered throughout the feeder, to the set of bus pseudomeasurements , will always improve the estimate of rele- vant quantities (bus voltage magnitudes and branch power flows) , which is true so long as the feeder is taken as a whole. However, depending on whether or not all loads downstream have coincident evolution patterns, branch current measurements may or may not be helpful to improve the esti^¬ mates of certain individual quantities when pseudomeasure- ments are not duly updated. As explained below, this limitation stems from the combina^¬ tion of two adverse factors: 1) low redundancy of RTU meas^¬ urements, clearly insufficient to render the network observable; 2) gradual obsolescence of barely critical pseudomeas- urements as time elapses, of particular relevance in periods when bus injections change at a fast rate.

In order to illustrate the analysis it is sufficient to con^¬ sider the two simplified radial feeders shown in Figure 2. In both cases, in addition to the head bus voltage, there is an Ampere measurement at the branch which is closer to the feeder head, while only P&Q pseudomeasurements are available at the two buses downstream. Notice that, in spite of their simplicity, such reduced feeders can be representative, in equivalent form, of different realistic situations, by simply playing with the relative sizes of Pi^~Qi and P2-Q2- For in^¬ stance, in case (a) each bus may represent the aggregated load of two main laterals downstream a bifurcation, both of similar size. In case (b) , bus 1 can be a single load while bus 2 may represent the aggregated load of the remaining buses downstream, etc. Instead of using an exact SE model, the same qualitative conclusions can be reached, with much less elaborated algebra, by adopting a lossless model with flat voltage profile in which active and reactive power in^¬ jections constitute the state variables.

Given the latest pseudomeasurement values P"—Q™ and P™—Q™ and the most recent current measurement, I_m, which is assumed to be much more accurate than power pseudomeasurements, the WLS estimates for PI, P2, Ql and Q2 can be analytically ob^¬ tained. As shown in the Appendix, the active power estimates are :

+ AP_i

(3)

+ AP₂ where

(Oi and a>2 represent the weights of P™ and P₂ ^m respectively and K_m is the ratio,

Notice that Km > 1 if the total load downstream has increased since slow-rate pseudomeasurements were updated, which is re^¬ flected in higher values of more recent I_m snapshots, while Km < 1 when the the total load has decreased. Similar expres^¬ sions are obtained for reactive powers by simply replacing P with Q (for this reason, only active powers will be paid at^¬ tention to in the sequel) .

In low-redundancy scenarios, like those considered in this work, the influence of the weighting coefficients (Oi and (O2 in the WLS estimates is crucial. In practice this poses a ma^¬ jor problem, since knowing at each time instant the real un^¬ certainty of pseudomeasurements is far from trivial. In this regard, it is worth considering the following two cases: Same weights adopted ( coi = ω₂ ) . This would lead to:

which means that both P™ and ₂ ^m will be corrected by the same amount to yield the respective estimates. In absence of any other information this will be acceptable provided both loads are of similar size. However, assume for instance that PI >> P2. Then, according to (6), even small changes in PI might lead to relatively high deviations in the smaller load, P2, irrespective of whether this load has really changed or not.

Weights inversely proportional to the pseudomeasurement ( a>i = 1 / P™ ) . In this case, it is easy to see that:

which means that each power will be corrected in proportion to its size. In other words, for low-redundancy sce- narios, like the simplified ones represented in Figure 2, the total load variation detected by an Ampere measurement at a given feeder section is prorated among the loads located downstream in proportion to their respective sizes.

Note that, irrespective of the weights adopted, the signs of both ΔΡι and ΔΡ₂ will be the same, according to (4), as determined by the value of Km. If the total load increases (decreases) then Km > 1 (Km < 1) and both estimates, P_l and P₂ , will be higher (lower) than the outdated pseudomeasurements , P™ and ₂ ^m . Indeed, this is an expected result for the low redundancy considered, since there is no way to know whether both loads have actually increased (decreased) or not (it is worth stressing that replacing Im by power flow measurements is not helpful in this regard) .

In feeder sections where all transformer loads downstream of an Ampere measurement evolve in an homogeneous way, which happens when most customers have similar patterns, this may not be a real limitation. However, in feeders comprising a mix of customers (residential, industrial, municipal, etc.) some transformer loads may be increasing while others are si^¬ multaneously decreasing, and combining few Ampere measure^¬ ments with critical pseudomeasurements can be counterproduc^¬ tive, particularly if sudden load changes take place. Figure 3 illustrates a case in which both PI and P2 decrease during the next 45 minutes. The power estimates obtained when Im is incorporated (in this particular example wl = w2) approach the actual load evolution better than in absence of Im (step^¬ wise solid grey line) .

The case in which both loads evolve in opposite direction is shown in Figure 4. In this case, note that, since the total aggregated load is decreasing, the addition of Im is benefi^¬ cial for the load which is actually decreasing, P2, but det- rimental to PI, whose increasing trend remains unnoticed when both loads are sensed upstream. From the point of view of the ^xlosing' load, PI, it would be preferable not to include Im in the model, since the persistent pseudomeasurement value (stepwise solid grey line) would better approximate the ac- tual load evolution.

This somewhat counterintuitive conclusion (i.e., adding an accurate measurement can be counterproductive in certain cases) will be reaffirmed by the results presented below. Needless to mention, such limitations vanish when sufficient redundancy levels are achieved. Solution enhancements

So far it has been implicitly assumed that pseudomeasurements are only updated every n measurement snapshots. In other words, at time instant tk, the TSSE combines the current measurement snapshot, z_m,_k, with a pseudomeasurement value, Zp,_k, given by the latest available pseudomeasurement, z_p,k = z_p,j k=j , j+n-1

This implies that pseudomeasurements are assumed to evolve in a stepwise fashion, as suggested by the uppermost diagram of Figure 5.

Depending on whether future pseudomeasurement values are available in advance or not, other strategies to generate in^¬ termediate pseudomeasurement values are possible, as dis- cussed below.

A. Pseudomeasurement extrapolations

The accuracy of the estimates can be frequently improved if, instead of keeping the components of z_p constant since the last update (stepwise evolution) , their values are obtained by linear extrapolation from the last two samples (higher- order extrapolation is also possible, but the results are usually worse owing to longer transient periods) . Mathemati^¬ cally,

= z P,J (^ZP -^ZP,J- k = j, ... ,j + n - 1 The middle diagram of Figure 5 shows two consecutive inter^¬ vals in which linear extrapolation behaves differently. At interval (a) the linearly extrapolated value approximates the actual evolution of the pseudo-measured quantity better than the latest available value, zl. At interval (b) , however, ow^¬ ing to the sudden change of slope, linearly extrapolating the pseudomeasurement is worse than just keeping the previous value, z2. Linear extrapolation is helpful in a majority of cases to improve the estimation provided otherwise by the va- nilla stepwise evolution. This happens when the time con^¬ stants characterizing the load evolution are large enough compared with the refreshing rate of pseudomeasurements .

B. Pseudomeasurement interpolations

Obtaining intermediate pseudomeasurements by extrapolation is the only choice when future information about the monitored quantity is missing. This is the case, for instance, of some distributed generators, usually burning fossil fuels, whose energy production is not forecasted but rather measured and collected at a relatively slow rate compared to regular SCADA measurements .

In practice, however, future pseudomeasurement values are al^¬ most always available, usually with decreasing accuracy as time elapses. For instance, the production of a wind genera^¬ tor for the next hour can be predicted with reasonable accu^¬ racy, and the same can be said of a PV farm. On the other hand, loads provided by service transformers can also be forecasted, usually within ±5% confidence intervals. In those cases, intermediate values can be easily obtained by linearly interpolating consecutive pseudomeasurements, as shown in the lower diagram of Figure 5. Mathematically, , t_k+t_j,

~^* p,k H -(z k = + n-\

Test results

The proposed TSSE model and solution refinements have been tested on a 15-kV, 100-bus distribution network (Figure 6) , made up of two feeders (11.8 and 8.7 km long) and 38 secon^¬ dary transformers. This real system delivers energy to a mix of residential, industrial and commercial loads, comprising a heterogeneous set of load patterns, some of which are shown for illustrative purposes in Figure 7. Twenty-four-hour ac^¬ tive power consumptions are known at all nodes, while reac^¬ tive powers are obtained by applying typical power factors for each customer type [15] . This leads to the branch power flow profiles shown in decreasing order in Figure 8. In addi- tion, the head voltage magnitude is assumed to be constant throughout the 24-hour period. This allows a load flow to be run, the results of which are considered as exact values for simulation purposes. In order to generate realistic sets of measurements (z_m) and pseudomeasurements (z_p), random errors have been added to the exact 24-hour quantities provided by the load flow solution (except for the 38 zero-injection buses corresponding to the 15-kV side of secondary transformers) . Each 24-hour error pattern is simulated by means of a sinusoidal wave of random amplitude and phase angle, spanning the 24-hour period, plus a random DC component, yielding together maximum peak errors of 10% for Zp and 1% for z_m. Snapshots of sets z_p and z_m are then obtained by sampling the 24-hour noisy curves at inter- vals Tp = 15 minutes and Tm = 1 minute, respectively (n = 15) . In future smart grids n can be significantly reduced, particularly if smart meter information is processed in a distributed manner, while the number of measurements in z_m will steadily increase as distribution automation devices proliferate . Pseudomeasurements in z_p comprise active and reactive power injections at all buses where loads are connected to (zero- injection buses are handled as very accurate, constantly available measurements) . In addition to the voltage magnitude at the head bus, fast-rate measurements (z_m) include sets of Ampere measurements, more or less uniformly distributed throughout the feeders. Three scenarios, labeled A, B, and C have been considered, including 8, 16 and 32 current measure^¬ ments respectively, placed as shown in Figure 6. Scenario A includes measurements numbered from 1 to 8 (4 of them in each feeder), scenario B measurements 1 to 16, and scenario C measurements 1 to 32, leading to really low redundancy levels (1.04, 1.08, and 1.16 respectively), in accordance anyway to what can be expected in future smart grids. The base-case scenario, in which only z_p and the head bus voltage magnitude are available (i.e., without current measurements), has been also analyzed. This is simply a load flow solution using a critical set of erroneous data (no possibility of filtering errors) , which will be useful to quantify the improvements brought about by the incorporation of z_m in the different scenarios.

In real life, determining the accuracy of pseudomeasurements is not a trivial task. For this reason, in absence of better alternative criteria, weights adopted for the TSSE model have been set in inverse proportion to the pseudomeasurement and measurement values, and then those of z_m are multiplied by 10 to reflect their higher accuracy compared with z_p. Simula^¬ tions for the base case and the three barely redundant see- narios have been performed and the results obtained are com^¬ pared with exact values. Solution enhancements described above (extrapolation and interpolation) have been also tested .

Figure 9 shows the 24-hour evolution of active power at a representative load bus, for both the base case (z_p only) and scenario A (8 branch measurements) . Exact values are also provided for comparison. The diagrams on top show the results obtained when z_p is kept constant between slow-rate snapshots (stepwise evolution assumed) . In average, estimates obtained in scenario A are better than those of the base case, par^¬ ticularly at the central hours. The zoomed image on the right, corresponding to the interval 13 : 45h-l 6 : 15h, clearly shows the stepwise, and sometimes sawtooth shape evolution of estimates, according to the behavior theoretically predicted in above for very low redundancy levels and loads with oppo^¬ site trends. The central curves in Figure 9 show the results obtained when extrapolation is applied to update future z_p values before they are available. In this case, the evolution of estimates is smoother than in the previous one. However, when there is a sudden slope change, the estimator needs a time interval of width Tp to get adapted to the new load trend.

The bottom diagrams show the results when interpolation of z_p is performed, which is possible only when future values are forecasted or somehow computed in advance. Compared to the two previous cases, this scheme clearly provides the best performance . Similar conclusions apply to the rest of nodes. For this rea^¬ son, since bus load and/or distributed generation forecasting is almost always available, considering the space limita^¬ tions, the analysis of results will be restricted in the se- quel to estimates obtained with interpolated z_p data.

The accuracy of estimates in scenarios A, B, and C has been numerically compared with that of the base case. Average ab^¬ solute values of errors (estimated minus exact) of power flow and voltage magnitude estimates (extended to all branches and nodes, respectively, for all minutes in 24 hours) , are pre^¬ sented in the following table which lists average absolute errors with pseudomeasurement interpolation:

Scenario

Base Case A B C

Redundancy 1 1.04 1.08 1.16

Pi_j (kW) 8.34 4.80 4.45 3.86

Qi_j (kvar) 4.47 3.28 3.01 2.85

Vi 0.16 0.11 0.10 0.08

(p.u. ) *10^"3 Clearly, as the number of current measurements increases, the average errors decrease. It is worth noting, however, that the larger relative improvement with respect to the base case is obtained with scenario A, comprising just 8 current meas^¬ urements .

Since adding more real-time measurements has a significant associated cost, each user of the TSSE should determine at the planning stage how many extra measurements are required to achieve the desired accuracy, for given sampling rates, pseudomeasurement quality and diversity of load evolution. If all loads have very homogeneous trends, as happens for in- stance when the feeders cover a purely residential area, then perhaps one or two Ampere measurements (at the head and mid^¬ dle of the feeder) , might suffice to complement forecasted load values (in the limit this reduces to a simple load allo- cation scheme) . However, as happens in the distribution system tested herein, if there is a mix of load types, some of them raising when others are decreasing, then it is not so obvious what is the optimal number and placement of real-time measurements (this constitutes an interesting optimization problem) .

The improvement brought about by the addition of 8 current measurements is better visualized in Figure 10, representing the 24-hour evolution of active power flow errors correspond- ing to the 5 branch sections carrying the largest apparent power (shown in boldface in Figure 8), both for the base case and scenario A. In the central hours, negative errors exceed^¬ ing 60 kW can be noticed for the base case, while errors arising with 8 real-time measurements are lower than 20 kW and clearly closer to zero in average. It is worth pointing out that, as theoretically predicted, error peaks (both posi^¬ tive and negative, of up to -80 kW in the base case) take place when there are sudden slope changes in the load evolu^¬ tion, which happens approximately at 5:30, 7:20, 8:20, 12:30 and 14:30 hours (the reader is referred to the quickly chang^¬ ing load patterns of Figure 7) .

Figure 11 shows an embodiment of a substation 500. The sub^¬ station 500 comprises a calculating unit 510. The calculat- ing unit 510 is configured to estimate a system value ESV which indicates a state of an electrical distribution system. The substation 500 further comprises a memory 520, which stores weighting coefficients WC that define or reflect the timely change rate of the measurement values RMV and/or the transmission rate of the measurement values RMV.

The calculating unit 510 comprises a computing unit CPU which is programmed to carry out the steps of receiving the meas^¬ urement values RMV which indicates one or more electrical quantities from a plurality of system nodes, and generating the system value ESV based on the measurement values RMV as well as on the weighting coefficients WC . The program PGM that defines the processing of the computing unit CPU may be stored in the memory 520. Conclusion

Based on an analysis (types, latencies and accuracy) of the information sources available at the distribution level, a state estimator in two time scales is proposed in this paper. It integrates a critical set of pseudomeasurements with very few redundant measurements, more accurate and captured n times faster. Limitations of the proposed model arising in low-redundancy environments, particularly when heterogenous load patterns coexist in the same feeder, are discussed, and several enhancements to deal with pseudomeasurement obsoles- cence are proposed.

Test results on a real distribution system, feeding a diversity of load patterns, fully confirms the suitability of the TSSE to both improve the accuracy and increase the latency of the load flow solutions that could otherwise be computed if only pseudomeasurements were used. In the midterm, forecasted pseudomeasurements will be gradually replaced by smart meter readings and the number of measurements will steadily in^¬ crease, but the need to handle two time scales will persist.

Appendix

In the simple networks of Figure 2, the measurement model comprises two pairs of power pseudomeasurements:

plus an Ampere measurement which, being much more accurate than the set of power pseudomeasurements, can be considered for our purposes as an equality constraint. Ignoring branch losses, with Vi = 1, it can be approximately expressed as follows :

In compact form, the objective function associated with the equality-constrained WLS SE can be written as:

^(ζ,-^(ζ, -.,<*)] (10,

with x given by:

χ = [^, ₂, ρ_{1 5} ρ₂]^Γ (ID

The estimate, , is the one satisfying the first-order opti- mality conditions:

W(_Zl-x)-S^T = 0 (12)

z₂-s(x) = 0 (13) Assuming the weighting coefficients of P and Q are the same, (12) can be rewritten as:

Q₂=Q₂ - 2a₂ ^L(Q_L+Q₂) (17)

Adding (14) to (15) and (16) to (17) yields

_{P L P}_ (Ρ:+Ρ₂Ί (18)

1 + 2λ{ω + ω₂ )

(19)

1 + 2λ(ω_ι +ω₂ )

Substituting (18) and (19) into (9) :

Let us define Km as:

r - ¹ (21)

1 + 2λ{α + ω₂ )

Then,

which allows rewriting (18) and (19) as:

0 +4 = β₂") (24)

Going back (14) -(15), they can be written in matrix form as (same for Q_t and Q ) :

ω, + 2λ 2λ ω_γΡΓ

(25)

2λ ω₂ + 2λ ω₂Ρ and, explicitly computing the inverse

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Claims

1. Method of estimating a system value which indicates a state of an electrical distribution system, the method com- prising the steps of:

measuring one or more electrical quantities at a plural^¬ ity of system nodes and providing measurement values, and generating the system value based on the measurement val^¬ ues as well as on weighting coefficients that define or reflect the timely change rate of the measurement values and/or the transmission rate of the measurement values.

2. Method according to claim 1, further characterized by: discriminating the measurement nodes into groups by the timely change rate of their measurement values and/or by their transmission rate of the measurement values, assigning a weighting coefficient to each group of said groups of measurement nodes depending on the timely change rate and/or the transmission rate of the measure- ment values of the respective group, and

estimating the system value based on the weighted meas^¬ urement values and the weighting coefficients.

3. Method according to any of the preceding claims wherein - the distribution system comprises a distribution feeder, and

the system value is estimated to define the electrical load connected to distribution feeder.

4. Method according to any of the preceding claims wherein the system value is estimated using a state estimation algo^¬ rithm.

5. Method according to any of the preceding claims wherein the measurement nodes are discriminated into two groups, namely a first group comprising frequently available measure^¬ ment values and a second group comprising less frequently available measurement values.

6. Method according to claim 5, wherein:

the system value is estimated through iteratively solving the normal equation

(H_p ^TW_p H_{p +} H_m ^TW_mH_m) Ax = H_p ^TW_p [z_p - h_p (x)]+ H_m ^TW_m [z_m - h_m (x)] wherein H_m designates a measurement matrix comprising the measure^¬ ment values of the first group,

H_m ^T designates the transposed matrix of H_m ,

H_p designates a measurement matrix comprising the measure^¬ ment values of the second group,

H_p designates the transposed matrix of H_p ,

W_m designates a weighting matrix comprising the weighting coefficient or coefficients assigned to the first group,

W designates a weighting matrix comprising the weighting coefficient or coefficients assigned to the second group,

Δχ designates an increment vector of calculated values, z_p designates a vector of measurement values assigned to the second group ,

h_p (x) designates a vector of functions that defines measure- ments of the second group, z_m designates a vector of measurement values assigned to the first group, and

h_m (x) designates a vector of functions that defines measure^¬ ments of the first group.

7. Method according to claim 6 wherein

Cholesky factorization is applied to the normal equation of claim 6.

8. Method according to any of the preceding claims wherein the weighting coefficients define or at least reflect the in^¬ formation uncertainty of the corresponding measurement val^¬ ues .

9. Method according to any of the preceding claims wherein the weighting coefficients define or at least reflect the hi^¬ erarchical position of the system nodes providing the meas^¬ urement values.

10. Method according to any of the preceding claims wherein the distribution system comprises a medium voltage grid and a low voltage grid.

11. Method according to claim 10

- wherein the measurement nodes are discriminated into a

first group and a second group,

- wherein the first group comprises the measurement values of the low voltage grid and the second group comprises the measurement values of the medium voltage grid,

- wherein one or more weighting coefficients are assigned to the first group of measurement nodes,

- wherein one or more weighting coefficients are assigned to the second group of measurement nodes, and - the system value is estimated based on the measurement values of the first and second group and the weighting co^¬ efficients of the first and second group.

12. Method according to any of the preceding claims 10-11 wherein a substation separates the medium voltage grid from the low voltage grid.

13. Method according to claim 12

wherein said step of estimating the system value is carried out in a substation, preferably the substation of claim 12.

14. Substation (500) comprising a calculating unit (510) configured to estimate a system value (ESV) which indicates a state of an electrical distribution system, the calculating unit further being configured to carry out the steps of:

receiving measurement values (RMV) indicating one or more electrical quantities from a plurality of system nodes, and

- generating the system value (ESV) based on the measure^¬ ment values (RMV) as well as on weighting coefficients (WC) that define or reflect the timely change rate of the measurement values and/or the transmission rate of the measurement values (RMV) .