A review of the enabling methodologies for PMUs ...

Electric Power Systems Research 152 (2017) 257–270

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Review

A review of the enabling methodologies for PMUs-based dynamic thermal rating of power transmission lines Guido Coletta, Alfredo Vaccaro ∗ , Domenico Villacci Department of Engineering (DING), University of Sannio, Piazza Roma 21, 82100 Benevento, Italy

a r t i c l e

i n f o

Article history: Received 13 March 2017 Received in revised form 15 May 2017 Accepted 11 July 2017 Keywords: Dynamic thermal rating Wide area monitoring Synchrophasors Phasor measurement unit

a b s t r a c t In the last years, dynamic thermal rating assessment of overhead lines has gained a critical importance in power system operation, since it allows transmission system operators to reliably increase the exploitation of existing infrastructures, avoiding the construction of new transmission assets, and increasing the hosting capacity of renewable power generators. Amongst the possible approaches that can be adopted to solve the thermal estimation problem, the one based on synchrophasor data processing is considered as one of the most promising enabling technologies, since it does not require the need for deploying dedicated sensing technologies distributed along the line route, but only the availability of synchronized measurements already available in the control centers for supporting wide area power system applications. Anyway, the deployment of this technology in real operation conditions is still at its infancy, and several open problems need to be addressed, such as the accuracy drop in low loading conditions, and the need for properly representing and managing the data uncertainties in the thermal estimation process. In trying to address these issues, this paper presents a comprehensive analysis of the most promising solution methods proposed in the literature, evaluating their performances on a real case-study based on a thermally constrained power transmission line located in the north of Italy. © 2017 Elsevier B.V. All rights reserved.

Contents 1. 2. 3.

4.

5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 PMU-based methods for DTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 3.1. Single and double measurement methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 3.2. Non-linear least square optimal estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 3.3. Calibration method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 3.4. Optimization-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 4.1. DTR accuracy for different levels of data uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 4.2. DTR accuracy in a real operation scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.2.1. Single measurement method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.2.2. Non-linear least square optimal estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.2.3. Calibration method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.2.4. Optimization-based method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.3. Load capability estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Result and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

∗ Corresponding author. E-mail address: [email protected] (A. Vaccaro). http://dx.doi.org/10.1016/j.epsr.2017.07.016 0378-7796/© 2017 Elsevier B.V. All rights reserved.

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List of acronyms OHL TSO DTR PMU PE NLLS TVE WAMS

Overhead transmission line Transmission system operator Dynamic thermal rating Phasor measurement unit Parameter estimation Non-linear least square Total vector error Wide area monitoring System

1. Introduction The increased loading of the power components, especially the overhead transmission line (OHL), is one of the most challenging issues to address in modern electrical transmission systems, which are being operated closest to their loadability limits, with severe impacts on the security and reliability of the entire power network. The main phenomena driving this trend include the increasing power transactions due to the difference in locational marginal prices, the massive penetration of renewable power generators, and the difficulties in upgrading the existing transmission assets due to social and environmental issues [1,2]. It is well known that reliable OHLs operation requires, amongst other things, the rigorous satisfaction of proper thermal constraints, which limit the conductor temperature in order to keep the line sag in fixed allowable ranges, avoiding the risk of ground faults. Traditionally, transmission system operators (TSOs) aimed at converting these thermal constraints into a maximum allowable value of the power transfer capability, which is obtained by solving the steady-state conductor thermal model using the worst-case values of the boundary conditions (i.e. wind speed 0.6 m/s, sun irradiation 1000 W/m2 , etc.) [3,4]. This conservative approach allows to reliably operate the OHLs, reducing the risk of faulty conditions, but at the cost of a sensible under-utilization of the transmission assets, since the real conductor temperature is extremely overestimated, especially when the weather conditions sensibly differ from the worst-case profiles assumed in solving the conductor thermal model. Hence, the definition of more effective OHLs loading policies become appealing in order to reliably improve the components explotation by a more accurate assessment of their real loadability margins [5–8]. In this domain, the application of advanced methodologies for dynamic thermal rating (DTR) represents a very promising research direction, since it could allow the on-line estimation of the actual conductor temperature, and the assessment of the corresponding load capability curves, supporting the TSOs in defining reliable and effective OHLs loading policies. Modern DTR architectures are based on pervasive and ubiquitous sensing technologies, which can directly measure the conductor temperature, or estimating it by periodically solving a calibrated thermal model with the actual values of the environmental variables measured along the line route [6,9,10]. Although the performance of these methods have been experimentally validated in different operation scenarios [11,12], several open problems need to be addressed in order to support their large-scale pervasion in existing power transmission networks. The need for deploying pervasive and reliable peer-to-peer networks, the difficulties arising in installing the sensors on the conductor surface, and the integrity assessment of the sensed data are some of the most critical issues to address in this domain. To face some of these limitations, alternative solution techniques based on synchronized data processing have been recently proposed in the literature. The insight of these methods is to try to

extract actionable intelligence from the large quantity of data made available by the large pervasion of wide area monitoring systems (WAMS) into transmission grids, which allow to reliable estimate the power system state by directly acquiring both current and voltage phasors in some strategical network buses [13,14]. Starting from these time synchronized measurements, these WAMS-based DTR techniques try to estimate the actual conductor thermal state by periodically identifying the electrical parameters of the power lines [15]. According to this computing paradigm, the average line temperature can be inferred from the knowledge of the actual values of the electrical line parameters, by using a proper regressive model, which correlates the variations of these parameters, especially the positive sequence resistance, to the conductor temperature. This thermal estimation process does not require the need for deploying dedicated sensing technologies distributed along the line route, since the only requested inputs for solving thermal estimation problem are the synchronized measurements of the voltage and current phasors at the both ends of the monitored line, which are already available in the control centers for implementing wide-area power systems applications [16]. Although this positive feature is considered a strategic benefit for the TSOs, the deployment of this technology in real operation conditions is still at its infancy, and several open problems need to be addressed, such as the accuracy drop in low loading conditions, and the need for properly representing and managing the data uncertainties. The latter represents one of the most challenging research objectives to address in the context of phasor measurement unit (PMU)-based applications [17,18], which has recently attracted the attention of wide-area monitoring, protection and control systems designers. In particular, paper [19] describes four methods for the solution of the OHL thermal estimation problem in the presence of measurement uncertainty, based on both single and double phasor measurements, and using both linear and non-linear least square-based solution methods. The results discussed in this paper demonstrate that the non-linear least square approach based on multiple measurements represents the most viable solution to mitigate the effects of measurement errors. This approach has been improved in [20], which allows improving the overall thermal estimation accuracy by defining proper bounds on the electrical parameters, in [21], where the non-linear regression technique is applied for estimating the electrical line parameters of both power lines and transformers, and in [22,23] where a new method for estimating the line conductor temperature and the corresponding line sag is proposed. An alternative solution technique for uncertainty management has been described in [24], which proposes a two-stage solution method to reduce the effects of systematic errors by identifying proper calibration coefficients, and applying a least square technique on the calibrated phasors to estimate the uncertain line parameters. Finally, Refs. [25,26] propose two reliable thermal identification techniques based on the solution of uncertain optimization problems, whose objective is to minimize the error between the measured and the estimated phasors. Although these optimizationbased methods exhibited good performances in several operation scenario, they suffer from severe limitations, mainly related to the difficulties in identifying the most suitable error function to minimize, and the need for large computational resources. On the basis of the results discussed in these papers, much of them based on simulation studies, it can be argued that the reliable solution of the DTR problem by PMUs data processing can be addressed according to different techniques, which sensibly differ in terms of accuracy, reliability, and promptness. Hence, the selection of the most effective technique to adopt in real operation scenarios is not straightforward, and it is further complicated by the lack of a comprehensive analysis aimed at comparing the

G. Coletta et al. / Electric Power Systems Research 152 (2017) 257–270

259

Fig. 2. -Circuit model of a short OHL. Fig. 1. -Equivalent circuit model of the OHL.

the average conductor temperature by the following approximated equation [6]:

performance of these techniques on a common experimental case study. In the light of these needs, in this paper the performance of the most advanced methods for PMUs-based DTR have been experimentally assessed in the task of solving the thermal estimation problem for a complex transmission asset, which is based on a 400 kV overhead line located in the north of Italy. A comparison between the average line temperature estimated by the analyzed methods for a 6h time horizon, and the real conductor temperature directly sensed by a measurement system installed in the critical line span, will be presented and discussed in order to outline the benefits and the limitations of the considered techniques.

r(T ) = r(Tref ) + ˛(T − Tref )

(3)

where ˛ is the temperature-resistance regression constant, r(T) is the series resistance at the temperature T and, Tref is a reference conductor temperature. Various techniques can be adopted to solve this parameter estimation (PE) problem in modern PMU-based DTR assessment procedures, and the most significant of them will be analyzed in the next section. 3. PMU-based methods for DTR 3.1. Single and double measurement methods

2. Mathematical preliminaries A fully-transposed electric power line can be modeled at the direct sequence through a concentrated parameters quadrupole, represented by the equivalent  circuit shown in Fig. 1. The electrical state of this circuit is described by the voltage and current phasors at the first line end, V¯ s and I¯s , which are related to the corresponding phasors at the second line end, V¯ r and I¯r , by the so called telegraphers equations: V¯ s = A˙ V¯ r + B˙ I¯r

(1)

I¯s = C˙ V¯ r + D˙ I¯r

where the transmission parameters for the power line can be defined as follows: A˙ = cosh B˙ = C˙ =





(r + jωl) (g + jωc)

(r + jωl)/(g + jωc) sinh



1 (r + jωl)/ (g + jωc)





sinh

(r + jωl) (g + jωc)







(r + jωl) (g + jωc)



The simpler approach for PMU-based DTR assessment [19] is based on the solution of the PE problem by manipulating the telegraphers’ equations. In particular, the main idea is to identify the quadrupole constants by inverting (1), using the voltage and current phasors measured at the both line ends. This solution paradigm is typically referred as double measurement method, since it requires the acquisition of two independent sets of phasor measurements, namely the voltage and current phasors at both line ends. An alternative solution approach, requiring only the knowledge of a single phasor measurement set, can be adopted for solving the PE problem for the so called short transmission lines, which are overhead lines characterized by a limited length, typically less than 100–150 km. In this case, it is possible to approximate the hyperbolic functions in the transmission parameter equations by their first order Taylor’s series terms, obtaining the simplified model shown in Fig. 2, which is described by the following equations:

(2)

D˙ = A˙ These parameters depend on the angular frequency ω = 2f, and the primary line constants per unit length, namely, the series resistance r, the series inductance l, the shunt conductance g, and the shunt susceptance c. The latter depend on both the overhead line characteristics, e.g. the pole geometry, and the conductor thermal state, especially the conductor temperature. Consequently, once the voltage and current line phasors are known, the electrical line parameters can be estimated by manipulating the model formalized in (1), and the corresponding conductor thermal state can be inferred by using proper regression models, describing the correlations of the electrical line parameters with the average conductor temperature. To this aim, the parameter characterized by the highest sensitivity to the conductor temperature is known to be the series resistance, which can be correlated to

V¯ s =

1+

Z˙ Y˙ 2



V¯ r + Z˙ I¯r



I¯s = Y˙ V¯ r +

1+

Z˙ Y˙ 2



(4) I¯r

where Z = (r + jωl) * L, Y = (g + jωc) * L and L is the line length. Since the obtained model is a two variables-two equations system, its inversion requires only the knowledge of 4 complex quantities, namely a single phasor measurement set. Hence, this approach is typically referred as single measurement method. Both the single and the double measurement methods allow obtaining reasonable estimation accuracy when input data are accurate enough, but their performances tend to degenerate in real operation scenario, due to the lack of intrinsic tools for data uncertainty modeling. Although the use of redundant measurements can be adopted in trying to increase the performances of these estimation algorithms, by reducing the effect of random measurement errors, the tolerance of the estimated line parameters could not be enough to reliably estimate the average conductor temperature with an acceptable level of confidence.

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G. Coletta et al. / Electric Power Systems Research 152 (2017) 257–270

3.2. Non-linear least square optimal estimator

3.3. Calibration method

A challenging idea aimed at solving the parameter estimation (PE) problem in the presence of data uncertainty by processing redundant measurements is to design estimation techniques based on the non-linear least square (NLLS) theory [19]. In particular, given a set of N measurements, Eq. (1) can be written as follows:

Errors in PMU measurements are not only induced by random errors, which can be effectively managed by the previously described techniques, but also by systematic errors, mainly affecting the measurement transformers. To face this issue, in [24] a solution method based on a calibration process is proposed. The insight principle is to model systematic errors in phasor measurements as an error in magnitude and an error in phase, hence expressing each measure phasor as follows:

⎧ ¯ ¯ V¯ r ) + R(B)R( ¯ ¯ I¯r ) R(V¯ s ) = R(A)R( V¯ r ) − I(A)I( I¯r ) − I(B)I( ⎪ ⎪ ⎪ ⎪ ⎨ I(V¯ s ) = R(A)I( ¯ V¯ r ) + I(A)R( ¯ ¯ I¯r ) + R(B)I( ¯ I¯r ) V¯ r ) + R(B)I(

(5)

⎪ ¯ ¯ ¯ ¯ R(I¯s ) = R(C)R( V¯ r ) − I(C)I( V¯ r ) + R(D)R( I¯r ) − I(D)I( I¯r ) ⎪ ⎪ ⎪ ⎩ ¯ ¯ ¯ ¯ I¯r ) + I(D)R( I¯r ) I(I¯s ) = R(C)I(V¯ r ) + I(C)R(V¯ r ) + R(D)I(

where R( · ) and I(·) denote the real and imaginary part of each complex quantity. Starting from these equations, and knowing the measured line phasors, it is possible to compute the corresponding residuals, namely the difference between the measured and the theoretical phasors for each set of line parameters, as follows:

⎧ ¯ ¯ V¯ r )−R(B)R( ¯ ¯ I¯r ) f1 (x, ) = R(V¯ s )−R(A)R( V¯ r ) + I(A)I( I¯r ) + I(B)I( ⎪ ⎪ ⎪ ⎪ ⎨ f2 (x, ) = I(V¯ s )−R(A)I( ¯ V¯ r )−I(A)R( ¯ ¯ I¯r )−R(B)I( ¯ I¯r ) V¯ r )−R(B)I(

P¯ = P¯ ∗ (1 + a)(1 + j) = P¯ ∗ (1 + a + j + ja) (6)

Then, once indicating with x and  the vectors of the synchrophasor measurements, and the line parameters, respectively, as follows:



= r

l

I(V¯ s ) c

g

R(I¯s )

I(I¯s )

R(V¯ r )

I(V¯ r )

R(I¯r )

I(I¯r )



⎡

P¯ = P¯ ∗ (1 + a)(1 + j) = P¯ ∗ (1 + a + j)

Starting form this result, all the phasors at the both line ends can be expressed as:

V¯ r = V¯ r∗ (1 + ar + jr )

⎤ ⎡

11 (x, )

I¯s = I¯s∗ (1 + bs + js )

f4N (x, )

and the total errors in phasor measurements can be expressed as: ıV¯ s = V¯ s∗ (as + js ) ıV¯ r = V¯ r∗ (ar + jr ) ıI¯s = I¯s∗ (bs + js ) (7)

Thanks to this mathematical formulation, it is possible to estimate the line parameters by identifying  that minimizes the residual of F(x, ), . This scalar minimization problem can be solved using a NLLS regression technique, by deploying the following iterative procedure:

 k = (HT H)

−1

HT (−F(x,  k ))

(15)

ıI¯r = I¯r∗ (br + jr ) If a single set of phasor measurements are available, the single measurement method is applied, in order to provide a first approximation of the line parameters, trough the inversion of the telegraphers’ equations:

N4 (x, )

 k+1 =  k +  k

(14)

I¯r = I¯r∗ (1 + br + jr )

⎤

⎢ f 1 (x, ) ⎥ ⎢ 1 (x, ) ⎥ ⎢2 ⎥ ⎢ 2 ⎥ ⎥ ⎢ ⎥ ⎢0⎥ ⎢ ⎢ ⎥ ⎢ 1 1 ⎢ ⎥ ⎢ f3 (x, ) ⎥ ⎢ 3 (x, ) ⎥ ⎥ ⎢0⎥ ⎢ ⎢ 1 ⎥ ⎢ ⎥ ⎢ f 1 (x, ) ⎥ ⎥ ⎢ ⎥  (x, ) ⎢0⎥ ⎢ 4 4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . . ⎥ ⎢ ⎥ = F(x, ) +  +⎢. ⎢ . ⎥ = ⎢ .. ⎥ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎢ N N (x, ) ⎥ ⎥ ⎢ (x, )  f ⎢ ⎥ ⎢1 1 ⎥ ⎢ ⎥ ⎢0⎥ ⎢ ⎢ N ⎥ ⎢ ⎥ ⎢ f N (x, ) ⎥ ⎥ ⎢ ⎥ ⎣0⎦ ⎢ 2 ⎥ ⎢ 2 (x, ) ⎥ ⎢ N ⎥ ⎢ N ⎥ ⎣ f3 (x, ) ⎦ ⎣ 3 (x, ) ⎦ 0

(13)

V¯ s = V¯ s∗ (1 + as + js )



f11 (x, )

(12)

and, by neglecting the second order error terms, it follows that:

Eq. (6) can be generalized for a set of N phasor measurements as follows:

⎡0⎤

(11)

Hence, Eq. (10) becomes:

¯ ¯ I¯r )−I(D)R( I¯r ) f4 (x, ) = I(I¯s )−R(C)I(V¯ r )−I(C)R(V¯ r )−R(D)I(



(10)

where P¯ ∗ ∈ C is the phasor measured by the PMU and a, ∈ R are the corresponding phase and magnitude errors, respectively. Since the phase errors are typically less than 0.530◦ , in order to satisfy the accuracy requirements of the IEEE Std. C37.118.1.2011 for synchrophasor measurements [27,28], small angle approximation for trigonometric functions can be used, obtaining: ej ≈ 1 + j

⎪ ¯ ¯ ¯ ¯ f3 (x, ) = R(I¯s )−R(C)R( V¯ r ) + I(C)I( V¯ r )−R(D)R( I¯r ) + I(D)I( I¯r ) ⎪ ⎪ ⎪ ⎩ ¯ ¯

x = R(V¯ s )

P¯ = P¯ ∗ (1 + a)ej

(8) (9)

where H is the Jacobian matrix of F(x,  k ). This iteration process is typically interrupted when the variable update, ␪k , results smaller than a specified tolerance.

Z˙

=

Y˙

=

V¯ s2 − V¯ r2 2 ¯ Vs I¯r2 + V¯ r2 I¯s2 2(I¯s − I¯r )

(16)

V¯ s + V¯ r

The first order approximation of the corresponding estimation errors can be expressed as:

∂Z˙ ¯ ∂Z˙ ¯ ıVs + ıVr ¯ ∂Vs ∂V¯ r ∂Y˙ ¯ ∂Y˙ ¯ ıY˙ = ıIs + ıIr ∂I¯s ∂I¯r ıZ˙

=

(17)

G. Coletta et al. / Electric Power Systems Research 152 (2017) 257–270

Hence, by assuming that the overall measurement errors can be defined as: a = ar − as  = r − s

(18)

b = br − bs  = r − s

where, according to the argumentation presented in paper, the maximum allowable ranges for these error functions can be estimated as: |a| < 0.02 || < 0.02

(19)

|b| < 0.02 || < 0.02



ıZ¯ Z¯ ıY¯ Y¯



app



Table 1 Main line characteristics. Phase conductors Conductor type Length Nominal voltage Conductor diameter Positive sequence resistance Positive sequence reactance Positive sequence susceptance

2 ACSR 112.2 km 400 kV 31.5 mm 3.15  38.2  2 × 191.5 × 10−6 s

of the squared errors between the measured and the estimated phasors, as detailed in the following equations: min [r,xl ,g,b]

[E (|V¯ |) + E (∠V¯ ) + E (|I¯ |) + E (∠I¯ )]

(24)

where

Starting from these results, it is possible to compute the corresponding relative errors in Z¯ and Y¯ estimation as follows:



261

−2(ar − as + jr − js )V¯ r∗2 = V¯ s∗2 − V¯ r∗2 =

app

−2(br − bs + jr − js )I¯r V¯ s∗ + V¯ r∗

i=1

(20)

= qR t + rR

fX (t)

= qX t + rX

fG (t)

= rG

fB (t)

= rB

E (∠V¯ ) =

(21)

where the quantities qR , qX , rR , rX , rG and rB are the coefficients of the line parameter models estimated through a least square procedure. Consequently, the unknown phasor calibration parameters can be estimated by solving the following non-linear, constrained optimization problems:

E (|I¯ |) =

∠V¯ s,i

2 N   |I|s,i − |I|s,i,estimated i=1

E (∠I¯ ) =

|V¯ |s,i

2 N   ∠V¯ s,i − ∠V¯ s,i,estimated i=1

This important result allows computing the values of the uncertain variables a, b,  and , which minimize, on a fixed time window, the time integral of the square error between the measured and the estimated phasors at both line ends. In solving this problem, a linear drift of both line resistance and reactance is assumed, while conductance and susceptance are assumed constant during the time window. Thanks to these hypothesis, which are consistent with the assumptions defined in IEEE Std 738-2012 [7], it is possible to represent the primary constants R = r * L, X = x * L, G = g * L and B = b * L as follows: fR (t)

2 N   |V¯ |s,i − |V¯ |s,i,estimated

E (|V¯ |) =

|I|s,i

2 N   ∠I¯s,i − ∠I¯s,i,estimated i=1

∠I¯s,i

(25)

(26)

(27)

(28)

and N is the number of measured phasors used for estimating each line parameter set. Similarly to the previously described NLLS methods, this solution approach tries to minimize the time integral of the squared error between the measured and the estimated phasors at both line ends, but rather than solving the PE problem by using a two-step optimization procedure, which firstly identifies the phasor calibration constants and then estimates the line parameters, it employs a single optimization procedure, without doing any assumption on the time evolution of the line parameters. 4. Case study

3.4. Optimization-based methods

In order to characterize the performances of the described methods in the task of solving complex thermal estimation problems in a real operation scenario, an experimental case study has been considered. To this aim, the data acquired from the Italian wide area monitoring system [13] for a 400 kV overhead line over a 6 h time window have been used. This is a strategic asset for the Italian power transmission network, since it interconnects two critical system sections, characterized by “large” power transactions. The main characteristics of the line under study are summarized in Table 1. The measured data include voltage and current phasors at both ends of the considered line sampled each 50 ms (Figs. 3 and 4), and the conductor temperature of the most critical line span sampled each 1 min by a high accurate temperature sensor. The available data are organized as:

Alternative solution approaches recently proposed in the literature [25,26] to solve the PE problem are based on the deployment of optimization frameworks aimed at identifying the set of line parameters, which minimize, on a fixed time window, the sum

- four vectors of complex number containing Vr , Vs , Ir and Is measurements, with a cardinality of [1,080,000 × 1]; - a vector of time tags associated with the PMU measurement, with the same cardinality.

min g Z (a, ) = SR + SX [a,]

{ |a| < 0.02

(22)

|| < 0.02 min g Y (b, ) = SG + SB [b,]

{ |b| < 0.02

(23)

|| < 0.02 Once a, b, ,  have been identified, the corrected phasors can be computed and a conventional least square procedure can be applied to solve the PE problem as described in the previous section.

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Fig. 3. PMU measurements-starting section.

Fig. 4. PMU measurements-receiving section.

The first issue to address in solving the PE problem is to fix the time window, namely the number of data samples that should be adopted in estimating the line parameter. To this aim, various options have been considered: - estimation of one parameter-set every N measurement; - estimation of one parameter-set every N measurement through a moving time window; - estimation of one parameter-set every Nk measurement through a moving time window, where Nk is a set of data containing the average phasors computed every k measurements. - hybrid approach – estimation of one parameter every N measurement, through a moving window, every k measurement. In this context, the firsts experimental results demonstrated that the application of the last approach is the most effective option, representing a good trade-off between estimation accuracy and computational burden. Hence, it has been selected for developing the experiments discussed in this paper. This conclusion can be confirmed by analyzing the data summarized in Table 2, which reports the computational times requested to solve the PE problem by using the analyzed solution techniques. These data have been observed on the same processing system,

Table 2 Computational time. Single measurement method NLLS method Calibration method Optimization method

0.15 6 1.6 13

Seconds Minutes Hours Minutes

which is based on the Intel(R) Core(IM) i7-4510 dual core processor, ® equipped by the Matlab environment. Starting from these data, two experimental sessions have been implemented: the first one was aimed at assessing the performance of the analyzed methods in the task of solving the PE estimation problem for different level of data uncertainty, which was generated by properly merging the real measurements with simulation data, while the second session was aimed at testing the DTR methods on a real case study, by using only the experimental data. 4.1. DTR accuracy for different levels of data uncertainty To generate realistic data-sets corrupted by a fixed level of data uncertainty, a reverse-engineering based procedure has been adopted. The main idea is to start from the conductor temperature profile measured by the sensor to obtain a realistic evolution of the line parameters for a fixed time window. Then, by using these pro-

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Fig. 5. TVE histogram (with 0.005 uncertainty level).

files and the measured phasors at one line end, which have been assumed with no uncertainty, the corresponding phasors at the other line end have been computed by using (1). These two phasor sets, which have been assumed as the “true” data-set on both line ends, have been corrupted by three levels of data uncertainty, applying the concept of total vector error (TVE). The latter represents one of the most important figures of merit characterizing PMUs accuracy. It is defined in [27] as:



TVE =

(R(At ) − R(A))2 + (I(At ) − I(A))2 R(At )2 + I(At )2

(29)

where At is the true synchrophasor, A is the sensed phasor, R( · ) and I(·) denotes the real and imaginary part, respectively. This parameter is used in the cited IEEE standard as a quality index of the synchrophasors measurements in power systems. Errors in measures have been modeled in both phase and magnitude, generating four independent random Gaussian variables, without doing any assumption on the correlation between the measurements errors of the same instrument channel. The considered level of uncertainty corresponds to average TVE of 0, 0.005 and 0.01, respectively. The corresponding histograms of the TVE are reported in Figs. 5 and 6, for TVE levels of 0.005 and 0.01, respectively. Starting from these data-sets, the analyzed methods have been applied obtaining the results summarized in Figs. 7–9, which report the conductor temperature measured at the critical span, and the corresponding estimated profiles, for average TVE of 0, 0.005 and 0.01 respectively. In this context, it is worth noting that, as expected, the calibration method failed to solve the PE problem, identifying inconsistent temperature profiles, for TVE greater than 0.005.

ing processing did not match the accuracy requirements of DTR applications. 4.2.2. Non-linear least square optimal estimator To test this method, a set of 50 samples for each phasor at both line ends is used for estimating a single line parameter set, using a moving window-based approach for solving the estimation problem at each time step. The corresponding results for the analyzed time window are organized in a matrix composed by 21,600 rows, each one representing a set of estimated line parameters, that are time averaged in order to estimate the conductor temperature every 5 min. Figs. 11 and 12 depict the line parameters and the corresponding conductor temperature estimated through the application of this methodology. 4.2.3. Calibration method This method allows managing both statistical and systematic errors affecting synchrophasor measurements. Correction constant identified for the 4 phasorial quantities are shown in Fig. 13 and the corresponding temperature profile, which has been averaged on 1 min time windows, is shown in Fig. 15. Line parameters, instead, are shown in Fig. 14. 4.2.4. Optimization-based method The optimization-based methodology has been applied in the task of solving the following NLP programming problem:

⎧   min ∗ [E (|V¯ |) + E (∠V¯ ) + E (|I¯ |) + E (∠I¯ )] ⎪ ⎪ [r,x ,b] ⎪ l ⎪ ⎨ 0 < r ∗ < inf ⎪ 0.9 xLnom < xl∗ < 1.1 xLnom ⎪ ⎪ ⎪ ⎩ ∗

(30)

0.9 bnom < b < 1.1 bnom

4.2. DTR accuracy in a real operation scenario 4.2.1. Single measurement method The result obtained by applying this method to solve the PE problem for the analyzed time window are shown in Fig. 10, which depicts the measured and the estimated temperature profiles. These data have been obtained by averaging the conductor temperature over a time window of 5 min, in order to compensate the errors by using redundant measurements. The deployment of proper data filtering functions is strictly recommended for the correct application of this estimation method on real measured data. This was also confirmed by the experimental results obtained on the analyzed case study, where it has been observed that the temperature profiles estimated without performing the averag-

where * is a proper scaling factor, and E (|V¯ |), E (|I¯ |), E (∠V¯ ) and E (∠I¯ ) are the error functions defined in Section 3.4. The latter optimization problem has been solved through an interior-point based method, using the built-in routines available ® in the optimization toolbox of the Matlab environment. Line parameter and conductor temperature estimated through this method are depicted in Figs. 16 and 17, respectively. 4.3. Load capability estimation The knowledge of the average conductor temperature is a prerequisite for assessing the real loadability margins of the monitored transmission asset on a short-medium time horizon. To this aim,

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Fig. 6. TVE histogram (with 0.01 uncertainty level).

Fig. 7. Estimated conductor temperature TVE = 0.

Fig. 8. Estimated conductor temperature TVE = 0.005.

the dynamic assessment of the load capability curve, reporting the maximum allowable time for each hypothetical line current, represents one of the most useful tools. Anyway, this computing process is a very complex issue to address, since it requires the knowledge of the actual conductor temperature, the worst-case estimation of the weather variables along the line route, and the repetitive solution of a detailed conductor thermal model.

To address this issue, an indirect method for weather parameters estimation has been applied in this paper. The rationale is to estimate the wind speed, the wind direction, and the ambient temperature by solving an optimization problem that minimizes, on a fixed time window, the prediction error between the estimated conductor temperature, and the corresponding one obtained by applying a first order thermal model [7]. Once the actual weather parameters have been estimated, and the maximum allowable con-

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Fig. 9. Estimated conductor temperature TVE = 0.01.

Fig. 10. Estimated conductor temperature – single measurement method.

Fig. 11. Estimated parameter – NLLS method.

ductor temperature has been fixed, the load capability curve is computed by solving the aforementioned thermal model for the overall set of hypothetical load currents according to the proce-

dure summarized in Fig. 18. The obtained results for a fixed thermal state, and a maximum conductor temperature of 90◦ , have been summarized in Fig. 19.

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Fig. 12. Estimated conductor temperature – NLLS method.

Fig. 13. Correction coefficients.

Fig. 14. Estimated parameters – calibration method.

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Fig. 15. Estimated conductor temperature – calibration method.

Fig. 16. Estimated parameters – optimization method.

Fig. 17. Estimated conductor temperature – optimization method.

These results confirmed the sensible upset of the conductor estimation accuracy on the loadability assessment. In particular, by observing the vertical asymptotes of the load capability curves, which represents the static thermal ratings, it can be noted as they

vary from 800 to 1200 A in function of the particular solution technique adopted to solve the PE problem. This effect is also evident for the other operation points in the load capability curve, hence

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Fig. 18. Capability curve assessment.

Fig. 19. Capability curve.

confirming the strategic importance of implementing a reliable and accurate method for DTR assessment.

5. Result and discussion Result obtained by the application of the various described methodologies allows us to make some useful considerations. In particular, by analyzing Fig. 7 it is worth observing the sensible impacts of input data uncertainty on the accuracy of the estimated line temperature profiles, which lead the calibration method to completely fail in identifying feasible solutions to the PE problem.

Moreover, in terms of conductor temperature accuracy, both Optimization-based and single measurement methods exhibited better performances compared to the other considered techniques, both in the simulated and the real-time application scenario, but as outlined in Fig. 20, their performances tend to deteriorate when line current drops below under certain threshold values. On the other hand, the numerical stability of both the NLLS and the calibration methods is less sensitive to the line current, but their estimation accuracy is lower. In particular, the calibration method does not provide feasible temperature values in the simulated

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Fig. 20. Temperature profiles.

scenario, giving acceptable, although overestimated, results in realcase applications. Another interesting argumentation emerging from the analysis of the obtained results concerns with the values of the estimated parameters. In this context, the calibration method provides estimations of the shunt susceptance, which are very close to the nominal value, and estimations of the series line parameters, which sensible differ from the corresponding nameplate values. This feature is consistent with previous experimental studies, which demonstrated that the series parameters exhibit larger sensitivity to the line temperature variations. The analyzed methodologies have been originally conceptualized for estimating the electrical line parameters. Anyway, as demonstrated by the experimental results obtained in this paper, and confirmed by other field studies reported in the literature, the deployment of these methodologies in real operation scenario is not straightforward, and requires the application of proper data miming techniques aimed at mitigating the effects of data uncertainty, such as missing data, outliers, incoherent samples and filtering other kind of defective data. These uncertainty sources can affect the accuracy of the estimated parameters, leading to unrealistic results. To address this issue, the role of data preprocessing techniques has been extensively emphasized in the literature only for specific power system application, such as PMUbased electromechanical oscillation analysis, while only standard outlier detection techniques has been deployed for PMU-based line parameter estimation, to the best of our knowledge. Anyway the adoption of these techniques did not allow us to obtain a reliable parameter estimation. Hence, this is an open problem, which asks for further research efforts. Finally, it is possible to note that the line temperature estimated by applying the calibration procedure is highly conservative, which can be considered as an advantage, since it reduces the risk of thermal line overloading, but at the cost of an asset under-utilization. 6. Conclusion In this paper a review of the most promising approaches for PMU-based conductor temperature estimation of overhead lines have been presented, and their performance have been assessed and compared by using experimental errors obtained from a 400 kV power line located in the North of Italy. The experimental sessions were aimed at both characterizing the uncertain propagation for fixed values of the total vector error, and the accuracy of the analyzed techniques in the task of estimating the average line tem-

perature by processing the real data sensed by the PMUs. The results obtained demonstrated that both systematic and random measured data influences the accuracy performances of the analyzed estimation algorithms to a considerable extent, and the selection of the most effective paradigm for data uncertainty management in PMU-based DTR is still an open problem, requiring further investigations. Further experimental activities, based on longer time horizons, higher loading levels, and extremely variable weather conditions, are currently under investigation by the Authors in order to confirm these conclusive remarks on different, and more complex, operation scenario.

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