| Sažetak (engleski) | Parameters of overhead transmission lines are defined by four electrical properties: series resistance and reactance and shunt conductance and capacitance. Values of these parameters depend on the line’s geometry and mechanical and thermal properties of the conductor material. These values are given by the manufacturer of the line and are not constant, i.e. they might change due to changes in ambient conditions, loading of the line or deterioration of the conductor material. This dissertation deals with determination of transmission line parameters form synchronized phasor measurements. In the first part of the dissertation a review of the loading conditions of the Croatian transmission system is presented. The loading conditions and power flows in the Croatian transmission system varies throughout the year depending on two non-electric factors, the ambient temperature and amount of the rain, i.e. hydro power commitment. The analysis gives an overview of how power system nodes (substations and power plants) change from highly- consumer types into highly-production types, depending on the amount of rain and ambient temperature. The analysis also pointed out the most critical transmission lines which during the year are highly loaded. The recognized lines are part of the south-north transmission corridor Konjsko – Velebit – Melina. This corridor is interesting to analyze with respect to the variation of transmission line parameters in time. For the purpose of calculating online transmission line parameters the phasor measurement units (PMU) technology proved to be of much value. In the Croatian transmission system the PMUs are installed in all 400 kV substations and measure the current in the 400 kV lines and also the line voltage. The current and the voltage phasor measurements are synchronized by the GPS system and delivered with 20 ms resolution to the phasor data concentrator (PDC) located in the National Dispatch Center. The PDC correlates all the synchrophasors with the same time-stamp. These feature enables the possibility of measurement comparison since they are GPS synchronized and taken in the exact same moment. The PMU device is the crucial part of every Wide Area Monitoring Systems (WAMS). WAM systems opened a whole new world of possibilities in detecting the behavior of the power system, form dynamic assessment (power swing detection, voltage stability monitoring) to enhancement of EMS systems (state estimators, linear state estimators). WAM systems nowadays are progressing towards more sophisticated and complex Wide Area Control and Protection Systems. In this dissertation the synchrophasor measurements are used for the development of online calculation methods of line parameters for pre-defined transmission lines. The pre- defined lines taken in consideration are critical lines in the network with respect to static voltage stability and branch parameters. A method for eigenvalue sensitivity analysis of the power flow Jacobian matrix is developed. The method is used to determine critical branches of a power system near the turning point of the system, i.e. the point where the power system is near the voltage collapse. Stability issues of an electric power system may originate from the transmission network elements and their parameters. The minimum eigenvalue sensitivity analysis method with respect to branch parameters is based on the power flow Jacobian matrix eigenvalue sensitivities derived with respect to the admittance matrix elements. The minimum eigenvalue is chosen because it indicates the proximity of the power system to instability. Ranking eigenvalue sensitivities by absolute value facilitates the detection of critical branches which mostly affect the minimal eigenvalue in the turning point (TP). Near the TP where the system is at the margin between stability and instability, the sensitivity analysis shows the effect that varying parameters, i.e. network admittance matrix elements, have on the power system behavior. The main contribution is the algorithm for sensitivity calculation of the minimal eigenvalue of the full power flow Jacobian by means of the corresponding left and right eigenvector, and the derivatives of the Jacobian matrix with respect to branch parameters. The sensitivities are ranked by severity and critical branches are identified. High sensitivity value of a branch indicate the most critical branch in the network with respect to static stability. From the eigenvalue sensitivity analysis of the Jacobian matrix based on branch parameters it can be determined how varying parameters affects the system variables such as the maximum loading factor, minimal voltage and minimal eigenvalue. The numerical experiments performed on the IEEE RTS-14 show that the method identifies the branches with the highest impact on the voltage stability point. Ranking branch sensitivities quantifies the determination of weak areas in the power system with low transmission capacity. The analysis of the change in the transmission network branch parameters confirmed that the change moves the turning point towards or away of maintaining system stability. The identified critical lines are the ones which are proposed for PMU device installation. For this purpose, two different sensitivity factors are defined: the branch and the bus sensitivity factor. The branch sensitivity factor is defined as the highest sensitivity factor of a branch in the power system and the matching branch is the first candidate for PMU installation. The bus sensitivity factor is defined as the sum of all sensitivity factors of branches connected to this bus. The bus with the highest sensitivity factor is the candidate for PMU installation. The wrong or erroneous parameters of transmission lines have influence in all the numerical and calculation programs for transmission system control (SCADA/EMS systems, i.e. state estimator, security analysis or dynamic modeling) and protection systems. To estimate how much these erroneous parameters affect the above mentioned systems, network analysis is performed in which the value of the transmission lines is varied from 0 to 50 %. Branches whose parameters (resistance and reactance) were changed were the ones detected as critical in the IEEE RTS-14 network model with the power flow Jacobian matrix eigenvalue sensitivity analysis. Results show the highest impact of the erroneous parameter on the state estimation is detected at the buses incident with the erroneous branch. The impact of the error decreases with the distance of the bus from the erroneous branch. Also, the highest impact on the state estimation results has the error in reactance, compared to the error in resistance. Finally, the analysis showed that the error in the branches detected as critical brought the highest cumulative error in the state estimation process. The results of this analysis represent a high motivation for establishing an algorithm for calculation of transmission line parameters from synchronized measurements. For the development of such algorithm, a few constraints have to be taken into account. The first one considers the error in the voltage and current transformer measurement. These errors have the highest influence in the transmission line parameter calculation in case the angle difference between two nodes is small. The second constraint is related to switching operations (short circuit or network manipulations) in the transmission network. The developed algorithms have to be able to reduce the effect of this constrains. Two different algorithms are developed based on the weighted least square method with variable number of phasor measurements. The first one is a modified WLS method with the augmentation of the measurement vector. The augmented measurement vector contains synchronized voltage phasor measurements on both sides of the line and the unknown line parameters (resistance, reactance and shunt admittance). The method is tested on the IEEE RTS-14 network for the lines detected as critical with the power flow Jacobian matrix eigenvalue sensitivities analysis. The results show that with increasing the number of measurement samples in the measurement vector the relative error in the parameter estimation is reduced below 1 %, if the voltage and current random measurement error is 5 %. The relative error doesn’t decrease if the number of measurement samples in the measurement vector is increased above 20. The simulation execution time is proportional to the number of measurement samples used, i.e. the higher the number of measurements, the longer is the simulation execution. The time of the simulation execution is independent of the measurement error. The main drawback of the method with measurement vector augmentation is the parameter covariance matrix which has to be a-priori known. Since these values are not known, their values are empirically detected. To overcome this drawback, a new method is developed based on the regularization methods. Regularization methods are used in optimization problems. The advantage of the regularization methods is that only the range of the expected result is needed. This feature is of special importance if the problem which one is trying to solve is an ill-conditioned problem. In this dissertation the regularization method used is the Tikhonov regularization with box constraints. The validity of the box constraint approach in the Tikhonov regularization method can be summarized with: - The transmission line parameters are assumed to be inside a pre-defined range which is determined with the characteristics of the conductor material and voltage level. The variation of the parameter values depends on the ambient conditions and/or loading of the line, but it can be assumed that these values do not exceed ±30 % of the initial (catalogue) value given by the manufacturer of the line. - The solution of the problem is the one which is the least distant (minimum norm) from the intuitive solution which derives from the physicality of the problem. - If the statistical variation of the parameters is unknown it is logical to assume their distribution will follow the Gaussian distribution. - The weighting for each parameter is given accordingly to its variance. Two different approaches of line parameter estimation with regularization methods are considered. The first one is the two-terminal Tikhonov regularization method which uses measurements from PMU devices installed on both sides of the line. The second one is the one-terminal Tikhonov regularization method which uses a reduced number of PMU devices. The first approach was tested on the critical transmission lines in the IEEE RTS-14 detected with the power flow Jacobian matrix eigenvalue sensitivities analysis . The approach was the same as in the augmented measurement vector method, i.e. the voltage and current phasor measurements were available on both sides of the line. Results showed that the parameter estimation error is minimal if the number of measurement samples is below 20 (the error is below 1 % for voltage and current measurement error of 5 %). Increasing the number of samples above 20 in the measurement vector the parameter estimation error increases. The simulation execution time, as in the augmented measurement vector method, depends on the number of measurement samples used in the simulation process. The Tikhonov regularization method is efficient in solving ill-conditioned problems. This feature was utilized for parameter estimation with a reduced number of PMUs in the “one terminal Tikhonov regularization method”. The network configuration tested in this case consisted of two transmission lines connected to the same terminal. The PMUs are installed at the adjacent buses,. i.e. at the beginning of the lines. In this approach, two PMUs are used to estimate unknown parameters of two transmission lines. The problem posed in such way is ill-conditioned. The mathematical infeasibility which results from lack of equations describing this network type can be overcome using the regularized least square method. The method was tested on the IEEE RTS-14 bus network. The simulations were performed with small load variations in the system. In this way a small value difference between two measurement samples is achieved. The method is sensitive to slow changes of measurement between two measurement samples. The increase in the number of measurement samples in the measurement vector reduces this influence. Results show the maximum parameter estimation error of 6 % for load variation of 0.05 %, and 3 % for load variation of 0.5 % and 5 %. In case a measurement error is introduced, increasing the number of measurement samples in the measurement vector the results are not improved. The augmented measurement vector method and the Tikhonov regularization method were tested on Croatian transmission system lines, with synchrophasor measurements from PMUs installed on the lines and collected in the phasor data concentrator. The chosen lines were the 400 kV Tumbri – Žerjavinec and 400 kV Melina – Velebit. The first line is chosen because of its relatively small length. For this line the mentioned constraint of the relatively small voltage angle difference between line ends is present. The second line is chosen because of its importance in energy transmission from the Croatian south to the north corridor, in respect to hydro power production. For the former line a network switching operation is captured in the synchrophasor measurements which were used for testing of the second constraint. Both methods were tested against the classical method of parameter calculation. The classical method is based on single measurement sample of voltage and current available from both sides of the line. Both methods show much less parameter variation and good results compared to the classical method and show very good behavior even during switching operations or low loading conditions. The Tikhonov regularization method showed very good results for parameter estimation for the short line. In the last paragraph of the dissertation, the transmission line parameters calculated with the Tikhonov regularization method of the 400 kV OHL Tumbri – Žerjavinec and 400 kV OHL Melina – Velebit were used in the improved state estimation process and distance protection setting enhancement. In the proposed hybrid state estimator, the estimated transmission line parameters for the above mentioned lines were updated in the network admittance matrix and used in the classical state estimation problem based on the WLS method. The state estimation results (node voltages) were compared to classical state estimation results, SCADA measurements and PMU measurements for a time period of one hour. The simulations were performed for night (low loading conditions) and morning hours (high loading conditions). The results showed the proposed hybrid estimator with the updated parameters calculated with the Tikhonov regularization method gave much better results compared to the classical approach (using the catalogue parameters given from the manufacturer of the line). The parameters were also updated in the distance protection device settings. The distance protection device reach for the first distance zone setting was improved in the case with updated parameters. The tests were made for three-phase faults along the line. In the dissertation a method of how to identify critical lines in the power system with respect to static voltage stability considering transmission line parameters is proposed. Two different factors of PMU location definition are given; the first factor based on the branch sensitivity and the second one based on the bus sensitivity. For the identified critical lines two different methods of parameter estimation is proposed, both based on the WLS method. The presented Tikhonov regularization method enables transmission line parameter estimation with a reduced number of PMUs. The calculated parameters were updated in the proposed algorithm for improved state estimation and distance protection device setting improvement. |
