Grigsby, L.L., Hanson, A.P., Schlueter, R.A., Alemadi, N." Power Systems The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Grigsby, L.L., Hanson, A.P., Schlueter, R.A., Alemadi, N. “Power Systems” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
63 Power systems 63.1 Power System Analysis Introduction. Types of Powe m Analyses. The Power Flow Problem. Formulation of the Bus Admittance Matrix. Example LL. Grigsby and Formulation of the Power Flow Equations. P-V Buses.Bus AP Hanson Classifications.Generalized Power Flow Development.Solution Auburn University Methods.componentPowerFlows R.A. Schlueter and 63.2 Voltage Instability Voltage Stability Overview. Voltage Stability Models and N. Alemadi Simulation Tools. Kinds, Classes, and Agents of voltage Michigan State University Instability. Proximity to Voltage Instability. Future Research 63.1 Power System analysis Introduction The equivalent circuit parameters of many power system components are described in Chapters 61, 64, and 66. The interconnection of the different elements allows development of an overall power system model. The stem model provides the basis for computational simulation of the system performance under a wide variety of projected operating conditions. Additionally, "post mortem"studies, performed after system disturbances or equipment failures, often provide valuable insight into contributing system conditions. The different types f power system analyses are discussed below; the type of analysis performed depends on the conditions to be Types of Power System Analyses Power Flow analysis Power systems typically operate under slowly changing conditions which can be analyzed using steady state analysis. Further, transmission systems operate under balanced or near balanced conditions allowing per phase point for many other analyses. For example, the small signal and transient stability effects of a given disturballs a analysis to be used with a high degree of confidence in the solution. Power flow analysis provides the starti are dramatically affected by the"pre-disturbance"operating conditions of the power system.(A disturbance resulting in instability under heavily loaded system conditions may not have any adverse effects under lightly loaded conditions. )Additionally, fault analysis and transient analysis can also be impacted by the "pre-distur bance"operating point of a power system(although, they are usually affected much less than transient stability and small signal stability analysis) Fault Analysis Fault analysis refers to power system analysis under severely unbalanced conditions. Such conditions inch downed or open conductors. Fault analysis assesses the system behavior under the high current and/or severely e 2000 by CRC Press LLC
© 2000 by CRC Press LLC 63 Power Systems 63.1 Power System Analysis Introduction • Types of Power System Analyses • The Power Flow Problem • Formulation of the Bus Admittance Matrix • Example Formulation of the Power Flow Equations • P-V Buses • Bus Classifications • Generalized Power Flow Development • Solution Methods • Component Power Flows 63.2 Voltage Instability Voltage Stability Overview • Voltage Stability Models and Simulation Tools • Kinds, Classes, and Agents of Voltage Instability • Proximity to Voltage Instability • Future Research 63.1 Power System Analysis Introduction The equivalent circuit parameters of many power system components are described in Chapters 61, 64, and 66. The interconnection of the different elements allows development of an overall power system model. The system model provides the basis for computational simulation of the system performance under a wide variety of projected operating conditions. Additionally, “post mortem” studies, performed after system disturbances or equipment failures, often provide valuable insight into contributing system conditions. The different types of power system analyses are discussed below; the type of analysis performed depends on the conditions to be assessed. Types of Power System Analyses Power Flow Analysis Power systems typically operate under slowly changing conditions which can be analyzed using steady state analysis. Further, transmission systems operate under balanced or near balanced conditions allowing per phase analysis to be used with a high degree of confidence in the solution. Power flow analysis provides the starting point for many other analyses. For example, the small signal and transient stability effects of a given disturbance are dramatically affected by the “pre-disturbance” operating conditions of the power system. (A disturbance resulting in instability under heavily loaded system conditions may not have any adverse effects under lightly loaded conditions.) Additionally, fault analysis and transient analysis can also be impacted by the “pre-disturbance” operating point of a power system (although, they are usually affected much less than transient stability and small signal stability analysis). Fault Analysis Fault analysis refers to power system analysis under severely unbalanced conditions. (Such conditions include downed or open conductors.) Fault analysis assesses the system behavior under the high current and/or severely L.L. Grigsby and A.P. Hanson Auburn University R.A. Schlueter and N. Alemadi Michigan State University
unbalanced conditions typical during faults. The results of fault analyses are used to size and apply syster protective devices(breakers, relays, etc. Fault analysis is discussed in more detail in Section 61.5. Transient Stability Analysis Transient stability analysis, unlike the analyses previously discussed, assesses the systems performance over a period of time. The system model for transient stability analysis typically includes not only the transmission network parameters, but also the dynamics data for the generators. Transient stability analysis is most often used to determine if individual generating units will maintain synchronism with the power system following a disturbance(typically a fault) Extended Stability analysis Extended stability analysis deals with system stability beyond the generating units' first swing. "In addition to the generator data required for transient stability analysis, extended stability analysis requires excitation system, d governor, and prime mover dynamic data. Often, extended stability analysis will also include dynamics data for control devices such as tap changing transformers, switched capacitors, and relays. The addition of these elements to the system model complicates the analysis, but provides comprehensive simulation of nearly all major system components and controls. Extended stability analyses complement small signal stability analyses by verifying the existence of persistent oscillations and establishing the magnitudes of power and/or Small Signal Stability Analysis Small signal stability assesses the stability of the power system when subjected to"small"perturbations. Small ignal stability uses a linearized model of the power system which includes generator, prime mover, and control device dynamics data. The system of nonlinear equations describing the system are linearized about a specific operating point and eigenvalues and eigenvectors of the linearized system found. The imaginary part of each eigenvalue indicates the frequency of the oscillations associated with the eigenvalue; the real part indicate damping of the oscillation. Usually, small signal stability analysis attempts to find disturbances and/or system conditions that can lead to sustained oscillations(indicated by small damping factors)in the power system. Small signal stability analysis does not provide oscillation magnitude information because the eigenvalues only indicate oscillation frequency and damping. Additionally, the controllability matrices(based on the linearized system)and the eigenvectors can be used to identify candidate generating units for application of new or improved controls(i.e, power system stabilizers and new or improved excitation systems) Transient analysis involves the analysis of the system (or at least several components of the system)when subjected to"fast"transients(i. e, lightning and switching transients). Transient analysis requires detailed component information which is often not readily available. Typically only system components in the immediate vicinity of the area of interest are modeled in transient analyses. Specialized software packages(most notably EMTP)are used to perform transient analyses perational analyses Several additional analyses used in the day-to-day operation of power systems are based on the results of the nalyses described above. Economic dispatch analyses determine the most economic real power output for each generating unit based on cost of generation for each unit and the system losses. Security or contingency analyses assess the systems ability to withstand the sudden loss of one or more major elements without overloading the remaining system. State estimation determines the"best"estimate of the real-time system state based on a redundant set of syster Im measuremen The power flow problem Power flow analysis is fundamental to the study of power systems. In fact, power flow forms the core of power system analysis. a power flow study is valuable for many reasons. For example, power flow analyses play a key role in the planning of additions or expansions to transmission and generation facilities. a power flow solution e 2000 by CRC Press LLC
© 2000 by CRC Press LLC unbalanced conditions typical during faults. The results of fault analyses are used to size and apply system protective devices (breakers, relays, etc.) Fault analysis is discussed in more detail in Section 61.5. Transient Stability Analysis Transient stability analysis, unlike the analyses previously discussed, assesses the system’s performance over a period of time. The system model for transient stability analysis typically includes not only the transmission network parameters, but also the dynamics data for the generators. Transient stability analysis is most often used to determine if individual generating units will maintain synchronism with the power system following a disturbance (typically a fault). Extended Stability Analysis Extended stability analysis deals with system stability beyond the generating units’ “first swing.” In addition to the generator data required for transient stability analysis, extended stability analysis requires excitation system, speed governor, and prime mover dynamic data. Often, extended stability analysis will also include dynamics data for control devices such as tap changing transformers, switched capacitors, and relays. The addition of these elements to the system model complicates the analysis, but provides comprehensive simulation of nearly all major system components and controls. Extended stability analyses complement small signal stability analyses by verifying the existence of persistent oscillations and establishing the magnitudes of power and/or voltage oscillations. Small Signal Stability Analysis Small signal stability assesses the stability of the power system when subjected to “small” perturbations. Small signal stability uses a linearized model of the power system which includes generator, prime mover, and control device dynamics data. The system of nonlinear equations describing the system are linearized about a specific operating point and eigenvalues and eigenvectors of the linearized system found. The imaginary part of each eigenvalue indicates the frequency of the oscillations associated with the eigenvalue; the real part indicates damping of the oscillation. Usually, small signal stability analysis attempts to find disturbances and/or system conditions that can lead to sustained oscillations (indicated by small damping factors) in the power system. Small signal stability analysis does not provide oscillation magnitude information because the eigenvalues only indicate oscillation frequency and damping. Additionally, the controllability matrices (based on the linearized system) and the eigenvectors can be used to identify candidate generating units for application of new or improved controls (i.e., power system stabilizers and new or improved excitation systems). Transient Analysis Transient analysis involves the analysis of the system (or at least several components of the system) when subjected to “fast” transients (i.e., lightning and switching transients). Transient analysis requires detailed component information which is often not readily available. Typically only system components in the immediate vicinity of the area of interest are modeled in transient analyses. Specialized software packages (most notably EMTP) are used to perform transient analyses. Operational Analyses Several additional analyses used in the day-to-day operation of power systems are based on the results of the analyses described above. Economic dispatch analyses determine the most economic real power output for each generating unit based on cost of generation for each unit and the system losses. Security or contingency analyses assess the system’s ability to withstand the sudden loss of one or more major elements without overloading the remaining system. State estimation determines the “best” estimate of the real-time system state based on a redundant set of system measurements. The Power Flow Problem Power flow analysis is fundamental to the study of power systems. In fact, power flow forms the core of power system analysis. A power flow study is valuable for many reasons. For example, power flow analyses play a key role in the planning of additions or expansions to transmission and generation facilities. A power flow solution
ogEn FIGURE 63.1 The one line diagram of a power system is often the starting point for many other types of power system analyses. In addition, power flow analysis and many of its extensions are an essential ingredient of the studies performed in power system operations. In this latter case, it is at the heart of contingency analysis and the implementation of real-time monitoring systems The power flow problem(popularly known as the load flow problem) can be stated as follows For a given power network, with known complex power loads and some set of specifications or restrictions on power generations and voltages, solve for any unknown bus voltages and unspecified generation and finally for the complex power flow in the network Additionally, the losses in individual components and the total network as a whole are usually calculated. Furthermore, the system is often checked for component overloads and voltages outside allowable tolerances Balanced operation is assumed for most power flow studies and will be assumed in this chapter. Consequently the positive sequence network is used for the analysis. In the solution of the power flow problem, the network element values are almost always taken to be in per unit. Likewise, the calculations within the power flow analysis are typically in per unit. However, the solution is usually expressed in a mixed format. Solution voltages are usually expressed in per unit; powers are most often given in kVA or MVA The " given network " may be in the form of a system map and accompanying data tables for the network components. More often, however, the network structure is given in the form of a one-line diagran Regardless of the form of the given network and how the network data are given, the steps to be followed in a power flow study can be summarized as follows 1. Determine element values for passive network components 2. Determine locations and values of all complex power loads. 3. Determine generation specifications and constraints. 4. Develop a mathematical model describing power flow in the network. 5. Solve for the voltage profile of the network. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC is often the starting point for many other types of power system analyses. In addition, power flow analysis and many of its extensions are an essential ingredient of the studies performed in power system operations. In this latter case, it is at the heart of contingency analysis and the implementation of real-time monitoring systems. The power flow problem (popularly known as the load flow problem) can be stated as follows: For a given power network, with known complex power loads and some set of specifications or restrictions on power generations and voltages, solve for any unknown bus voltages and unspecified generation and finally for the complex power flow in the network components. Additionally, the losses in individual components and the total network as a whole are usually calculated. Furthermore, the system is often checked for component overloads and voltages outside allowable tolerances. Balanced operation is assumed for most power flow studies and will be assumed in this chapter. Consequently, the positive sequence network is used for the analysis. In the solution of the power flow problem, the network element values are almost always taken to be in per unit. Likewise, the calculations within the power flow analysis are typically in per unit. However, the solution is usually expressed in a mixed format. Solution voltages are usually expressed in per unit; powers are most often given in kVA or MVA. The “given network” may be in the form of a system map and accompanying data tables for the network components. More often, however, the network structure is given in the form of a one-line diagram (such as shown in Fig. 63.1). Regardless of the form of the given network and how the network data are given, the steps to be followed in a power flow study can be summarized as follows: 1. Determine element values for passive network components. 2. Determine locations and values of all complex power loads. 3. Determine generation specifications and constraints. 4. Develop a mathematical model describing power flow in the network. 5. Solve for the voltage profile of the network. FIGURE 63.1 The one line diagram of a power system
FIGURE 63.2 Off nominal turns ratio transformer flows and losses in the network. 7. Check for constraint violations Formulation of the bus admittance matrix The first step in developing the mathematical model describing the power flow in the network is the formulation of the bus admittance matrix. the bus admittance matrix is an nxn matrix(where n is the number of buses in the system) constructed from the admittances of the equivalent circuit elements of the segments making up the power system. Most system segments are represented by a combination of shunt elements(connected between a bus and the reference node) and series elements(connected between two system buses). Formulation of the bus admittance matrix follows two simple rules: 1. The admittance of elements connected between node k and reference is added to the(k, k)entry of the 2. The admittance of elements connected between nodes j and k is added to the(])and(k, k)entries of the admittance matrix. The negative of the admittance is added to the (i, k)and (k, j) entries of the admittance matrix Off nominal transformers(transformers with transformation ratios different from the system voltage bases at the terminals) present some special difficulties. Figure 63. 2 shows a representation of an off nominal turns ratio transformer The admittance matrix mathematical model of an isolated off nominal transformer is Y -C"Y. Yelv where e is the equivalent series admittance (referred to node j c is the complex(off nominal)turns ratio , is the current injected at node j is the voltage at node j(with respect to reference) Off nominal transformers are added to the bus admittance matrix by adding the corresponding entry of the isolated off nominal transformer admittance matrix to the system bus admittance matrix. Example Formulation of the Power Flow Equations Considerable insight into the power flow problem and its properties and characteristics can be obtained by consideration of a simple example before proceeding to a general formulation of the problem. This simple case will also serve to establish some notation e 2000 by CRC Press LLC
© 2000 by CRC Press LLC 6. Solve for the power flows and losses in the network. 7. Check for constraint violations. Formulation of the Bus Admittance Matrix The first step in developing the mathematical model describing the power flow in the network is the formulation of the bus admittance matrix. The bus admittance matrix is an n×n matrix (where n is the number of buses in the system) constructed from the admittances of the equivalent circuit elements of the segments making up the power system. Most system segments are represented by a combination of shunt elements (connected between a bus and the reference node) and series elements (connected between two system buses). Formulation of the bus admittance matrix follows two simple rules: 1. The admittance of elements connected between node k and reference is added to the (k, k) entry of the admittance matrix. 2. The admittance of elements connected between nodes j and k is added to the (j, j) and (k, k) entries of the admittance matrix. The negative of the admittance is added to the (j, k) and (k, j) entries of the admittance matrix. Off nominal transformers (transformers with transformation ratios different from the system voltage bases at the terminals) present some special difficulties. Figure 63.2 shows a representation of an off nominal turns ratio transformer. The admittance matrix mathematical model of an isolated off nominal transformer is: (63.1) where – Ye is the equivalent series admittance (referred to node j) – c is the complex (off nominal) turns ratio – Ij is the current injected at node j – Vj is the voltage at node j (with respect to reference) Off nominal transformers are added to the bus admittance matrix by adding the corresponding entry of the isolated off nominal transformer admittance matrix to the system bus admittance matrix. Example Formulation of the Power Flow Equations Considerable insight into the power flow problem and its properties and characteristics can be obtained by consideration of a simple example before proceeding to a general formulation of the problem. This simple case will also serve to establish some notation. FIGURE 63.2 Off nominal turns ratio transformer. I I Y cY -c*Y c Y V V j k e e e e j k = − 2