Load Flow Analysis For Electricity Supply Engineering Essay
Power flow studies, commonly referred to as load flow, are essential of power system analysis and design. Load flow studies are necessary for planning, economic operation, scheduling and exchange of power between utilities. "Load flow study is also required for many other analyses such as transient stability, dynamic stability, contingency and state estimation."
Network equations can be formulated in a variety of forms. However, node voltage method is commonly used for power system analysis. "The network equations which are in the nodal admittance form results in complex linear simultaneous algebraic equations in terms of node currents." The load flow results give the bus voltage magnitude and phase angles and hence the power flow through the transmission lines, line losses and power injection at all the buses.
1.1 BUS Classification
Four quantities are associated with each bus. These are voltage magnitude, phase angle Î´, real power P and reactive power Q. In a load flow study, two out of four quantities are specified and the remaining two quantities are to be obtained through the solutions of equations. The system buses are generally classified into three categories.
Slack bus: Also known as swing bus and taken as reference where the magnitude and phase angle of the voltage are specified. This bus provides the additional real and reactive power to supply the transmission losses, since there are unknown until the final solution is obtained.
Load buses: Also know as PQ bus. At these buses the real and reactive powers are specified. The magnitude and phase angle of the bus voltage are unknown until the final solution is obtained.
Voltage controlled buses: Also known as generator buses or regulated buses or P- buses. At these buses, the real power and voltage magnitude are specified. The phase angles of the voltages and the reactive power are unknown until the final solution is obtained. The limits on the value of reactive power are also specified.
The following table summarizes the above discussion:
1.2 BUS Admittance Matrix
In order to obtain the bus-voltage equations, consider the sample 4-bus power system as shown in Fig. 1.1
1.1 The impedance diagram of sample 4-bus power system
For simplicity resistances of the lines are neglected and the impedances shown in Fig.1.1 are expressed in per-unit on a common MVA base.
Now impedances are converted to admittance, i.e,
Fig.1.2 shows the admittance diagram and transformation to current sources and injects currents at buses 1 and 2 respectively. Node 0 (normally ground) is taken as reference.
1.2 the admittance diagram of 1.1
Applying KCL to the independent nodes 1,2,3,4 we have
Rearranging the above equations, we get
The node equations reduce to
Note that ,in Fig.1.2, there is no connection between bus 1 and bus 4, so
Above equations can be written in matrix form,
or in general
vevtor of the injected currents (the current is positive when flowing into the bus and negative when flowing out of the bus)
Diagonal element of Y matrix is known as self-admittance or driving point admittance, i.e. 1.4
Off-diagonal element of y matrix is known as transfer admittance or mutual admittance, i.e. 1.5
can be obtained from equation (1.3), i.e. 1.6
From Fig.1.2, elements of Y matrix can be written as:
1.3 BUS Loading Equations
Consider i-th bus of a power system as shown in Fig.7.4. transmission lines are represented by their equivalent Ï€ models. is the total charging admittance at bus i.
Fig 1.4: i-th bus of a power system
Net injected current into the bus I can be written as :
Let us define
The real and reactive power injected at bus is is
From equations 7.9 and 7.10 we get
1.4 BUS Impedance Matrix
The bus impedance matrix for en t 1T nodes can be written as
Unlike the bus admittance matrix, the bus impedance matrix cannot be formed by simple examination of the network circuit. The bus impedance matrix can be formed by the following methods:
â-Inversion of the admittance matrix
â-By open circuit testing
â-By step-by-step formation
â-From graph theory
Direct inversion of the Y matrix is rarely implemented in computer applications. Certain assumptions in forming the bus impedance matrix are:
1. The passive network can be shown within a closed perimeter, (Fig.1.3). It includes the impedances of all the circuit components, transmission lines, loads, transformers, cables, and generators. The nodes of interest are brought out of the bounded network, and it is excited by a unit generated voltage
Fig.1.3 Representation of a network as passive elements with loads and faults excluded. The nodes of interest are pulled out of the network and unit voltage is applied at the common node.
2. The network is passive in the sense that no circulating currents flow in the network. Also, the load currents are negligible with respect to the fault currents. For any currents to flow an external path (a fault or load) must exist.
3. All terminals marked 0 are at the same potential. All generators have the same voltage magnitude and phase angle and are replaced by one equivalent generator connected between 0 and a node. For fault current calculations a unit voltage is assumed
1.5 POWER IN AC CIRCUITS
The concepts of instantaneous power, average power, apparent power, and reactive power are fundamental and are briefly discussed here. Consider lumped impedance Z, excited by a sinusoidal voltage E < 0° at constant frequency. The linear load draws a sinusoidal current. The time varying power can be written as
The first term is the average time-dependent power, when the voltage and current waveforms consist only of fundamental components. The second term is the magnitude of power swing. Equation (1.2) can be written as
The first term is the power actually exhausted in the circuit and the second term is power exchanged between the source and circuit, but not exhausted in the circuit.
The active power is measured in watts and is defined as
The reactive power is measured in var and is defined as:
These relationships are shown in Fig. 1.4; cosÎ¸ is called the power factor (PF) of the circuit, and Î¸ is the power factor angle.
The apparent power in VA is given by
The power factor angle is generally defined as
If cosÎ¸=1, Q=0. Such a load is a unity power factor load. Except for a small percentage of loads, i.e., resistance heating and incandescent lighting, the industrial, commercial, or residential loads operate at lagging power factor. As the electrical equipment is rated on a kVA basis, a lower power factor derates the equipment and limits its capacity to supply active power loads. The reactive power flow and control is one important aspect of power flow. The importance of power factor (reactive power) control can be broadly stated as:
â-Improvement in the active power handling capability of transmission lines.
â-Improvement in voltage stability limits.
â-Increasing capability of existing systems: the improvement in power factor for release of a certain per unit kVA capacity can be calculated from Eq. (10.6):
where PFimp is improved power factor, PFext is existing power factor, and
kVAava is kVA made available as per unit of existing kVA.
â-Reduction in losses: the active power losses are reduced as these are proportional to the square of the current. With PF improvement, the current per unit for the same active power delivery is reduced. The loss reduction is
given by the expression:
Where Lossred is reduction in losses in per unit with improvement in
power factor from PFext to PFimp. An improvement of power factor
from 0.7 to 0.9 reduces the losses by 39.5%
â-. Improvement of transmission line regulation: the power factor improvement
improves the line regulation by reducing the voltage drops on load flow.
All these concepts may not be immediately clear and are further developed.
1.5.1 Complex Power
If the voltage vector is expressed as A t jB and the current vector as C t jD, then by convention the volt-ampe`res in ac circuits are vectorially expressed as
E= (A +jB) (C- jD)
= AC +BD +j(BC-AD)
= P+ jQ (1.20)
where P = AC t BD is the active power and Q BC _ AD is the reactive power; I_
is the conjugate of I. This convention makes the imaginary part representing reactive power negative for the leading current and positive for the lagging current. This is the convention used by power system engineers. If a conjugate of voltage, instead of current, is used, the reactive power of the leading current becomes positive. The power factor is given by
1.5.2 Conservation of Energy
The conservation of energy concept (Tellegen's theorem) is based on Kirchoff laws
and states that the power generated by the network is equal to the power consumed by the network (inclusive of load demand and losses). If i1; i2; i3; . . . ; in are the currents and v1; v2; v3; . . . ; vn the voltages of n single-port elements connected in any manner:
This is an obvious conclusion.
Also, in a linear system of passive elements, the complex power, active power,
and reactive power should summate to zero:
1.6 POWER FLOW IN A NODAL BRANCH
The modeling of transmission lines is unique in the sense that capacitance plays a significant role and cannot be ignored, except for short lines of length less than approximately 50 miles (80 km). Let us consider power flow over a short transmission line. "As there are no shunt elements, the line can be modeled by its series resistance and reactance, load, and terminal conditions. Such a system may be called a nodal branch in load flow or a two-port network." The sum of the sending end and receiving end active and reactive powers in a nodal branch is not zero, due to losses in the series admittance Ysr (Fig. 1.5). Let us define Ysr, the admittance of the series elements= - j or Z= zl= l(+j)= + =1/Ysr, where l is the length of the line. The sending end power is
Where is conjugate.This gives
where sending end voltage is Vs < _s and receiving end voltage is Vr < Î¸r. The complex power in Eq. (1.17) can be divided into active and reactive power components: At the sending end:
and, at the receiving end:
If is neglected:
where Î´ in the difference between the sending end and receiving end voltage vector angles= (. For small values of delta, the reactive power equation can be written as
Fig1.5 Power flow over a two-port line.
where is the voltage drop. For a short line it is
Therefore, the transfer of real power depends on the angle Î´, called the transmission angle, and the relative magnitudes of the sending and receiving end voltages. As these voltages will be maintained close to the rated voltages, it is mainly a function of Î´. The maximum power transfer occurs at Î´=90(steady-state stability limit).
The reactive power flows is in the direction of lower voltage and it is independent of Î´. The following conclusions can be drawn:
1. For small resistance of the line, the real power flow is proportional to sin Î´. It is a maximum at Î´=90Ëš. For stability considerations the value is restricted to below Î´=90Ëš. The real power transfer rises with the rise in the transmission voltage.
2. The reactive power flow is proportional to the voltage drop in the line, and is independent of Î´. The receiving end voltage falls with increase in reactive power demand.
2.1 Practical Load Flow
The requirements for load flow calculations vary over a wide area, from small industrial systems to large automated systems for planning, security, reactive power compensation, control, and on-line management. The essential requirements are:
â-High speed, especially important for large systems
â-Convergence characteristics, which are of major consideration for large systems, and the capability to handle ill-conditioned systems.
â-Ease of modifications and simplicity. i.e. adding, deleting, and changing system components, generator outputs, loads, and bus types.
â-Storage requirement, which becomes of consideration for large systems
The size of the program in terms of number of buses and lines is important.
Practically, all programs will have data reading and editing libraries, capabilities of manipulating system variables, adding or deleting system components, generation, capacitors, or slack buses. Programs have integrated databases, i.e., the impedance data for short-circuit or load flow calculations need not be entered twice, and graphic user interfaces. Which type of algorithm will give the speediest results and converge easily is difficult to predict precisely. Table.2.1 shows a comparison of earlier Z and Y matrix methods. Most programs will incorporate more than one solution method. While the Gauss-Seidel method with acceleration is still an option for smaller systems, for large systems some form of the NR decoupled method and fast load-flow algorithm are commonly used, especially for optimal power flow studies. Speed can be accelerated by optimal ordering .In fast decoupled load flow the convergence is geometric, and less than five iterations are required for practical accuracies. "If differentials are calculated efficiently the speed of the fast decoupled method can be even five times that of the NR method. Fast decoupled load flow is employed in optimization studies and in contingency evaluation for system security."
The preparations of data, load types, extent of system to be modeled and specific problems to be studied are identified as a first step. The data entry can be divided into four main categories: bus data, branch data, transformers and phase shifters, and generation and load data. Shunt admittances, i.e., switched capacitors and reactors in required steps, are represented as fixed admittances. Apart from voltages on the buses, the study will give branch power flows; identify transformer taps, phase-shifter angles, loading of generators and capacitors, power flow from swing buses, load demand, power factors, system losses, and overloaded system components.
Digital computer memory requirements
Sparse matrix techniques easily applied to Y matrix
Software programs can basically operate from the same data input
Slow, may not converge at all
Both methods may slow down on large systems
2.2 Y-Matrix Method
The Y-matrix iterative methods were the very first to be applied to load flow calculations on the early generation of digital computers. This required minimum storage, however, may not converge on some load flow problems. This deficiency in Y-matrix methods led to Z-matrix methods, which had a better convergence, but required more storage and slowed down on large systems.
"Some buses may be designated as PQ buses while the others are designated as PV buses. At a PV bus the generator active power output is known and the voltage regulator controls the voltage to a specified value by varying the reactive power output from the generator. There is an upper and lower bound on the generator reactive power output depending on its rating, and for the specified bus voltage, these bounds should not be violated. If the calculated reactive power exceeds generator
Qmax, then Qmax is set equal to Q. If the calculated reactive power is lower than the generator Qmin, then Q is set equal to Qmin."
At a PQ bus, neither the current, nor the voltage is known, except that the load demand is known. A "mixed" bus may have generation and also directly connected loads. The characteristics of these three types of buses are shown in Table 2-1.
Active and reactive power
Current and voltage
Active power and voltage
Current and reactive power
Current, active and reactive power
2.2.1 GAUSS AND GAUSS-SEIDEL Y-MATRIX METHODS
The principal of Jacobi iteration is shown in Fig. 2.1. The program starts by setting initial values of voltages, generally equal to the voltage at the swing bus. In a well-designed power system, voltages are close to rated values and in the absence of a better estimate all the voltages can be set equal to 1 per unit. "From node power constraint, the currents are known and substituting back into the Y-matrix equations, a better estimate of voltages is obtained. These new values of voltages are used to find new values of currents. The iteration is continued until the required tolerance on power flows is obtained." This is diagrammatically illustrated in Fig. 2.1. Starting from an initial estimate of, the final value of x* is obtained through a number of iterations. The basic flow chart of the iteration process is shown in Fig. 2.2
Fig2.1 Illustration of numerical iterative process for final value of a function
Fig. 2.2 Flow chart of basic iterative process of Jacobi-type iterations
2.2.2 Gauss Iterative Technique
Consider that n linear equations in n unknowns () are given. The a coefficients and b dependent variables are known:
These equations can be written as
An initial value for each of the independent variables is assumed. Let these values be denoted by
The initial values are estimated as
These are substituted into Eq. (2.1), giving
These new values of
are substituted into the next iteration. In general, at the k-th iteration:
2.2.3 Gauss-Seidel Y-Matrix Method
In load flow calculations the system equations can be written in terms of current, voltage, or power at the k-th node. We know that the matrix equation in terms of unknown voltages, using the bus admittance matrix for n+ 1 node, is
Although the currents entering the nodes from generators and loads are not known, these can be written in terms of P, Q, and V:
The convention of the current and power flow is important. Currents entering the nodes are considered positive, and thus the power into the node is also positive. A load draws power out of the node and thus the active power and inductive vars are entered as:-p - j (-Q) =-p + j Q. The current is then (-P + j Q)/. The nodal equal of current at the k-th node becomes:
In general, for the k-th node:
The k-th bus voltage at r + 1 iteration can be written as
The voltage at the k-th node has been written in terms of itself and the other voltages.
The first equation involving the swing bus is omitted, as the voltage at the swing bus is already specified in magnitude and phase angle.
The Gauss-Seidel procedure can be summarized for PQ buses in the following steps:
1: Initial phasor values of load voltages are assumed, the swing bus voltage is known, and the controlled bus voltage at generator buses can be specified.
Though an initial estimate of the phasor angles of the voltages will accelerate the final solution, it is not necessary and the iterations can be started with zero degree phase angles or the same phase angle as the swing bus. A flat voltage start assumes 1 + j0 voltages at all buses, except the voltage at the swing bus, which is fixed.
2: Based on the initial voltages, the voltage at a bus in the first iteration is calculated using Eq. (2.2)
3: The estimate of the voltage at bus 2 is refined by repeatedly finding new values of by substituting the value of into the right-hand side of the equation.
4: The voltages at bus 3 are calculated using the latest value of found in step 3 and similarly for other buses in the system.
This completes one iteration. The iteration process is repeated for the entire network
till the specified convergence is obtained.
A generator bus is treated differently; the voltage to be controlled at the bus is specified and the generator voltage regulator varies the reactive power output of the generator within its reactive power capability limits to regulate the bus voltage:
where stands for the imaginary part of the equation. The revised value of is found by substituting the most updated value of voltages:
For a PV bus the upper and lower limits of var generation to hold the bus voltage constant are also given. The calculated reactive power is checked for the specified limits:
If the calculated reactive power falls within the specified limits, the new value of voltage is calculated using the specified voltage magnitude and. This new value of voltage is made equal to the specified voltage to calculate the new phase angle.
If the calculated reactive power is outside the specified limits, then,
This means that the specified limits are not exceeded and beyond the reactive power bounds, the PV bus is treated like a PQ bus. A flow chart is shown in Fig. 2.3
2.3 Newton-Rapson Method
Newton-Raphson method is an iterative method which approximates the set of non-linear simultaneous equations to a set of linear equations using Taylor's series expansion and the terms are restricted to first order approximation.
2.3.1 Simultaneous Equations
The Taylor series is applied to n nonlinear equations in n unknowns,
As a first approximation, the unknowns represented by the initial values can be substituted into the above equations,
where are the first estimates of n unknowns. On transposing
Where is abbreviated as
The original nonlinear equations have been reduced to linear equations in
The subsequent approximations are
Or in matrix form:
The matrix of partial derivatives is called a Jacobian matrix. This result is written as
This means that determination of unknowns requires inversion of the Jacobian
2.3.2 Rectangular Form of Newton-Rapson Method of Load Flow
The power flow equation at a PQ node is
Voltage can be written as:
Thus, the power is
Equating the real and imaginary parts, the active and reactive power at a PQ node is:
where and are functions of and . Starting from the initial values, new values are found which differ from the initial values by and
For a PV node (generator bus) voltage and power are specified. The reactive power equation is replaced by a voltage equation:
2.3.3 Polar Form of Jacobian Matrix
The voltage equation can be written in polar form:
Thus the power is
Equating real and imaginary terms:
The Jacobian in polar form for the same four-bus system is
The slack bus has no equation, because the active and reactive power at this bus is unspecified and the voltage is specified. At PV bus 4, the reactive power is unspecified and there is no corresponding equation for this bus in terms of the variable.
The partial derivatives can be calculated as follows:
2.3.4 Calculation Procedure of Newton-Raphson Method
The procedure is summarized in the following steps, and flow charts are shown in Figs 2.4 and 2.5
â-Bus admittance matrix is formed.
â-Initial values of voltages and phase angles are assumed for the load (PQ) buses. Phase angles are assumed for PV buses. Normally, the bus voltages are set equal to the slack bus voltage, and phase angles are assumed equal to 0Ëš, i.e., a flat start.
â-Active and reactive powers, P and Q, are calculated for each load bus
â-P and Q can, therefore, be calculated on the basis of the given power at the buses
â-For PV buses, the exact reactive power are not specified, but its limits are known. If the calculated value of the reactive power is within limits, only P is calculated. If the calculated value of reactive power is beyond the specified limits, then an appropriate limit is imposed and Q is also calculated by subtracting the calculated value of the reactive power from the maximum specified limit. The bus under consideration is now treated as a PQ (load) bus.
â-The elements of the Jacobian matrix are calculated
â-This gives and
â-Using the new values ofand, the new values of voltages and phase angles are calculated.
â-The next iteration is started with these new values of voltage magnitudes and phase angles.
â-The procedure is continued until the required tolerance is achieved. This is generally 0.1kW and 0.1 kvar.
Fig 2.4 Flow chart for NR method of load flow for PQ buses.
Fig.2.5Flow chart for NR method of load flow for PV buses
2.3.5 Impact Loads and Motor Starting
Load flow presents a frozen picture of the distribution system at a given instant, depending on the load demand. While no idea of the transients in the system for a sudden change in load application or rejection or loss of a generator or tie-line can be obtained, a steady-state picture is presented for the specified loading conditions.
Each of these transient events can be simulated as the initial starting condition, and the load flow study rerun as for the steady-state case. Suppose a generator is suddenly tripped. Assuming that the system is stable after this occurrence, we can calculate the redistribution of loads and bus voltages by running the load flow calculations afresh, with generator 4 omitted. Similarly, the effect of an outage of a tie-line, transformer, or other system component can be studied.
Table 2-2 Representation of Load Models in Load Flow
Load flow is a solution of the steady-state operating conditions of a power system. It presents a ''frozen'' picture of a scenario with a given set of conditions and constraints. This can be a limitation, as the power system's operations are dynamic. In an industrial distribution system the load demand for a specific process can be predicted fairly accurately and a few load flow calculations will adequately describe the system. For bulk power supply, the load demand from hour to hour is uncertain, and winter and summer load flow situations, though typical, are not adequate. A moving picture scenario could be created from static snapshots, but it is rarely adequate in large systems having thousands of controls and constraints. Thus, the spectrum of load flow (power flow) embraces a large area of calculations, from calculating the voltage profiles and power flows in small systems to problems of on-line energy management and optimization strategies in interconnected large power systems.
By the load flow studies which performed using digital computer simulations. I have a main idea of how a power network's power flow calculation operation, planning, running, and development of control strategies. Applied to large systems for optimization, security, and stability, the algorithms become complex and involved. While the study I have done above just a small part of the research and I think the treatment of load flow, and finally optimal power flow, will unfold in my following study.
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