The Load Flow Problem and Bus Classification

Chapter 16 finished with the network in matrix form, \(\mathbf{I}_{bus}=\mathbf{Y}_{bus}\mathbf{V}_{bus}\), and Chapter 17 inverted it. Both are linear relations, and if the injected currents were known the bus voltages would follow in one matrix multiplication. Nothing about the network stands in the way. The obstacle is the data.

Consider what is actually known about a power system at, say, six o'clock on a winter evening. A substation feeds an industrial estate drawing \(40\) MW at \(0.85\) power factor lagging. A generating station is scheduled to deliver \(250\) MW with its terminal voltage held at \(1.02\) per unit. A grid interconnection is contracted for \(150\) MW. Not one of these is a current, and not one is an impedance. Every quantity that is dispatched, metered, contracted or controlled in a power system is a power.

That is not a matter of convention but of physics. A large industrial or domestic load is not a fixed impedance. If the supply voltage sags by five per cent, an induction motor does not obligingly draw five per cent less power — it draws very nearly the same power, because the mechanical torque its shaft is delivering has not changed, and it therefore draws more current. Thermostatically controlled heating behaves the same way over a cycle. A power-electronic converter with a regulated output does so instantaneously and exactly. Over the minutes-long timescale that a load flow study describes, constant power is the right model and constant impedance is not.

Now see what constant power does to the equations. The current injected at bus \(i\) is fixed by the complex power there:

The injected current depends on the bus voltage, which is exactly the quantity being sought. Substituting into the nodal equation gives

It is worth being precise about what the nonlinearity costs, because "nonlinear" is not by itself alarming. It costs three things. There is no closed-form solution, so the answer must be obtained by iteration. There is no guarantee of a unique solution — Section 18-2 exhibits a two-bus system with two perfectly valid voltage answers. And there is no guarantee that a solution exists at all: load a network beyond a certain point and the equations simply have no real solution, which is the mathematical signature of voltage collapse.

Before building the general machinery it pays to solve the smallest possible case completely, because everything that makes the large problem awkward is already visible in it.

Take a generator bus holding \(V_S\) at a fixed magnitude, feeding a single load bus through a line of series reactance \(X\) — the short-line model of Chapter 10, with resistance neglected for clarity. The load draws constant power \(P+jQ\). Choose the receiving-end voltage as the angle reference, so \(V_R = |V_R|\angle 0^\circ\), and the load current is

Walking from the load back to the source through the reactance,

The real part is the sending-end magnitude reduced by the reactive drop; the imaginary part is the quadrature component that produces the angle. Take the squared magnitude of both sides and multiply through by \(|V_R|^2\):

Writing \(u = |V_R|^2\) turns this into an ordinary quadratic:

Three lessons come out of this one quadratic, and all three survive into the \(n\)-bus problem where no closed form exists.

First, the nonlinearity was unavoidable. It entered when the current was written as \(S^{*}/V_R^{*}\); everything after that was algebra. Second, the solution is not unique. The load flow equations of a real system have many solutions, of which one is the sensible operating point, and an iterative method must be started close enough to that one to find it rather than another. Third, and most important for operations, a solution may fail to exist. The discriminant is

At unity power factor this simplifies to \(|V_S|^4 \ge 4X^2P^2\), giving a maximum deliverable power of \(P_{max}=|V_S|^2/2X\), reached when \(u=|V_S|^2/2\), that is, when the receiving voltage has fallen to \(|V_S|/\sqrt2 = 0.707|V_S|\). Beyond that point the two roots have merged and vanished into the complex plane: there is no steady state at all. This is the nose of the \(P\)–\(V\) curve, and passing it is voltage collapse — a load flow that will not converge is often reporting exactly this, and not a numerical failure. Chapter 34 returns to it as a problem of reactive power support.

A bus, or busbar, is the arrangement of conductors that serves as the common connection point for two or more circuits in a substation. Generators feed energy into it, feeders draw energy out of it, and lines carry energy between it and other buses. In the single-line diagram of Chapter 3 it is drawn as a short heavy bar; in the network model of Chapter 16 it is simply a node, and every bus of a system has a single voltage that all equipment connected there shares.

Four real quantities are associated with each bus \(i\):

"Net injected" needs a convention, and the one used throughout is that current and power entering a bus are positive. Generation therefore enters with a plus sign and load with a minus:

A pure load bus has \(P_{Gi}=Q_{Gi}=0\) and therefore a negative \(P_i\); a generator bus supplying more than its local load has a positive one. Care with this sign is the difference between a converged study and a plausible-looking nonsense.

The four quantities are not independent. The complex power at a bus is \(S_i = V_iI_i^{*}\), and the network fixes \(I_i\) once every voltage is known — so \(P_i\) and \(Q_i\) are determined by \(|V_i|\), \(\delta_i\) and the voltages of every other bus. What can be done is to specify some of them and let the equations produce the rest. The bookkeeping is uniform and exact:

Three classes cover every bus in a conventional study, and each corresponds to a real control action.

The load bus, or PQ bus. No generator is connected, so \(P_{Gi}=Q_{Gi}=0\) and the injections are simply the negatives of the load. Both are known from metering or forecasting, and nothing at the bus controls its voltage — the voltage is whatever the network delivers. So \(P_i\) and \(Q_i\) are specified; \(|V_i|\) and \(\delta_i\) are computed. In a real system the overwhelming majority of buses are of this type, often eighty per cent or more.

The generator bus, or PV bus, also called a voltage-controlled bus. Here two independent controls act on the same machine, and they are worth separating because they explain the whole classification. The turbine governor sets how much mechanical power the prime mover delivers, and in the steady state that fixes the real power output \(P_{Gi}\) — the operator dispatches it, and Chapter 31 decides its economic value. The automatic voltage regulator adjusts the field excitation to hold the terminal voltage at a set point, and in doing so it fixes \(|V_i|\). What the excitation actually varies is the reactive output \(Q_{Gi}\), which takes whatever value the network demands and cannot be known in advance. So \(P_i\) and \(|V_i|\) are specified; \(Q_i\) and \(\delta_i\) are computed.

The slack bus, also called the swing or reference bus. One generator bus in the system is treated differently: its voltage magnitude and angle are specified, conventionally \(1.0\angle0^\circ\) or the operating value \(1.05\angle 0^\circ\), and neither \(P\) nor \(Q\) is scheduled. Both are computed at the end, and the machine is understood to supply whatever real and reactive power the rest of the solution leaves over. Section 18-5 explains why such a bus is not a convenience but a necessity.

Two refinements appear in practice. A generator bus may also carry local load; the specified injection is then the net \(P_{Gi}-P_{Li}\), and the machine's own output is recovered afterwards. And a bus with neither generation nor load — a pure junction — is a PQ bus with \(P_i=Q_i=0\), which is exactly the zero-injection condition that made Kron reduction possible in Chapter 16.

The slack bus looks at first like an evasion — one machine exempted from scheduling so that the arithmetic works out. It is not. Two independent arguments force it, and each would require it on its own.

The losses are not known in advance. Conservation of complex power over the whole system says

Suppose every generator's real output were scheduled. Then the left side is fixed, the load is fixed, and the equation asserts a specific value for the loss. But the loss depends on the current in every line, which depends on the bus voltages and angles — which are precisely what the study has not yet computed. In a transmission system the loss is a few per cent of the load, small but never negligible and never known beforehand. If every generator is scheduled, the equations are over-specified and generally have no solution. Leaving one machine's real power free lets it take up the difference: the slack bus supplies the losses, whatever they turn out to be.

A generator bus is chosen for this role rather than a load bus for the obvious reason that only a generator can absorb the assignment — it must be able to increase its output to cover the shortfall. In practice the largest machine, or the one with the most headroom, is selected, since it must be able to swing without hitting a limit. Where no slack is nominated, a program will normally pick the generator bus with the largest scheduled real power.

An angle datum is needed. The second argument is independent of losses and would apply even to a lossless network. Examine the load flow equation of Section 18-1: every voltage enters as \(V_i^{*}V_k\), so shifting every angle in the system by the same amount \(\alpha\) leaves \(\delta_k-\delta_i\) unchanged and therefore leaves every injected power unchanged. The solution is determined only up to an arbitrary common rotation. To pin it down, one angle must be declared to be zero, and it is natural to declare it at the same bus that is already exempt from real-power scheduling. Hence \(\delta_{\text{slack}}=0^\circ\), and every other angle in the converged solution is measured relative to it.

The equations now need to be written in a form that can be worked with. Start from the current injected at bus \(i\), which Chapter 16 gave as a row of the admittance matrix:

and combine it with \(S_i = V_iI_i^{*}\). It is slightly more convenient to conjugate the whole product, so that the sum is left untouched:

Now introduce polar forms, \(V_i=|V_i|\angle\delta_i\) and \(Y_{ik}=|Y_{ik}|\angle\theta_{ik}\). A single term of the sum becomes

because magnitudes multiply and angles add. Separating the real and imaginary parts of the whole sum, and remembering that the imaginary part of \(P_i-jQ_i\) is \(-Q_i\), gives the equations in the form the professor's lectures and most textbooks use:

An equivalent form splits the admittance into its rectangular parts, \(Y_{ik}=G_{ik}+jB_{ik}\), and writes the angle difference as \(\delta_{ik}=\delta_i-\delta_k\). Expanding the same product and collecting terms:

The two forms are the same equations; the second is the one differentiated in Chapter 20 to build the Newton-Raphson Jacobian, because \(G\) and \(B\) are stored directly and no polar conversion is needed at every iteration.

The rectangular form also exposes a structural fact that dominates the rest of Part 4. In a transmission network \(X \gg R\), so \(|B_{ik}|\gg|G_{ik}|\); and angle differences across a line are small, so \(\cos\delta_{ik}\approx1\) and \(\sin\delta_{ik}\approx\delta_{ik}\). Under those approximations \(P_i\) is governed by the \(B\sin\delta\) terms and \(Q_i\) by the \(B\cos\delta\) terms — real power follows angle, reactive power follows magnitude. This near-decoupling is what the fast decoupled method of Chapter 20 exploits, and it is the reason an operator adjusts governors to redistribute megawatts and excitation or capacitors to correct voltages.

The power flow equations offer \(2n\) relations in \(2n\) unknowns, but not all of them are used, because the bus classification has already declared some of the unknowns to be known. Counting properly is the check that the problem is correctly posed, and it is a standard examination question.

Let the system have \(n\) buses, of which one is the slack, \(n_g\) are PV buses and the remaining \(n_p = n-1-n_g\) are PQ buses.

The unknowns. At each PQ bus both \(|V|\) and \(\delta\) are unknown: \(2n_p\) quantities. At each PV bus the magnitude is fixed and only the angle is unknown: \(n_g\) quantities. At the slack bus nothing is unknown. Total:

The equations. A power flow equation is usable only where the power on its left-hand side is a specified number. Real power is specified at every bus except the slack, giving \(n-1\) equations in \(P\). Reactive power is specified only at the PQ buses, giving \(n_p\) equations in \(Q\); at a PV bus the reactive injection is an unknown, so that equation carries no information and is set aside until the end, when it is used to compute \(Q\). At the slack bus both equations are set aside. Total:

Having as many equations as unknowns does not make the system solvable in closed form — that was settled in Section 18-1 — but it does confirm that the problem is neither over- nor under-specified, and it tells the analyst how large the iteration is going to be. For a \(300\)-bus system with \(40\) generators the count is \(2(300)-2-39 = 559\), and the Jacobian is \(559\times559\): large, but sparse in exactly the pattern inherited from \(\mathbf{Y}_{bus}\).

The data a load flow study needs is short and entirely conventional. Line data — from-bus, to-bus, \(R\), \(X\) and total charging susceptance for each circuit, all in per unit on the common base of Chapter 4 — feeds the construction of \(\mathbf{Y}_{bus}\) by the rules of Chapter 16. Transformer data adds the leakage impedance and the off-nominal tap. Bus data lists, for each bus, its type and the two quantities specified there, together with an initial estimate of the two that are not.

That initial estimate matters. The universal default is the flat start: every unknown magnitude set to \(1.0\) per unit and every unknown angle to zero. It is a good guess precisely because a well-designed power system operates near nominal voltage with small angle spreads, so the flat start is already close to the answer, and — a point worth noticing — it is close to the high-voltage solution of Section 18-2 rather than the low-voltage one. At PV buses the given magnitude is used instead of \(1.0\), which is closer still.

With voltages estimated, the equations of Section 18-6 can be evaluated to give the powers those voltages would produce. Comparing them with the powers that were specified gives the mismatch:

Mismatches are computed at every bus where the corresponding power is specified — \(\Delta P\) at all non-slack buses, \(\Delta Q\) at PQ buses only — which is the same count arrived at in Section 18-7, as it must be. They are the residuals of the nonlinear system, and every solution method is a rule for adjusting the voltages so that they shrink. Gauss-Seidel (Chapter 19) rearranges each equation to update one voltage at a time; Newton-Raphson (Chapter 20) linearises the mismatch about the present iterate and solves for the correction. Neither changes what a mismatch is.

The iteration stops when the residuals are small enough:

On a \(100\) MVA base, a tolerance of \(10^{-4}\) per unit is \(10\) kW — far below the accuracy of any load forecast, so tightening it further buys nothing physical. A tolerance on the voltage change per iteration is sometimes used instead, but it is the weaker test: an iteration that is converging slowly can produce small voltage changes while the mismatches are still large, and a method may then report success prematurely. Testing the mismatch tests the equations themselves.

Two failures must be distinguished when the iteration does not converge. It may be diverging numerically — a poor starting point, a data error such as an impedance entered in ohms instead of per unit, or an isolated bus that makes \(\mathbf{Y}_{bus}\) singular. Or the case may genuinely have no solution, the system having been loaded past the nose of Section 18-2. The two look identical on screen, and telling them apart is a matter of re-running with the load scaled down: if convergence returns at eighty per cent of load, the original case was beyond the limit and no amount of numerical care will recover it.

The converged solution is a list of \(|V_i|\) and \(\delta_i\), which is not yet what anyone wanted to know. The quantities of interest are recovered from it by direct calculation, with no further iteration.

Slack bus output and PV bus reactive output. The equations set aside in Section 18-7 are now evaluated with the known voltages, giving \(P\) and \(Q\) at the slack bus and \(Q\) at every PV bus.

Line flows. Take a line between buses \(i\) and \(k\), modelled by the nominal \(\pi\) of Chapter 11 with series admittance \(y_{ik}\) and half-charging admittance \(y_{ik}^{sh}\) at each end. The current leaving bus \(i\) along that line has a series part and a charging part:

The same calculation from the other end gives \(S_{ki}\), and the two do not simply differ in sign — the difference between them is what the line consumes:

For a line with no charging this reduces to \(|I_{ik}|^2(R_{ik}+jX_{ik})\), so the real loss and the reactive loss stand in the ratio \(X/R\) — a check worth applying to every line in a hand calculation. With charging included the reactive term may come out negative, the line generating more reactive power than it absorbs, which is the Ferranti behaviour of Chapter 14 appearing in load flow output. Summing over all lines gives the system loss, which must equal the total generation minus the total load.

Limits. A converged load flow is not automatically an acceptable one. Three checks follow every study. Bus voltages must lie within their permitted band, typically \(0.95\) to \(1.05\) per unit. Line and transformer flows must not exceed thermal ratings. And the reactive output computed at each PV bus must lie within what the machine can actually deliver.

That last one feeds back into the solution itself, and it is the one refinement to the bus classification that every practical program implements. A synchronous generator's reactive capability is bounded — by field heating above and by stator-end heating and steady-state stability below — so the study carries limits \(Q_i^{min} \le Q_i \le Q_i^{max}\) at every PV bus. If the reactive power computed during the iteration exceeds a limit, the machine cannot hold its scheduled voltage: the AVR has run out of excitation. The correct response is to hold the reactive output at the violated limit and let the voltage go:

The voltage at a bus that has switched will settle below its set point if the upper limit was hit, and above it if the lower limit was hit. Either outcome is a genuine result and not a numerical artefact: it is the study reporting that the reactive support at that point of the system is insufficient. The remedy is a shunt capacitor, a tap change, or a static VAr compensator — the subject matter of Chapters 34 and 38.

All quantities are per unit on a \(100\) MVA base unless stated. Take the injection sign convention of Section 18-3 throughout: power entering a bus is positive.