How many readings a maximum takes

The load a curve recommends found that the maximum power theorem survives a photovoltaic panel only with the incremental resistance in place of the internal one: at the maximum the load $V/I$ equals $-dV/dI$, 3.6717 ohms on both sides for the solved panel, while the straight-line estimate $V_{oc}/I_{sc}$ says 3.9709. The condition is $I + V\,dI/dV = 0$, and it is satisfied at the answer rather than being a recipe for finding it.

It closed on the recipe. Every maximum-power tracker reaches the fixed point by measuring the panel — moving its operating point, reading the power or the current, moving again — and what a solve can say that a tracker cannot is how many readings the fixed point costs, and from how far away a method can start before the shape of the characteristic breaks it. That is a question about algorithms on a curve, and the curve is here.

The panel and the three trackers #

The panel is the earlier essays’ single-diode model: a photocurrent of five amperes, a junction of twenty-four cells, 0.02 ohms in series and two hundred ohms across, its current at each terminal voltage found by bisection on the implicit diode equation. Its maximum, golden-sectioned on the power itself, is 80.99 watts at 17.244 volts.

Every tracker starts at open circuit, or just below it, because that is where a panel sits before anything draws on it. The three are the ones in use. Perturb-and-observe moves the voltage by a fixed step, reads the power, and reverses whenever the power fell. Newton’s method reads the current and its slope at the present voltage, forms $g = I + V\,dI/dV$, and moves to where a straight line through $g$ with its own slope would cross zero; each step needs the current and the slope, counted as two readings. Fractional open-circuit reads the open-circuit voltage once and sets the operating point at a fixed fraction of it, here 0.8686, the fraction at which this panel’s maximum sits in full light.

Thirty-nine readings and an oscillation #

Perturb-and-observe in 0.2-volt steps walks from 19.75 volts down to the maximum at a step a reading, so it needs about thirteen readings just to cover the distance, and it is within two parts in a thousand only from reading 39, because the last approach is made with a step that overshoots and comes back. It never stops. Settled, it visits the point nearest the maximum, steps across it, finds the power fell, steps back, steps the other way, and repeats.

The loss in the oscillation has a closed form. Settled, the tracker visits the point nearest the maximum twice in every four readings and the points a step either side once each, so its mean square distance from the maximum is $\delta^2/2$ plus the square of where its grid happens to sit, $\delta^2/12$ on average. Near the maximum the power is a parabola with curvature $P'' = -6.20$ watts per volt squared, so the loss is $\tfrac12|P''|$ times that mean square: $7|P''|\delta^2/24P_{max}$. Averaged over sixty-four starting voltages the measured loss sits within four per cent of the law at every step up to half a volt. At a volt it runs nine per cent above, because a volt either side of the maximum is outside the parabola: the characteristic bends faster on its high side.

The law is the design rule. The loss falls as the square of the step and the readings needed to walk in from open circuit rise as one over it, so halving the step quarters the steady loss and doubles the walk. A step of 0.1 volts on this panel costs two parts in ten thousand forever; a step of half a volt costs six parts in a thousand and arrives in a quarter of the readings. No step is good at both, which is why practical trackers shrink their step as they close in.

Ten readings, and the digits double #

Newton’s method needs the slope of $g$ as well as $g$, and with the panel’s slope known in closed form — $dI/dV = -G/(1 + R_sG)$, $G$ the junction’s and the shunt’s conductance together — its convergence is the textbook’s. From 18.5 volts the distance from the answer goes 1, 0.6, 0.2, 0.02, $2\times10^{-4}$, $4\times10^{-8}$: once it is close, each error is about a fixed multiple of the square of the one before, and the number of correct digits doubles every step. From open circuit it is within a part in a million in five steps, ten readings, against perturb-and-observe’s thirty-nine to reach two parts in a thousand.

The root it converges to is a second route to the maximum. The golden section finds the voltage where the power is largest by comparing powers; Newton finds the voltage where the stationary condition is zero without comparing any powers at all. They agree to $7.4\times10^{-8}$ volts, which is the golden section’s own resolution on a maximum that is flat to second order — Newton’s root is the more precise of the two, because it finds a zero rather than a flat top.

The window Newton converges from #

Newton is fast only from somewhere near the answer, and the essay that raised the question asked how far away it may start. The answer is lopsided.

Across most of its range the panel is a current source: its current hardly changes with voltage, so $V\,dI/dV$ is small and $g$ is nearly the short-circuit current, 4.91 amperes at half the maximum’s voltage, and nearly flat. Newton divides $g$ by its slope, so from the flat it takes a step of five amperes over a few milliamperes a volt — hundreds of volts — and leaves the characteristic in one step. Near the maximum $g$ falls steeply through zero and then plunges toward open circuit, and there its own curvature carries every step inward.

The window is everything above 15.82 volts, 1.42 volts below the maximum, up to open circuit: 4.03 volts of the panel’s 19.85. A tracker that starts at open circuit, as a panel does, is inside it and converges in five steps. A tracker that resumes from a point left over from a cloud passing, when the maximum has moved up with returning light, can be outside it and must fall back to something slower. The amplitude nothing linear predicts marched an oscillator with Newton’s method at every time step and never met this, because a time step starts each solve beside its answer; a tracker that has lost its place does not.

Why the window is lopsided #

The window runs from 1.42 volts below the maximum all the way up to open circuit, four volts above it on one side and less than a volt and a half on the other, and the asymmetry is the shape of $g$. Above the maximum the junction’s exponential dominates: $g$ falls steeply and curves so that a straight line from any point on it crosses zero short of the root, and Newton approaches from above one side at a time — from 19.5 volts the errors are 2, 1, 0.7, 0.3, 0.04, falling monotonically with no overshoot. Below the maximum the curvature has the other sign, so a step from slightly below overshoots to the far side, and from there the descent from above takes over. Further below, $g$ is flat at the short-circuit current, and the step is wild.

A tracker that knows this approaches from the side Newton likes. Starting at open circuit, as a panel does, is starting on the good side; resuming from a voltage below the maximum, after a dip in the light has dragged the operating point down, is starting on the bad side. The method’s speed and its fragility are the same fact about one curve read from its two sides, and a damped Newton — one that refuses any step larger than a volt — trades a little of the speed for convergence from the whole range, because from the flat it then walks rather than jumps.

Incremental conductance is Newton without its step #

The tracker sold as “incremental conductance” compares the measured $dI/dV$ with $-I/V$, which is exactly the sign of $g$, and moves a fixed step in the direction the sign indicates. It reads the same quantity Newton reads and throws away its size, keeping only its sign. That makes it as robust as perturb-and-observe — it works from anywhere, since the sign of $g$ is right everywhere — and, in exact arithmetic, it stops dead when $g$ is zero rather than oscillating, because it does not need to step past the maximum to know it has arrived.

In practice it oscillates too: $g$ is read from noisy differences and is never exactly zero, so the tracker dithers by a step about the root, and its steady loss is the same $7|P''|\delta^2/24$ as perturb-and-observe’s with the same step. Newton is the version that uses the size of $g$ as well as its sign, and it is ten times faster for it from inside the window. One step, computed twice found a numerical method’s error falling as a power of its step set by how many terms it cancels; trackers rank the same way, with the sign-only methods first order and Newton second.

A worked system #

A small off-grid charger uses an eighty-watt panel of this kind into a twelve-volt battery through a converter, and spends a reading every millisecond. From a cold start at dawn the panel is at open circuit. A fixed fraction puts the operating point within a few tenths of a per cent of the maximum in the first reading. A perturb-and-observe tracker with a 0.2-volt step then settles in forty milliseconds and gives up nine parts in ten thousand for the rest of the day — seven hundredths of a watt, continuously. Newton from the fixed fraction converges in ten milliseconds and gives up nothing measurable, until a cloud moves the maximum and the operating point finds itself outside the window, at which point it must fall back to stepping.

So the system that would be built uses all three, in the order the figures recommend, and spends its complexity on deciding which to use when. The load for a source that will not hold still found a conjugate match losing a known fraction as the frequency moved; a tracker’s equivalent is the fraction it loses as the light moves, and the next measurement is how fast the light may move before the slow methods’ losses overtake the fast one’s fallbacks.

One reading, and a fraction that depends on the light #

The one-reading rule is exact at the light it was set at and wrong elsewhere, because the maximum’s fraction of the open-circuit voltage is not a constant. At full light it is 0.8686; at a fifth of full light 0.8634; at a twentieth 0.8408; at a fiftieth 0.6428. The knee of the characteristic softens as the open-circuit voltage falls, and the shunt carries a growing share of a shrinking current. The constant that is a window found a junction’s ideality depending on the decade of current it is read over; a panel in dim light is a junction read in a lower decade, and the maximum’s fraction follows the junction’s shape there. A fraction fixed at full light loses under four parts in ten thousand down to a fifth of full light, 0.83 per cent at a twentieth and 17 per cent at a fiftieth.

It is the cheapest tracker by far — one reading of the open-circuit voltage, which needs the load disconnected for an instant — and the figures say where it belongs: a system that runs mostly in good light and can afford to lose a few per cent in poor light. Combined with either of the others it is the natural first guess. At full light 0.87 of the open-circuit voltage is the maximum itself, deep inside Newton’s window, so one reading followed by Newton reaches the exact maximum in a handful more; how far down in light that remains true depends on how the window moves with the light, which this panel’s figures do not measure.

What the three cost, side by side #

Against readings, the three methods sit in a clear order on this panel. The fixed fraction is one reading and an error that depends on the light. Newton is ten readings to a part in a million from inside its window and useless outside it. Perturb-and-observe is tens of readings, works from anywhere, and never stops costing $7|P''|\delta^2/24$ of the maximum. The practical trackers combine them for exactly these reasons: a fixed fraction to land inside the window, a few Newton or incremental-conductance steps to converge, and a small perturbation to follow the maximum as it drifts.

The load that takes the most found the maximum power theorem for a linear source as an exact match between two resistances. For a panel the match is a condition on a slope, and finding it is an iteration whose cost depends on where it starts. The source that is not a source found the ideal voltage source’s model ending at a current set by a resistance nobody draws; the panel’s model is a current source over most of its range, and that is precisely what breaks Newton there.

What the comparison leaves out #

The panel holds still. A real panel’s characteristic moves with the light and with its temperature while the tracker hunts, and a method that takes forty readings is chasing a target that may have moved by more than its own step; two currents with one name has the junction’s temperature law that moves the open-circuit voltage. The readings are exact. A real tracker reads current and voltage with noise, and Newton’s slope — a difference of two noisy readings — is the most sensitive of the three methods to it; perturb-and-observe’s comparison of two powers is the least. And the panel is uniform. A partly shaded panel with bypass diodes has several local maxima, and every method here converges to whichever one it starts nearest.

The number to keep #

From open circuit, perturb-and-observe in 0.2 V steps takes 39 readings to come within two parts in a thousand of a panel’s maximum and then loses $7|P''|\delta^2/24P_{max}$ for good; Newton on $I + V\,dI/dV$ takes 10 readings to a part in a million, converging quadratically, but only from a window above 15.82 V on a panel whose maximum is at 17.24 V, because below it the condition is nearly the short-circuit current and nearly flat. A fixed fraction of the open-circuit voltage costs one reading and loses nothing at the light it was set at and 17 per cent at a fiftieth of it.

Still open: the maximum that moves, the noisy slope, and the second maximum #

A maximum that moves while it is tracked. The light changes in seconds and the temperature in minutes, and a tracker hunting a moving fixed point has an error that depends on how fast it moves against how many readings the tracker takes per step. Putting a ramp in the photocurrent and finding the rate at which each method’s tracking error exceeds its own steady loss would turn these reading counts into a bandwidth.

A slope read from noise. Newton’s step divides by a slope estimated from two noisy readings, and near the maximum the slope’s change is small. Adding seeded noise to the readings and finding the noise level at which Newton’s quadratic convergence turns into a random walk would say when the fast method is worth its sensitivity.

A shaded panel. Bypass diodes give a partly shaded string several maxima, and only one is the global one. Which of the three methods finds which maximum from open circuit, and what a sweep to find the global one costs in readings, is the comparison that matters on any roof with a chimney.