Core loading patterns

Teodora shows Hana a quarter-core map of one of TVA's reactors with each assembly marked F, 1 or 2 for fresh, once-burned or twice-burned. "This is the loading pattern," she says. "Before every refueling outage, we decide where each of 193 assemblies goes. The same fuel arranged differently can change the cycle length by weeks and the peak power by ten percent or more. Finding a good pattern is one of the hardest optimization problems in the industry."

What a loading pattern controls

A loading pattern is the arrangement of fuel of different reactivities in the core. Its goals pull in different directions. Placing the most reactive fuel where neutrons are least likely to leak, near the center, gives the best neutron economy and the longest cycle, but tends to concentrate power and raise the peaking factor. Placing fresh fuel near the edge flattens the power but wastes neutrons through leakage and exposes the reactor vessel to more fast flux. A good pattern meets every safety limit on power peaking and reactivity coefficients with margin, achieves the required cycle length and burnup, and does so with the least fuel cost.

Three classic strategies

Batch-loaded cores have used three broad strategies over the history of the industry. In an out-in pattern, fresh fuel is loaded at the periphery and moved inward in later cycles. This flattens the radial power shape, because the highly reactive fresh fuel offsets the leakage at the edge, but it sends many neutrons into the reflector and vessel. In a scatter or checkerboard pattern, fresh and burned assemblies are mixed throughout the core, so that each fresh assembly is surrounded by burned ones that absorb some of its excess neutrons. In a low-leakage pattern, fresh fuel is placed mostly in the interior, and twice-burned fuel, with the lowest reactivity, is placed at the periphery to act partly as a reflector.

Since the 1980s, low-leakage patterns have become standard for light-water reactors. They extend the cycle by improving neutron economy and reduce the fast flux on the reactor vessel, which slows embrittlement. Their disadvantage is higher power in the fresh assemblies in the interior, which must be controlled with burnable absorbers. The development of integral burnable absorbers and gadolinia rods, described in lesson 6, is what made low-leakage patterns practical.

A one-dimensional loading study

The program below compares three patterns in a one-dimensional, one-group model of a core of eight zones from center to edge, with a reflector outside. Each zone holds fresh, once-burned or twice-burned fuel with an illustrative k_inf. The program solves for k and the relative power in each zone.

# Core loading patterns in a one-dimensional, one-group model of a core with a
# reflector. Each of 8 fuel zones (center to edge) holds fresh (F), once-burned (1)
# or twice-burned (2) fuel. Batch k_inf values are illustrative. We compare the
# radial power peaking and leakage for three common strategies.
def run(pattern, kinf={'F': 1.20, '1': 1.10, '2': 1.02}, width=20.0, refl=3, n_per=6):
    zones = list(pattern) + ['R'] * refl
    D, sa, nsf = [], [], []
    for z in zones:
        for _ in range(n_per):
            if z == 'R': D.append(1.2); sa.append(0.004); nsf.append(0.0)
            else: D.append(1.5); sa.append(0.008); nsf.append(0.008 * kinf[z])
    n = len(D); h = width / n_per
    phi, k = [1.0] * n, 1.0
    for _ in range(400):
        src = [nsf[i] * phi[i] / k for i in range(n)]
        # tridiagonal: reflective (symmetry) at center, zero flux at outer edge
        a = [0.0] * n; b = [0.0] * n; c = [0.0] * n
        for i in range(n):
            Dw = 2 * D[i] * D[i - 1] / (D[i] + D[i - 1]) if i else 0.0
            De = 2 * D[i] * D[i + 1] / (D[i] + D[i + 1]) if i < n - 1 else 2 * D[i]
            a[i] = -Dw / h**2; c[i] = -De / h**2 if i < n - 1 else 0.0
            b[i] = (Dw + De) / h**2 + sa[i]
        cp, dp = [0.0] * n, [0.0] * n
        for i in range(n):
            m = b[i] - (a[i] * cp[i - 1] if i else 0.0)
            cp[i] = c[i] / m; dp[i] = (src[i] - (a[i] * dp[i - 1] if i else 0.0)) / m
        new = [0.0] * n
        for i in reversed(range(n)):
            new[i] = dp[i] - (cp[i] * new[i + 1] if i < n - 1 else 0.0)
        k = k * sum(nsf[i] * new[i] for i in range(n)) / sum(nsf[i] * phi[i] for i in range(n))
        phi = new
    nf = len(pattern) * n_per
    power = [nsf[i] * phi[i] for i in range(nf)]
    zp = [sum(power[j * n_per:(j + 1) * n_per]) for j in range(len(pattern))]
    avg = sum(zp) / len(zp)
    return k, [p / avg for p in zp]
for name, pat in (("out-in (fresh at the edge)", "22211FFF"),
                  ("checkerboard scatter",       "F2F1F2F1"),
                  ("low-leakage (fresh inside)", "2F1F1F22")):
    k, rel = run(pat)
    print(f"{name:28s} k = {k:.4f}  peak zone power = {max(rel):.2f}  zones: " + " ".join(f"{x:.2f}" for x in rel))

out-in (fresh at the edge)   k = 1.0876  peak zone power = 1.53  zones: 0.51 0.57 0.72 1.01 1.24 1.53 1.42 1.00
checkerboard scatter         k = 1.1191  peak zone power = 1.59  zones: 1.59 1.22 1.38 1.13 1.05 0.68 0.61 0.34
low-leakage (fresh inside)   k = 1.1101  peak zone power = 1.40  zones: 1.07 1.40 1.30 1.38 1.09 0.94 0.52 0.29

The out-in pattern has the lowest k, because the fresh fuel sits where leakage is highest, but its power peaks near the edge. The checkerboard pattern, which here puts fresh fuel at the center, has the highest k but also a high central peak. The low-leakage pattern has an intermediate k and the lowest peak. In this crude model, the low-leakage pattern wins on peaking; in real cores with burnable absorbers, it wins on both peaking and cycle length, which is why it is the industry standard.

Loading patterns in a CANDU core, for comparison

A CANDU core is never reloaded all at once, so its loading pattern takes a different form. The core is divided into regions, typically an inner region and an outer region, with different target exit burnups. Fuel in the inner region is left in the channel longer, reaching higher burnup and lower reactivity, while the outer region is refueled more often. This differential fueling flattens the radial power distribution in the same way as a low-leakage pattern does in a light-water reactor, but it is achieved by the timing of refueling rather than the placement of fresh assemblies. Adjuster rods, placed near the center of the core, add further flattening. The result is a radial power distribution flat enough that a large fraction of channels run near the maximum channel power.

The direction of refueling also matters. In a CANDU reactor, adjacent channels are refueled in opposite directions, so that fresh fuel enters at one face of the core in one channel and at the other face in its neighbors. This bidirectional fueling makes the axial power distribution symmetric when averaged over the core, even though each channel individually has fresh fuel at one end and highly burned fuel at the other.

Searching for the best pattern

For a 193-assembly core with eighth-core symmetry, the number of possible arrangements of fuel is astronomically large, far too many to evaluate one by one. Designers reduce the search with experience, rules and computer optimization. Rules include keeping fresh assemblies from touching each other face to face, placing twice-burned fuel at the periphery and keeping the highest-burnup assemblies away from the core center. Optimization methods include simulated annealing, genetic algorithms and more recent machine learning methods, all of which use a fast core simulator to evaluate thousands of candidates against the objectives and constraints. The final pattern is always checked with the full licensing methods.

At TVA, as at most utilities, reload design starts about a year before each refueling outage. The designers receive the projected end-of-cycle burnups of the current core, the number and enrichment of fresh assemblies planned, and any assemblies that must be discharged for inspection or damage. They search for a pattern, then confirm it with licensing calculations of peaking, coefficients, shutdown margin and accident analyzes. Lesson 12 follows such a reload design from start to finish.

Shuffling, symmetry and practical limits

Several practical points shape real patterns. Most patterns are designed with quarter-core or eighth-core rotational or mirror symmetry, so that the core simulator can model a fraction of the core and the power distribution is easier to monitor with in-core detectors. Assemblies may be rotated when moved, to even out burnup gradients across them. Some locations, such as those with in-core instruments or neutron sources, have restrictions. Assemblies with known defects or those being kept for inspection programs may be placed in low-power positions or removed. The schedule of the outage itself, which may move dozens of assemblies, must also be planned so that the core remains subcritical and well monitored at every step of fuel movement.

Reading a loading pattern map

A loading pattern is usually shown as a quarter-core or full-core map, with each assembly or channel marked by its batch, burnup, enrichment or burnable absorber loading. Experienced designers read such maps for warning signs: two fresh assemblies sharing a face, a high-burnup assembly at the core center where it would have little reactivity left, a fresh assembly at a corner of the core where its neutrons would leak into the reflector, or an asymmetric arrangement that could tilt the power. The predicted power map is then laid over the loading map to show where peaks occur. Peaks often appear in once-burned assemblies next to fresh ones, because the burned assembly has lost its burnable absorber while still receiving many neutrons from its fresh neighbor.

In a CANDU core, the equivalent map shows each channel's exit burnup, time since last refueling and current power. The fueling engineer reads it for clusters of recently refueled channels, which could raise local power, and for regions where many channels are approaching high burnup at the same time, which would require concentrated refueling in the coming days.

Loading patterns also have to respect practical constraints. Some assemblies have known minor defects or inspection findings that limit where they may go; some locations hold instrumentation that requires particular assembly types; and the refueling sequence must allow the crew to move fuel safely and efficiently. A pattern that is optimal on paper but needs many extra moves may cost outage time, so the final choice balances physics with practicality.