How to optimize hydro storage and trading into multiple electricity markets?

Results

The model is handed published prices, so the question is not whether it reproduces them but whether the schedule it builds resembles the one the system actually ran. Three things can be checked against something published: the prices it realises, its monthly volumes, and — for Uruguay alone — the water value itself.

British Columbia. Its monthly hydro shape lands within 0.79 percentage points of StatCan's, inside the 1.05 points by which 2022 and 2023 differ from each other — and the within-year shape is the one series no input could have fitted, since only the annual totals are measured. Monthly net exports correlate at 0.582, at −9.1 TWh against a province-wide −7.8 TWh. Price is where it is plainly beaten: it sells into the US at $58.32/MWh against BC's realised $85.52, and buys at $23.35 against $33.87. Three markets cannot reach most of the counterparties BC actually trades with, and the capture price is where that shows up. The actuals here are province-wide rather than BC Hydro's alone, so read the levels as indicative and the shapes as the test.

Uruguay is the case with a measured counterpart for the payoff itself. Its hourly price tracks ADME's sanctioned spot at r = 0.853, averaging $90.88/MWh against $87.61, and monthly net interchange correlates at 0.829. The water value — the number the model exists to produce — tracks ADME's own published one at r = 0.938, but sits $28.80/MWh under it on average: through the drought the model holds water too cheaply. Right shape, wrong level, is the honest summary.

British Columbia's run returns a trading margin of $116M over the year. There is no reported figure for the same year to set beside it: BC Hydro's F2025 trade income was removed from the Revenue Requirements line by regulation, and the filing's system-trade schedule stops a year earlier. The nearest published comparator is the regulator's own five-year average of Powerex trade income, about $399M — a normal year rather than this one, and a figure the model comes in well under, which is the direction to expect from a three-market system that cannot reach most of the counterparties BC actually sells to.

Uruguay has no published margin at all, and in 2023 the model's objective is −$150M — a cost rather than a profit, once thermal generation is paid for in a drought year. Neither figure is what the model is scored on; the comparisons above are.

Across 8,760 hours British Columbia's water value never moves. It is one number, $41.61/MWh, for the whole year, while the system's own marginal price runs from −$34.00 to $155.20. A reservoir this large is a flywheel: it absorbs almost everything the market does, and the opportunity cost of water never notices.

That is the useful finding, and it is a trading rule rather than a curiosity. For 773 hours of the year the marginal price sits above that value. Those are the hours when releasing water is unambiguously right, and they can be identified without re-running anything.

It never moves because it is never forced to. Storage touches neither the ceiling nor the floor in any of the 8,760 hours — it runs between 6.8 and 20.8 TWh inside walls of 3 and 25 — so nothing in the year prices the water except the requirement that it end where it began. Move that target and the whole level moves with it, which is why the storage path is a consequence of an assumption rather than a prediction of one. Uruguay, where the reservoir really was scarce, is the counter-example: there the same dual moves across the year.

Methodology

The LP model is a stylized representation of hydro systems that trade into neighboring markets. Results are illustrative rather than predictive. The model objective is to maximize annual trading margin over 8,760 hourly periods. This is achieved by scheduling hydro generation, other generation, imports and exports, and storage levels while respecting physical and operational constraints. In total, the model has 78,840 variables, and a full year solves in 1.8 to 4.5 seconds across the five full-year runs behind this case study and its companions.

Model formulation

The model formulation is:

subject to StorageBalance {t in T}:
    S[t] = (if t = 1 then S0 else S[t-1]) + inflow[t] - G[t] - Spill[t];

subject to EnergyBalance {t in T}:
    G[t] + sum {u in U} Th[u,t] + surplus[t] - Curtail[t]
      + sum {m in M} Y[m,t] = load[t] + sum {m in M} X[m,t];

subject to PathExport {p in P, t in T}:
    sum {m in M: path_of[m] = p} X[m,t] <= path_exp_cap[p];
subject to PathImport {p in P, t in T}:
    sum {m in M: path_of[m] = p} Y[m,t] <= path_imp_cap[p];

subject to RampUp {t in T: t > 1 or has_prev = 1}:
    G[t] - (if t = 1 then G_prev else G[t-1]) <= ramp_max;
subject to RampDown {t in T: t > 1 or has_prev = 1}:
    (if t = 1 then G_prev else G[t-1]) - G[t] <= ramp_max;

var Short >= 0, <= S_end_min;
subject to EndStorage:
    S[N] + Short >= S_end_min;

var Below >= 0, <= reserve;
subject to ReserveFloor {t in T}:
    S[t] + Below >= Smin + reserve;

var Curtail {t in T} >= 0, <= surplus[t];

var Th {u in U, T} >= 0, <= th_cap[u];

subject to OfferLink {m in M, t in T: card(K) > 0}:
    X[m,t] = sum {k in K} Xk[m,k,t];
subject to BidLink {m in M, t in T: card(KB) > 0}:
    Y[m,t] = sum {kb in KB} Yk[m,kb,t];

subject to TransitLimit {t in T}:
    sum {m in M} Y[m,t] - load[t] <= transit_cap;

var Spill {T} >= 0, <= spill_max;

var Tv <= (if card(C) > 0 then Infinity else 0);
subject to CutCurve {c in C}:
    Tv <= cut_slope[c] * S[N] + cut_int[c];

Data sources

Our LP model is built on and verified against published data, using synthetic data where necessary. British Columbia's within-year inflow shape is synthetic, and its reservoir bounds, intertie ratings, wheeling charges and 3,000 MW generation floor are illustrative parameters rather than measurements. Uruguay's inflow is back-calculated from observed levels, generation and spill. Our model is built from operator and regulator data availalbe at the sources in the list below. here a series is published, back-calculated from the observed balance where it is not (inflow, in all three), and synthetic only in BC's within-year shapes, whose annual totals come from the audited Annual Report. Assumed parameters are labelled as assumptions and are never fitted to the series used to validate. The model code is checked against thirteen fixtures with answers computed outside AMPL; the results are scored against published prices, published monthly volumes, and — for Uruguay — ADME's own published water value.