Scene 7 — Hydro and dynamic programming

1) The hydro problem in desk language

Lukas splits the whiteboard into two columns.

Trader language | Model language

He fills the trader side first.

• We choose when to generate.
• We keep water for high-value periods.
• We respect environmental flows and turbine limits.
• We do not know inflows. We do not know prices.
• We manage shortage risk.

Then he mirrors it in model language.

Lukas: “Formally, it’s dynamic programming.”

He writes:

• State: reservoir level $R_t$ (sometimes also snowpack, soil moisture, constraints status)
• Action: generation $g_t$ (and spill $sp_t$ if forced)
• Transition: $R_{t+1} = R_t + \mathrm{inflow}_t - g_t - sp_t$
• Reward: $\mathrm{reward}_t = P_t \cdot g_t - \mathrm{operating\ costs} - \mathrm{penalties}$

Anna leans forward.

Anna: “This is already better than ‘it’s a call’. It shows the conservation law.”

Lukas nods and adds constraints underneath, because that is where hydro becomes real.

• $0 \le R_t \le R_{\max}$
• $0 \le g_t \le g_{\max}$
• minimum environmental release
• ramping limits
• outages and maintenance schedules

Lukas: “These constraints create non-linearities and kinks. They are not an add-on. They are the asset.”

2) Bellman recursion, explained as water value

Lukas writes a clean recursion, then immediately translates it.

$$V_t(R_t) = \max_{g_t \in \mathcal{A}(R_t)} \Big(P_t \cdot g_t - c(g_t) + \mathbb{E}[V_{t+1}(R_{t+1})]\Big)$$

He circles $V_t(R_t)$.

Lukas: “This value function is what you call water value. It’s the shadow value of one more unit of water in the reservoir.”

Anna nods.

Anna: “Yes. We literally say ‘the water value is high today’ meaning: do not generate unless price beats the future value of water.”

Lukas draws a simple inequality.

Generate if: $P_t \ge \mathrm{water\ value}_t(R_t)$

Lukas: “Management hears this and immediately understands. Hydro is not calls. It’s: compare spot price to opportunity cost of water.”

Anna taps the board.

Anna: “That’s the story. Opportunity cost. Not strike price.”

3) Uncertainty: why scenarios beat single-factor intuition

Anna points at the printed inflow scenarios.

Anna: “And inflows. That’s where non-hydro people get lost. They assume inflows are some average.”

Lukas shakes his head.

Lukas: “We never have a single inflow. We have a distribution. That’s why we use scenarios: trees or Monte Carlo paths.”

He draws a small tree: wet vs dry, then normal vs extreme across months.

Lukas: “The DP uses price scenarios and inflow scenarios. The optimal policy changes across regimes. In a dry regime, the water value rises and the strategy becomes conservative. In a wet regime, you may spill unless you generate. That creates conditional optionality.”

Anna: “So the hydro book’s risk is not just price risk. It’s joint price–inflow risk.”

Lukas: “Yes. Correlation matters too. Cold winters often mean higher demand and can come with different inflow patterns. Your stress ‘dry winter + high demand’ is exactly the right combined scenario.”

4) Risk measures: when expected value is not enough

Anna flips to a management slide draft.

Anna: “They also asked: why don’t we just maximise expected revenue?”

Lukas smiles.

Lukas: “Because you’ll blow up in a dry year.”

He adds a small box:

Risk-aware hydro optimisation

• CVaR constraint: limit worst-case revenue shortfall
• Chance constraint: keep probability of reservoir depletion below a threshold
• Reliability constraint: ensure ability to generate during scarcity windows

Lukas: “If management cares about security-of-supply or reputational risk, you encode it. That changes the policy. And it changes valuation.”

Anna nods.

Anna: “That’s useful. It shows why hydro value is partly a risk-management asset.”

5) How it shows up in risk reports and hedging

Anna returns to her earlier pain point: the risk pack.

Anna: “So how does this show in risk reports without oversimplifying?”

Lukas answers with three layers.

Lukas: “Hydro produces exposures at three levels.”

1. Forward price sensitivities

• delta to power forwards across months and seasons
• strong shape exposure (peak/base, winter premium)

2. Optionality and non-linearity

• meaningful vega and convexity effects
• boundary effects near full or empty reservoir

3. Scenario PnL

• dry winter / wet summer
• low inflows + high demand
• outage + cold snap
• policy regime shifts

Anna nods.

Anna: “So it’s not a call on Cal-26. It’s a set of conditional rights to generate, constrained by water, with path risk.”

Lukas: “Yes. And if we want to be honest, we should report a hydro policy summary: expected generation schedule under base case, plus how it shifts under stress.”

Anna: “That is what management actually wants. An operational narrative.”

Lukas: “Exactly. And it also explains hedging: you hedge the forward exposure and you manage the path risk with scenario ladders, not with a single delta.”

6) Anna’s management-ready story

Anna picks up the marker and writes her version.

Anna: “Here is the story I want to tell.”

• We manage an inventory of optionality.
• The inventory is water.
• Value depends on paths, constraints, and decisions.
• Hydro is a dynamic programme, not a static option.
• We hedge forward exposure in hedgeable instruments.
• We manage path and regime risk with scenarios and stress.
• We use limits and risk-aware optimisation to reflect risk appetite.

Lukas reads it and nods.

Lukas: “That’s correct. And it’s defendable. And it avoids pretending the book is linear.”

Anna caps the marker.

Anna: “Good. Now: I want two artefacts from you for management. One page. No equations. One figure. And a sentence that explains water value in English.”

Lukas nods.

Lukas: “Done. I’ll give you:

• a diagram of the DP loop (state → decision → transition → value),
• a chart showing water value versus reservoir level across seasons,
• and a scenario table: dry vs normal vs wet.”

Anna stands up.

Anna: “Perfect. That’s how we stop calling hydro ‘some calls’.”