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1. The Big Picture

Peaky Flexers is a welfare-maximising linear program (LP) for an electricity market. It answers a deceptively simple question: if we had to design an electricity system from scratch, what mix of generation and storage capacity would maximise the total value that consumers get from electricity, minus the cost of providing it?

The model works with five supply-side elements—renewables (zero variable cost), nuclear (firm baseload), gas peakers, battery storage, and transmission & distribution—and a demand side that can flex: consumers can shift demand to cheaper periods, expand consumption when prices are low, or curtail when the cost of supply exceeds the value of the electricity to them.

The key output is a set of market-clearing prices—one for each time period. These prices emerge naturally from the optimisation: they are the shadow prices on the energy-balance constraints, representing the cost to the system of serving one more megawatt-hour of demand in that period. In a well-functioning market, these are the prices that would prevail.

2. Understanding the Inputs

The model page has four input sections. Each represents a distinct part of the electricity system. Hover over the ⓘ button next to any section heading on the model page to see a condensed version of the guide for that section. Below is the full detail for each input.

2.1 Load Duration Curve Blocks

Real electricity demand varies hour by hour across the year. Rather than modelling all 8,760 hours individually, the model uses a Load Duration Curve (LDC)—a standard tool in energy economics that sorts hours by demand level.

The year is divided into three blocks:

You only set the hours for each block. The total should sum to 8,760 (one year). The demand levels come from the tier definitions below.

2.2 Renewables Availability Profile (Sub-Blocks)

The wind does not blow constantly during winter, nor does the sun always shine in summer. To capture this intermittency, each LDC block is split into two sub-blocks:

• Low Renewables—hours when renewable output is scarce (e.g. calm, overcast days in winter)
• High Renewables—hours when renewable output is plentiful (e.g. windy afternoons)

For each sub-block you set two parameters:

For example, the default Winter Peak is split 10/90: for just 10% of winter peak hours (30 hours), only 15% of installed renewables capacity is available. For the remaining 90% (270 hours), 80% is available. This creates a large price differential between the two sub-blocks, and it is precisely this differential that makes storage valuable within a block—storage can charge when renewables are plentiful and discharge when they are scarce, even within the same season.

2.3 Demand Tiers & Expandable Demand

Not all electricity demand is equally valuable. A hospital's life-support system has a very different willingness to pay than a dishwasher that could run at 3am instead of 6pm. The model captures this by dividing demand in each block into three tiers:

• High-value demand — critical loads that consumers would pay a very high price to keep running (e.g. hospitals, data centres, industrial processes). These have a high Value of Lost Load (VoLL) and a high cost to shift.
• Mid-value demand — important but somewhat flexible loads (e.g. commercial heating, manufacturing). Moderate VoLL and shift cost.
• Low-value demand — deferrable or interruptible loads (e.g. EV charging, water heating, non-urgent industrial). Low VoLL and low shift cost.

For each tier you set three parameters:

Expandable demand represents new consumption that would switch on if prices fell low enough—think of energy-intensive industries that would locate near cheap power, or electric heating that becomes attractive when electricity prices drop. You specify a quantity (GW) and an activation value (£/MWh). If the clearing price drops below this value, the expandable demand activates.

2.4 Supply-Side Technology Costs

The model does not pre-set any generation capacities. Instead, it decides how much of each technology to build, balancing capital costs against the value of the electricity they produce. There are five supply-side elements:

Renewables (Zero Variable Cost)

Wind and solar. Once built, they produce electricity at zero fuel cost, but their output depends on weather conditions (captured by the availability profile above). The only input is the annualised capital cost (£/kW/year).

Nuclear

Firm baseload capacity with zero carbon emissions. You set:

• Capital cost (£/kW/year) — the annualised cost of building the plant. Default £600/kW/yr.
• Variable cost (£/MWh) — fuel and O&M costs. Default £10/MWh.
• Planned outage (%) — fraction of time offline for scheduled maintenance, applied uniformly across all periods. Default 10%.
• Peak unavailability (%) — additional probability of being unavailable during winter peak, on top of planned outages. This captures unplanned or forced outage risk at the time it matters most. Default 5%.

Unlike renewables, nuclear is not affected by the renewables availability profile. Its effective capacity in non-peak periods is capacity × (1 − planned outage); during winter peak sub-blocks it is further derated to capacity × (1 − planned outage) × (1 − peak unavailability).

Gas Peaker

A flexible thermal plant that can run whenever needed but burns gas. You set:

• Capital cost (£/kW/year) — the annualised cost of building the plant
• Variable cost (£/MWh) — the fuel and O&M cost of running it
• Carbon price (£/tCO₂) — a carbon tax or shadow price, multiplied by the emission factor to add to variable cost. Default 50.
• Emission factor (tCO₂/MWh) — carbon intensity of the gas plant

Battery Storage

Storage is specified in terms of physical CapEx parameters that are familiar to anyone looking at battery project costs:

• Duration (hours) — how many hours the battery can discharge at full power (e.g. 4h). This fixes the ratio of energy to power capacity.
• Connection cost (£/MW) — the one-off capital cost of the inverter and grid connection
• Cell cost (£/MWh) — the one-off capital cost of the battery cells
• Asset life (years) — economic life of the asset
• Discount rate (%) — used in the capital recovery factor to annualise CapEx. Higher rates increase the annual cost (time value of money). Default 10%.
• Cycles / year — how often the battery is assumed to cycle. The annualised cell cost is spread over this many cycles, so a battery cycling daily (365) has a much lower effective cost per MWh than one cycling weekly (52).
• Round-trip efficiency (%) — default 95%. For every MWh charged, only 0.95 MWh is available to discharge. The 5% is lost as heat.

The model converts these internally using the capital recovery factor (CRF):

• CRF = r(1+r)ⁿ ÷ ((1+r)ⁿ − 1), where r = discount rate (decimal), n = asset life
• Power cost (£/kW/yr) = (connection cost ÷ 1000) × CRF
• Energy cost (£/kWh/yr) = (cell cost ÷ 1000) × CRF
• Effective energy cost = energy cost ÷ cycles per year

When the discount rate is zero, CRF = 1/n (straight-line). For r > 0, CRF > 1/n, so annual costs reflect the time value of money.

A duration constraint ensures that energy capacity (GWh) = power capacity (GW) × duration (h), so the optimiser determines how much storage to build as a single decision, with the duration as a design parameter.

Because the LDC representation can only track one storage cycle through the sub-block sequence, the model scales the storage energy level constraints by cycles per year. This gives the LP a realistic annual throughput budget (physical energy capacity × cycles) while keeping full annual capital costs in the objective. The result is that the LP makes correct investment decisions: it compares the full annual CapEx against the full annual arbitrage value from repeated cycling.

Transmission & Distribution (T&D)

T&D capacity represents the cost of the wires, transformers, and infrastructure needed to deliver electricity from generators to consumers. You set a single parameter:

• T&D capacity cost (£/kW/year) — the annualised cost of network capacity

T&D capacity is determined endogenously by the model. The constraint binds at peak total generation dispatched (renewables + gas + storage discharge) across all sub-blocks, so the system must build enough T&D to handle the highest simultaneous power flow.

3. How the Optimisation Works

3.1 In Plain Language

The model asks: what system design and operating pattern makes society best off?

It simultaneously decides:

1. How much capacity to build for each technology (renewables, nuclear, gas, storage, T&D)
2. How to dispatch each technology in every sub-block (6 sub-blocks total)
3. How much demand to serve, shift, or curtail in each tier and sub-block
4. How to operate storage—when to charge, when to discharge, and how much energy to hold

The objective is to maximise net welfare: the total value that consumers place on the electricity they receive, minus the total cost of providing it (both capital and operating costs).

Think of it as a social planner trying to make the best possible use of available resources. The planner values each MWh of served demand at its VoLL, credits shifted demand at VoLL minus the shift cost, and charges for nuclear and gas generation at their variable costs and all technologies at their capital costs. Internally, the model converts GW × hours into MWh (×1000) and GW/GWh into kW/kWh (×1,000,000), so all welfare and cost outputs are reported in full annual currency units rather than “thousands”.

The optimisation is subject to physical and economic constraints:

• Energy balance: In every sub-block, total supply (renewables + gas + storage discharge) must be at least as large as total demand (served tiers + shifted-in demand + expandable demand + storage charging).
• Capacity limits: Nothing can dispatch more than its installed capacity. Renewables are further limited by the availability fraction in each sub-block.
• T&D capacity: Total generation dispatched in every sub-block must not exceed T&D capacity.
• Storage duration: Energy capacity (GWh) = power capacity (GW) × duration (hours).
• Demand limits: You cannot serve more than the available quantity of each tier, and you cannot shift more than is available.
• Shifting rules: Demand can only shift downward (from expensive to cheaper blocks). Shifting conserves total energy: the GWh shifted out of the source block must equal the GWh arriving in destination sub-blocks.
• Storage physics: Energy stored cannot exceed energy capacity. The storage level evolves sequentially through all 6 sub-blocks (charge adds, discharge subtracts, efficiency applies to charging). The system is cyclical: the level at the end of the year must equal the level at the start.

The market-clearing prices are not directly chosen. They emerge as shadow prices on the energy-balance constraints—they tell you the marginal value to society of one additional MWh of supply in each sub-block. In an efficient market, these would be the wholesale electricity prices.

3.2 The Formal Problem

For those who want the mathematics, here is the LP formulation. The model maximises:

Where b indexes blocks, i indexes sub-blocks within each block, t indexes demand tiers, h are hours, K are capacity variables, and C are capital costs. C_{sto_pw} and C_{sto_en} are derived from the connection and cell CapEx inputs using the CRF. The factor 1000 converts GWh into MWh in the flow terms; the factor 10⁶ converts GW/GWh into kW/kWh in the capital terms. The variable cost of gas includes any carbon adder: c_gas^var = variable_cost + carbon_price × emission_factor.

Subject to (for all sub-blocks b,i):

• Energy balance:   ren_b,i + gas_b,i + sto_dis_b,i − sto_chg_b,i ≥ ∑_t served_b,i,t + shift_in_b,i + expand_b,i
• Capacity:   ren_b,i ≤ avail_b,i · K_ren,   gas_b,i ≤ K_gas,   sto_dis_b,i ≤ K_{sto_pw},   sto_chg_b,i ≤ K_{sto_pw}
• T&D:   ren_b,i + gas_b,i + sto_dis_b,i ≤ K_td
• Duration:   K_{sto_en} = duration · K_{sto_pw}
• Demand:   served_b,i,t + shift_out_b,t ≤ quantity_b,t
• Energy conservation:   shift_out_src,t,dest · h_src = ∑_i shift_in_src,t,dest,i · h_dest,i
• Storage dynamics:   level_b,i = level_prev + h_b,i · (η · sto_chg_b,i − sto_dis_b,i)
• Cyclical:   level_last = level_start
• Storage capacity:   level_b,i ≤ K_{sto_en}

Shadow prices on the energy-balance constraints, divided by the sub-block's hours, give clearing prices in £/MWh.

4. Reading the Outputs

After you click Optimise the System, the results page shows several sections. Here is what each one tells you.

4.1 Market Equilibrium Summary

A row of metric cards showing the headline results at a glance:

• Sub-block prices — the market-clearing price for each of the 6 sub-blocks. Higher prices indicate scarcity; lower prices indicate surplus. Compare Winter Peak Low Renewables (typically the highest) with Low Demand High Renewables (typically the lowest) to see the range of price differentials driving storage and shifting.
• Load-Weighted Average Price — the average price weighted by GWh consumed in each sub-block. This is what a "typical" MWh costs.
• Capacity built — GW of renewables, gas, storage power, and T&D, plus GWh of storage energy.
• Welfare metrics — net welfare, consumer value, variable cost, and capital cost, shown in millions of currency units (for example, £m). Net welfare = consumer value − variable cost − capital cost.

4.2 Optimal Capacity Mix

Two charts side by side:

• Bar chart — the GW of power capacity for each technology. This shows the physical scale of the system.
• Pie chart — the annual capital cost split. Storage power and energy costs are combined into a single "Storage" wedge to avoid confusing energy and capacity concepts.

Together, these tell you how the system spends its investment budget. A system dominated by renewables will have high renewables capital cost but zero variable cost. A system with more gas will have lower capital cost but higher variable cost. T&D costs appear as a separate wedge.

4.3 Input Load Duration Curve

This shows the raw demand profile before any optimisation. Sub-blocks are sorted from highest to lowest GW, and the three demand tiers are stacked within each sub-block (low-value at the bottom, high-value at the top). This is what the demand side "looks like" before the model decides what to do with it.

Use this chart as a reference point. After you look at the optimised results, come back here to see how much has changed—how much demand was shifted, curtailed, or served by different sources.

4.4 Generation Duration Curve

This is the post-optimisation counterpart of the input LDC. It shows total generation in each sub-block, broken down by source (renewables in green, gas in red, storage discharge in blue). Sub-blocks are sorted by total generation from left to right.

Hover over any block in the chart to see a pop-out with the sub-block name, total generation, supply stack, and clearing price. Hatched areas indicate curtailed demand.

4.5 Price Duration Curve

This chart sorts sub-blocks from highest to lowest clearing price, with price on the Y-axis and cumulative hours on the X-axis. Hover over any block to see a pop-out summarising the key activity in that sub-block:

• Demand served (GW)
• Shift in / shift out (GW, if any demand is shifting)
• Curtailed demand (GW, if any)
• Storage discharge and charge (GW, if active)

4.6 Demand & Supply by Sub-Block (GW)

A 2×3 grid of stacked bar charts (one per sub-block). Each chart has two bars:

• Demand bar (left) — stacked from bottom: High/Mid/Low-value served demand, shifted-in demand, expandable demand, and storage charging.
• Supply bar (right) — stacked from bottom: renewables, gas, and storage discharge.

All six charts share the same Y-axis so you can visually compare the scale of demand and supply across sub-blocks. The title of each chart shows the clearing price.

Storage charging appears as demand (it consumes electricity) while storage discharging appears as supply (it provides electricity). This is physically correct—charging is a load on the system.

4.7 Energy by Sub-Block (GWh)

The same structure as the GW charts, but multiplied by hours to show energy (GWh). Each sub-block has its own Y-axis scale because the blocks have very different durations (300 hours vs. 5,460 hours), which would make the shorter blocks invisible if they shared an axis.

These charts are important for understanding the total energy balance. A sub-block might have low GW but high GWh if it lasts many hours. Storage is particularly visible here: you can see how much energy in GWh is being moved around the system.

4.8 Demand Flexibility (GW)

Per-block Sankey diagrams showing what happens to demand in each block. Left-side nodes are demand sources (tiers + shifted-in demand + storage charging). Right-side nodes are destinations:

• Served by Gen — met directly by renewables or gas generation
• Served by Storage — met by battery discharge
• → [Block] — shifted to a later block (e.g. → Shoulder means this demand moved from Winter Peak to Shoulder)
• Not Served — curtailed (VoLL was too low relative to clearing price)
• Into Storage — electricity consumed for charging batteries

The width of each flow band is proportional to the GW. This chart only appears when there is flexibility activity (shifting, curtailment, or storage) in the system.

4.9 Energy Transfers (GWh)

A system-level Sankey diagram showing energy transfers between sub-blocks. This is fundamentally different from the flexibility Sankeys above: while those show what happens within a block, this shows what moves between blocks.

Left nodes are energy exporters (sub-blocks that charge storage or shed demand). Right nodes are energy importers (sub-blocks that discharge storage or receive shifted demand), plus efficiency losses.

The diagram visually distinguishes two types of flows:

• Battery storage (solid rectangles, more opaque bands) — energy physically moved via charging and discharging
• Behavioural demand shift (dashed, hatched rectangles, lighter bands) — demand moved by consumer behaviour change, including curtailment ("Not Served")

This is perhaps the most informative single chart. It reveals the full pattern of energy arbitrage: where is energy coming from, where is it going, and how much is lost to efficiency?

4.10 Storage Activity

Two charts plus summary metrics:

• Summary cards — total GWh charged and discharged, efficiency loss, average charge and discharge prices, and the implied cycling cost.
• Power chart (top) — bar chart of charge/discharge GW in each sub-block. This shows when storage is active.
• Energy level chart (bottom) — line chart tracking the GWh stored across the year. The dashed line shows total energy capacity. The cyclical constraint means this line starts and ends at the same level.

4.11 Detailed Sub-Block Results

The final section gives you the full numerical detail for each of the 6 sub-blocks: every demand tier (available, served, lost, shifted out), every supply technology (dispatched vs available), expandable demand activation, storage level, and the clearing price.

This is where you go to verify specific numbers or understand exactly what is happening in a particular sub-block. If a Sankey diagram shows an unexpected flow, the detail tables will tell you exactly which tiers and technologies are involved.

4.12 Household Bill Optimisation (Optional)

This optional section lets you model a single household as a price-taker in the wholesale market. After the system optimisation determines clearing prices, the household minimises its annual electricity bill given those prices.

Enable it by expanding the “Household Bill Optimisation” section in the inputs and ticking the checkbox. You specify:

• Demand per block (kW) — the household’s average load in each LDC block (e.g. 2 kW in winter, 1 kW in summer).
• Value of lost load (£/MWh) — the price above which the household would rather curtail than consume. If the system price in a sub-block exceeds this, the household cuts demand.
• Home battery — capacity (kWh), power (kW), round-trip efficiency (%), and annualised cost (£/year). The household can charge when prices are low and discharge when prices are high.

The output shows total annual spending (energy cost and battery cost separately), average unit cost (p/kWh), and a per-sub-block breakdown of consumption, curtailment, battery activity, and spend. This is useful for understanding how retail consumers would respond to the wholesale price signals the model produces.

5. Using the AI Chat Assistant

Peaky Flexers includes a built-in AI assistant powered by Anthropic's Claude. Click the AI Chat button in the bottom-right corner to open the chat panel.

What can you ask?

The AI has access to the complete model source code, your current input values, and your latest optimisation results. This means it can reason about your specific scenario—not just generic electricity economics. Good questions include:

• "Why is the Winter Peak Low Renewables price so much higher than the other sub-blocks?"
• "What is driving the storage investment? Where is the arbitrage opportunity?"
• "If I doubled the carbon price, what would you expect to happen to the gas capacity?"
• "Why is demand being shifted from Shoulder to Low Demand? That seems counterintuitive."
• "Can you explain the energy balance for the Shoulder High Renewables sub-block?"

Here is an example of the kind of conversation you might have:

Getting an API key

To use the AI chat, you need an Anthropic API key. The model defaults to Claude Opus 4.6, which provides excellent analysis of the model's economics. To get a key:

1. Visit Claude API Keys
2. Create an account (or sign in)
3. Click Create Key, give it a name, and copy the key
4. Paste it into the API key field in the chat panel

Your API key is stored in your browser's local storage and is only sent to the Peaky Flexers server to proxy requests to Anthropic. It is never stored on the server.

The Clear Chat button resets the conversation history, which is useful when you change input parameters and run a new optimisation—the old context will no longer match the new results.