Skill: wu-uk/energy-market-pricing-locational-marginal-prices (version 0.1.0)

--- name: locational-marginal-prices description: "Extract locational marginal prices (LMPs) from DC-OPF solutions using dual values. Use when computing nodal electricity prices, reserve clearing prices, or performing price impact analysis." --- # Locational Marginal Prices (LMPs) LMPs are the marginal cost of serving one additional MW of load at each bus. In optimization terms, they are the **dual values** (shadow prices) of the nodal power balance constraints. ## LMP Extraction from CVXPY To extract LMPs, you must: 1. Store references to the balance constraints 2. Solve the problem 3. Read the dual values after solving ```python import cvxpy as cp # Store balance constraints separately for dual extraction balance_constraints = [] for i in range(n_bus): pg_at_bus = sum(Pg[g] for g in range(n_gen) if gen_bus[g] == i) pd = buses[i, 2] / baseMVA # Create constraint and store reference balance_con = pg_at_bus - pd == B[i, :] @ theta balance_constraints.append(balance_con) constraints.append(balance_con) # Solve prob = cp.Problem(cp.Minimize(cost), constraints) prob.solve(solver=cp.CLARABEL) # Extract LMPs from duals lmp_by_bus = [] for i in range(n_bus): bus_num = int(buses[i, 0]) dual_val = balance_constraints[i].dual_value # Scale: constraint is in per-unit, multiply by baseMVA to get $/MWh lmp = float(dual_val) * baseMVA if dual_val is not None else 0.0 lmp_by_bus.append({ "bus": bus_num, "lmp_dollars_per_MWh": round(lmp, 2) }) ``` ## LMP Sign Convention For a balance constraint written as `generation - load == net_export`: - **Positive LMP**: Increasing load at that bus increases total cost (typical case) - **Negative LMP**: Increasing load at that bus *decreases* total cost Negative LMPs commonly occur when: - Cheap generation is trapped behind a congested line (can't export power) - Adding load at that bus relieves congestion by consuming local excess generation - The magnitude can be very large in heavily congested networks (thousands of $/MWh) Negative LMPs are physically valid and expected in congested systems — they are not errors. ## Reserve Clearing Price The reserve MCP is the dual of the system reserve requirement constraint: ```python # Store reference to reserve constraint reserve_con = cp.sum(Rg) >= reserve_requirement constraints.append(reserve_con) # After solving: reserve_mcp = float(reserve_con.dual_value) if reserve_con.dual_value is not None else 0.0 ``` The reserve MCP represents the marginal cost of providing one additional MW of reserve capacity system-wide. ## Finding Binding Lines Lines at or near thermal limits (≥99% loading) cause congestion and LMP separation. See the `dc-power-flow` skill for line flow calculation details. ```python BINDING_THRESHOLD = 99.0 # Percent loading binding_lines = [] for k, br in enumerate(branches): f = bus_num_to_idx[int(br[0])] t = bus_num_to_idx[int(br[1])] x, rate = br[3], br[5] if x != 0 and rate > 0: b = 1.0 / x flow_MW = b * (theta.value[f] - theta.value[t]) * baseMVA loading_pct = abs(flow_MW) / rate * 100 if loading_pct >= BINDING_THRESHOLD: binding_lines.append({ "from": int(br[0]), "to": int(br[1]), "flow_MW": round(float(flow_MW), 2), "limit_MW": round(float(rate), 2) }) ``` ## Counterfactual Analysis To analyze the impact of relaxing a transmission constraint: 1. **Solve base case** — record costs, LMPs, and binding lines 2. **Modify constraint** — e.g., increase a line's thermal limit 3. **Solve counterfactual** — with the relaxed constraint 4. **Compute impact** — compare costs and LMPs ```python # Modify line limit (e.g., increase by 20%) for k in range(n_branch): br_from, br_to = int(branches[k, 0]), int(branches[k, 1]) if (br_from == target_from and br_to == target_to) or \ (br_from == target_to and br_to == target_from): branches[k, 5] *= 1.20 # 20% increase break # After solving both cases: cost_reduction = base_cost - cf_cost # Should be >= 0 # LMP changes per bus for bus_num in base_lmp_map: delta = cf_lmp_map[bus_num] - base_lmp_map[bus_num] # Negative delta = price decreased (congestion relieved) # Congestion relieved if line was binding in base but not in counterfactual congestion_relieved = was_binding_in_base and not is_binding_in_cf ``` ### Economic Intuition - **Relaxing a binding constraint** cannot increase cost (may decrease or stay same) - **Cost reduction** quantifies the shadow price of the constraint - **LMP convergence** after relieving congestion indicates reduced price separation


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