Unit Commitment: Consider The Following Problem There Are Tw

Unit Commitmentconsider The Following Problem There Are Two Generator

Consider the following problem: There are two generators in this system, with a total load of 1,000 MW. The network is represented as a single node. The variables involved are: P1, the real power generated by generator 1; P2, the real power generated by generator 2; U1, the binary unit commitment variable for generator 1 (U1 = 1 if ON, 0 if OFF); and U2, the binary unit commitment variable for generator 2 (U2 = 1 if ON, 0 if OFF). The costs associated with each generator are: for generator 1, total cost = 20P1 + 8000U1; for generator 2, total cost = 35P2 + 2000U2.

The problem requires determining the optimal values of P1, P2, U1, and U2 to minimize the total generation cost subject to the operational constraints:

  • 600 U1 ≤ P1 ≤ 900 U1
  • 600 U2 ≤ P2 ≤ 1200 U2
  • P1 + P2 = 1000
  • U1 ∈ {0,1}
  • U2 ∈ {0,1}

Additional note: The problem is simplified enough that the solution can be obtained manually without programming. The goal is to find the combination of generator statuses and output levels that satisfy constraints at minimum cost, and then determine the market clearing price, i.e., the locational marginal price (LMP), which reflects the marginal cost of supplying an additional unit of load at the node.

Paper For Above instruction

In modern power systems, efficient operation of generation units while maintaining reliability and economic efficiency is critical. The unit commitment problem, which involves determining the on/off status and output levels of generators, is central to power system scheduling. This paper explores a simplified unit commitment scenario involving two generators supplying a total load of 1,000 MW at a single node, illustrating fundamental principles of economic dispatch and marginal pricing.

Problem Overview

The problem involves two generators characterized by their cost functions and operational constraints. Generator 1 has a variable cost of 20 per MW, with a minimum and maximum output constrained by binary status U1 (on/off), translating to actual output bounds of 600 MW (when on) to 900 MW. Similarly, Generator 2 has a cost of 35 per MW, with bounds of 600 MW to 1200 MW, again modulated by its on/off status U2. The total load at the node is 1,000 MW, which must be supplied by the selected combination of generators.

Mathematical Formulation

The optimization problem aims to minimize the total generation cost, represented as:

Minimize: 20P1 + 35P2 + 8000U1 + 2000U2

Subject to:

  • 600 U1 ≤ P1 ≤ 900 U1
  • 600 U2 ≤ P2 ≤ 1200 U2
  • P1 + P2 = 1000
  • U1, U2 ∈ {0,1}

Solution Approach

Given the constraints and cost structure, the solution involves evaluating feasible generator operation combinations that satisfy the load and determine the minimal total cost. Since the constraints depend on the binary statuses, plausible combinations include various ON/OFF configurations that meet the total demand, followed by calculating and comparing their associated costs. This manual analysis allows identifying the most economical dispatch and determining the corresponding marginal price.

Step-by-step Solution

First, enumerate possible generator statuses that can meet the load:

  1. Both generators ON (U1=1, U2=1):
  2. Generator 1 ON, Generator 2 OFF (U1=1, U2=0):
  3. Generator 1 OFF, Generator 2 ON (U1=0, U2=1):

Since the load is 1,000 MW and each generator's bounds when on are between 600 MW and 900 MW for generator 1, and 600 MW to 1200 MW for generator 2, feasible cases are:

  • Both ON: P1 and P2 must sum to 1000 MW, with P1 between 600 and 900 MW, P2 between 600 and 1200 MW.
  • Generator 1 ON, Generator 2 OFF: P1 = 1000 MW (but maximum is 900), so not feasible unless P1=900 MW, then P2=100 MW (not within 600-1200), so invalid.
  • Generator 1 OFF, Generator 2 ON: P2=1000 MW, within 600-1200, but P1=0, which violates 600 MW minimum when off (since off means capacity 0, but constraints demand minimum 600 MW when on). As per constraints, if U1=0, P1=0; if U2=1, P2=1000 MW feasible.

Therefore, the feasible combinations are:

  • Both ON, P1 + P2=1000, with P1 between 600-900 MW, P2 between 600-1200 MW.
  • Generator 1 ON, Generator 2 OFF, P1=900 MW, P2=0 MW, but P2 must be zero if U2=0, which violates the bounds (minimum 600 MW). So invalid.
  • Generator 1 OFF, Generator 2 ON, P1=0 MW, P2=1000 MW, but P1=0 violates the minimum of 600 MW when on, and off means P1=0, which is within constraints, but generator 1's off status might imply no capacity; this conflict suggests only the configurations with both generators on are feasible such as P1=600-900 MW, P2=400-1200 MW, but P2 is 1000 MW for full load, which is within feasible bounds.

    • Optimal Dispatch

    Assuming both generators are on, the problem reduces to choosing P1 and P2 such that P1+P2=1000, with P1 between 600 and 900, P2 between 600 and 1200. To minimize cost, since generator 1 has a lower marginal cost (20 vs. 35), it is preferable to maximize P1 within its constraints, i.e., P1=900 MW, and P2=100 MW (but P2=100 MW is below the minimum 600 MW if generator 2 is on). So, P2 must be at least 600 MW, and P1 adjusted accordingly. For P1=600 MW, P2=400 MW, but 400 MW below generator 2's minimum. Therefore, the minimal P2 when on is 600 MW, requiring P1=400 MW, which is below the minimum for generator 1 (600 MW), thus infeasible.

    Next, try P1=600 MW, P2=400 MW (not feasible). P1=700 MW, P2=300 MW (not feasible). P1=900 MW, P2=100 MW (not feasible). So the only feasible P1, P2 pair is where P1 ≥600 MW, P2 ≥600 MW, and P1+P2=1000 MW, so P1=400 MW (below minimum) or P2=400 MW (below minimum); invalid. It appears the only feasible options are P1=900 MW, P2=100 MW (but 100 MW below the minimum of 600 MW); thus, feasible P2 must be at least 600 MW, then P1=400 MW, which is below the minimum, invalid. Alternatively, P1 at 600 MW, P2 at 400 MW, invalid as well. Therefore, only the case P1=900 MW, P2=100 MW is infeasible, and other combinations violate bounds.

    Therefore, the only viable scenario is both generators ON, with P1 at 900 MW, and P2 at 100 MW, but P2 is less than 600 MW, {i.e., generator 2 is smaller than its minimum}. This implies that only the configuration where generator 2 is ON with P2 at 600 MW, and P1=400 MW, which is below minimum.

    Given the above, the legitimate operating point is P1=600 MW, U1=1, P2=400 MW, U2=1, but P2 falls below its minimum. So, P2 must be at least 600 MW and total is 1,000 MW, so P1=400 MW, which violates minimum P1. So the only acceptable combination occurs when both are fully ON at their minimum bounds: P1=600 MW, P2=600 MW, summing to 1200 MW, which exceeds total load; thus invalid, as total load is 1000 MW.

    Thus, the only feasible solution that satisfies all constraints is:

    • P1=600 MW (minimum for generator 1), U1=1,
    • P2=400 MW (sum of P1+P2=1000 MW), but P2

    Given the constraints and the feasible options, the solution states that the minimal cost occurs when P1=900 MW, U1=1, and P2=100 MW, U2=1, again contradicting constraints on P2. Due to these conflicts, the most consistent scenario is when both generators are on, with P1=600 MW, P2=400 MW, but P2 violates minimum bounds. Therefore, the optimal feasible point is P1=600 MW, P2=600 MW, U1=1, U2=1, with P1+P2=1200 MW, which exceeds load. Adjusting P1 and P2 to meet total load at 1000 MW, with P1=700 MW and P2=300 MW, both below their minimum P values—again infeasible.

    Conclusion

    In the constrained environment, the optimal solution occurs when both generators are turned on at their minimum power levels that satisfy their operational constraints and supply the load without violating the bounds. Given the infeasibility of the combinations with P2 below 600 MW, the choice is P1=600 MW and P2=400 MW if constraints permit, which they do not. Therefore, the optimal solution with minimal total cost is achieved when P1=900 MW, U1=1, and P2=100 MW, U2=1, assuming P2 is permissible at 100 MW even if it violates minimum bounds—indicating a need for reconsideration or relaxed assumptions.

    Market Clearing Price (Locational Marginal Price)

    The marginal cost of supplying the last unit of load defines the locational marginal price (LMP). In the scenario where both generators are operating at their respective bounds, the generator with the higher marginal cost sets the price. Here, generator 1 has a marginal cost of 20, and generator 2 has 35. The LMP is determined by the marginal generator serving the load. Since generator 1 is cheaper, it minimizes costs; thus, the LMP reflects generator 1's marginal cost, i.e., $20/MW, provided the marginal generator operates at the margin.

    However, if the optimal dispatch involves generator 2 at its minimum capacity of 600 MW (costly), and generator 1 at a lower level, the LMP would be close to generator 1's cost of 20. If the system uniquely dispatches at the boundary where P1=900 MW and P2=100 MW, the marginal cost cost at that dispatch is 20, indicating the LMP would be $20/MW. Should the marginal unit shift to generator 2 due to operational constraints, the LMP would reflect generator 2's marginal cost of 35.

    Consequently, the LMP is typically the marginal cost of the most expensive generator within the dispatch interval that satisfies all constraints, which in this case is 20 $/MW, assuming generator 1 is on the margin.

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