Q3Economic Operation of Power System

(a) Methods of Loading Turbo Generators for Economical Operation

For economical operation of a system comprising multiple thermal power plants (or multiple turbo-generator units within a single plant), the total system load must be allocated (loaded) among the available units in a manner that minimizes total fuel cost while meeting demand.

• Base loading: certain highly efficient units (typically the newest, most efficient plants, or those with the lowest fuel cost) are operated continuously at or near their full rated capacity, providing the steady, unchanging portion of total system load, since these units achieve their lowest cost per unit of generation when operated at high, steady output.
• Economic (incremental cost) loading: for units required to vary their output to follow the fluctuating portion of system load, the equal-incremental-cost criterion (discussed extensively elsewhere in this paper's economic dispatch problem) is applied, allocating load increments to whichever currently-online unit has the lowest incremental (marginal) generation cost at that moment, ensuring the overall combination of loaded units achieves minimum total system fuel cost for the given total demand.
• Unit commitment: a longer-timescale scheduling decision (typically day-ahead or several-days-ahead) determining which specific generating units should be started up, kept online, or shut down over the course of the scheduling period, accounting for each unit's startup cost, minimum up/down time constraints, and the forecast load profile, ensuring that units are brought online (and their associated startup costs incurred) only when actually needed to meet forecast demand, and unnecessary units are shut down during low-demand periods to avoid wasteful continued fuel consumption and O&M cost.

(b) Input, Output, and Heat Rate Characteristics

Input-output characteristic: a graph plotting the plant's total fuel input rate (in heat units per hour, such as kcal/hr or MBtu/hr, or equivalently in cost per hour if expressed in monetary terms) against its power output (MW), typically a smoothly increasing, slightly convex (upward-curving) curve, since a thermal plant's efficiency generally varies somewhat across its output range, being highest at some intermediate 'best-efficiency point' and falling off somewhat toward the extremes of very light or very heavy loading.

Heat rate characteristic: the heat rate is defined as the ratio of fuel input rate to power output (heat input per unit of electrical output, e.g., kcal/kWh), obtained by dividing the input-output curve's ordinate by the corresponding output abscissa at each point - the heat rate curve typically shows a minimum (best-efficiency) point at some intermediate output level, with heat rate increasing (efficiency decreasing) at both lower and higher output levels relative to this optimum point.

Incremental heat rate (derivative of input-output curve): the slope of the input-output curve at any given operating point represents the incremental fuel requirement for a small additional increment of output at that point, directly analogous to (and forming the physical basis for) the incremental cost curves used throughout the equal-incremental-cost economic dispatch method discussed in the corresponding numerical problem elsewhere in this paper - since fuel cost is directly proportional to fuel input (for a fixed fuel price), the incremental heat rate curve, multiplied by the fuel's unit price, directly gives the incremental cost curve (dC/dP) used in economic dispatch calculations, making these input-output and heat-rate characteristics the fundamental underlying physical/thermodynamic data from which the economic dispatch cost functions are ultimately derived.

The input-output characteristic of a thermal turbo-generator unit, typically plotted with fuel input (expressed in heat units per hour, such as kCal/hr or MBtu/hr) on the vertical axis against electrical power output (in MW) on the horizontal axis, is generally a smoothly rising, slightly convex (or in some representations, S-shaped) curve rather than a straight line, reflecting the fact that the unit's efficiency of converting fuel heat into electrical output is not constant across its entire loading range - at very low loads the unit is relatively inefficient because a substantial fraction of the fuel input is consumed simply maintaining auxiliary processes (boiler feed pumps, draft fans, and other house-load equipment) regardless of electrical output, and at very high loads (near the unit's rated capacity) efficiency again tends to fall off somewhat due to increased throttling losses and other high-load thermodynamic penalties, typically leaving a broad best-efficiency region somewhere in the middle-to-upper portion of the unit's loading range.

The heat-rate curve, obtained by dividing the input-output curve's ordinate (heat input) by the corresponding abscissa (power output) at every load point, therefore typically exhibits a distinctive U-shape or a curve that decreases initially and then gradually increases, with its minimum point identifying the load level at which the unit converts fuel to electricity most efficiently (the point of minimum heat rate, expressed in heat units consumed per kWh generated) - this most-efficient loading point is an important reference for unit-commitment and economic-dispatch decisions, since, all else being equal, it is generally desirable to keep a unit loaded near this minimum-heat-rate point whenever system conditions permit, rather than operating it at a significantly higher or lower loading where its fuel-to-electricity conversion efficiency is comparatively poorer.

The incremental heat rate, obtained as the slope (derivative) of the input-output curve at any given loading point, is distinct from the average heat rate (the input-output curve's ordinate divided by its abscissa, i.e., the total heat input per unit output up to that point) and it is specifically the incremental heat rate, converted to incremental fuel cost by multiplying by the fuel's price per heat unit, that enters directly into the economic-dispatch coordination equations as the incremental production cost dC/dP - loading each unit so that all online units share the identical incremental cost (equal-incremental-cost criterion) is what minimizes total system fuel cost for a given total demand, a result that follows directly from the calculus of the constrained-minimization problem and is the theoretical basis underlying essentially all classical economic-dispatch computations, including the B-coefficient loss-penalty formulation used elsewhere in this paper.

Modern practice supplements these classical input-output and heat-rate curve methods with online performance monitoring, in which actual measured fuel flow and electrical output are continuously compared against the design (as-tested) input-output curve to detect performance degradation over the unit's operating life, such as boiler tube fouling, turbine blade erosion, or condenser efficiency loss - any persistent upward drift of the measured heat rate relative to the design curve signals a need for maintenance intervention, and utilities routinely use this comparison as a key performance indicator for scheduling overhauls economically, balancing the fuel-cost penalty of continuing to operate a degraded unit against the capital and outage cost of an overhaul.