A steam demand step that a steady-state dispatch treats as instantaneous lowers header pressure by 37 kPa before control restores it
A synthetic single-header steam system simulated through six disturbances under two pressure controllers, quantifying the transient that a steady-state dispatch model does not represent.
Keywords: steam header; pressure control; anti-windup; feedforward; dynamic simulation; Modelica
Abstract
Steady-state dispatch models determine how much steam each unit should raise but say nothing about the transient between operating points. We simulated a synthetic single-header steam system with one supply and one process demand through six disturbances under two pressure controllers: a proportional-integral controller and the same controller with anti-windup and partial demand feedforward. A step doubling steam demand returns the header to its set-point, as a steady-state balance predicts, but under the proportional-integral controller pressure first falls by 37 kPa and re-enters a 2 percent band 11.4 seconds after the step. Anti-windup and feedforward reduce the dip by 70 percent and keep pressure within the band. For a demand surge beyond the boiler's maximum supply, pressure falls by about 525 kPa or more under either controller. A component-based reference model with IAPWS-IF97 water reproduces the lumped model's peak deviations to within 1.9 percent.
1. Introduction
Dispatch optimization of industrial steam systems is almost always steady-state: it balances mass and energy at an operating point and moves instantaneously to the next [1]. Real headers store steam, and when demand changes faster than boiler firing can follow, header pressure moves until supply catches up. Whether those excursions stay within process tolerances is a control question that a steady-state model cannot answer.
This study quantifies such transients for a minimal synthetic system and a set of declared disturbances, and evaluates a standard controller improvement [2]. The system is a generic header, not a dynamic model of any particular site.
2. Methods
2.1 System model
The system comprises one supply, one header of 20 m3 and one process demand. The header operates at 1.1355 MPa and 700 K. Steam is modeled as an ideal gas with constant heat capacity in a lumped volume, with mass and internal-energy balances. The controller output sets the inlet mass flow directly, bounded between zero and 3.0 kg/s; the actuator has no lag or ramp limit, so the boiler's own thermal dynamics are not represented. Each simulation covers 300 seconds, and every disturbance begins at 60 seconds.
The state variables are the header mass m and internal energy U, with dm/dt = min − md and dU/dt = mincpTb − mdcpT, where T = U/(m cv), p = mRT/V, cv = cp − R, and supply enters at the boiler temperature Tb, equal to the initial temperature. The heat capacity is cp = 2,081.5 J/(kg K), the ideal-gas value for water vapor at 700 K, and R = 461.5 J/(kg K). The system starts in equilibrium at the set-point with the initial demand.
2.2 Controllers
The controller output is u = kffmd + Kpe + Ki∫e dt, with e the pressure error, and the supply is u limited to the actuator bounds. The integral state is initialized so that supply equals the initial demand. Both controllers use the same proportional and integral gains, 2 × 10-5 kg/(s Pa) and 2 × 10-6 kg/(s2 Pa); the baseline uses no feedforward. The improved controller adds conditional-integration anti-windup, so that the integral term stops accumulating while the actuator is at a limit, and feedforward of measured demand with a gain of 0.7.
2.3 Disturbances and metrics
Six disturbances were applied: demand steps up and down, a ten-second demand pulse, a demand ramp, a set-point step, and a thirty-second demand surge exceeding the maximum supply. For each, the peak absolute deviation from set-point and the time after the disturbance at which pressure last re-enters a 2 percent band were recorded, together with the fraction of time the actuator was saturated.
2.4 Reference model and verification
The system was rebuilt from Modelica Standard Library components [3] and all 12 disturbance and controller pairs were simulated with each of 2 water models: the same ideal-gas water and IAPWS-IF97 water [4]. The lumped simulations were checked against an analytical response of the reduced model, for integrated mass and energy closure, and for sensitivity to the solver's maximum step.
3. Results
3.1 Response to a demand step
When process demand doubles, a steady-state model moves directly to the new balance at the same pressure. In simulation the header supplies stored steam while the controller raises supply (Figure 1). Under the proportional-integral controller pressure falls by 37 kPa and re-enters the band 11.4 seconds after the step; the stored-energy swing in the header is 2.57 MJ. Under the improved controller the dip is 11 kPa and pressure remains within the band.
3.2 All disturbances
Table 1. Peak absolute pressure deviation, and time after the disturbance to re-enter the band, for each disturbance.
| Disturbance | Baseline peak deviation, kPa | Baseline time to settle | Improved peak deviation, kPa | Improved time to settle |
|---|---|---|---|---|
| Demand step up | 37 | 11.4 s | 11 | within band |
| Demand step down | 22 | within band | 7 | within band |
| Demand pulse | 44 | 20.1 s | 13 | within band |
| Demand ramp | 8 | within band | 3 | within band |
| Set-point step | 114 | 3.0 s | 114 | 3.1 s |
| Surge beyond actuator capacity | 674 | 123.3 s | 525 | 43.2 s |
For the four demand disturbances within the supply limit, the improved controller reduces the peak deviation to about a third (Table 1, Figure 2). The step down produces a rise in pressure rather than a dip, 22 kPa under the baseline controller. For the set-point step the initial error, 114 kPa, is imposed by the step itself and is identical for both controllers; the improved controller overshoots the new set-point by 12.6 kPa against 14.6 kPa.
The surge, which exceeds the maximum supply, is a deliberate failure case. Under the baseline controller the actuator is saturated for 33 percent of the run; pressure first falls by 556 kPa and later overshoots by 674 kPa, consistent with integral windup during saturation. Under the improved controller the actuator is saturated for 13 percent of the run, pressure falls by 525 kPa with no comparable overshoot, and the peak deviation is 22 percent smaller. No controller can compensate for demand above the supply limit.
3.3 Model adequacy and numerical verification
At the operating point the steam is 241 K superheated, and IAPWS-IF97 differs from the ideal-gas idealization by 1.3 percent in density and 2.9 percent in heat capacity. The component reference reproduces the lumped model's peak deviation to 0.07 percent with ideal-gas water and to 1.9 percent with IF97 water. The lumped simulations agree with the analytical response to a relative error of 1.9 × 10-9; integrated mass and energy close to 3.8 × 10-7 and 4.3 × 10-7; and halving the maximum step twice changes the result by 6.5 × 10-11 of the pressure span.
4. Discussion
The steady-state solution is correct about where the system ends and silent about how it gets there. For the declared step the excursion is moderate and brief, and a standard controller improvement keeps it within tolerance. Anti-windup matters most when supply saturates: without it, the surge was followed by a large overshoot consistent with integral windup. For disturbances beyond supply capacity, control cannot substitute for supply. Because the actuator here has no lag, adding boiler response dynamics with the other conditions unchanged may increase the excursions; actual boiler and header responses require simulation with those dynamics or measurement. Dispatch decisions that move boiler output quickly, or leave little reserve capacity, should be checked with such a model.
5. Limitations
- A single synthetic header is modeled; real networks have several headers, letdown valves and turbines that interact.
- Supply follows the controller instantaneously, with no boiler thermal lag or ramp limit; the effect of those dynamics on the excursions was not evaluated.
- Steam properties are idealized in the lumped model; the idealization is assessed against IF97 water, not removed.
- Disturbances and controller gains are synthetic, chosen to exercise the controller rather than tuned to a plant.
- Equipment trips and startup transients are not considered.
6. Conclusions
A doubling of steam demand that a steady-state dispatch treats as instantaneous produces a 37 kPa pressure dip that returns to the band 11.4 seconds after the step under proportional-integral control. Anti-windup with demand feedforward reduces the dip by 70 percent and reduces the overshoot that follows actuator saturation from 674 to 11.5 kPa. Transient behavior should be assessed with a dynamic model when dispatch changes are fast or reserve capacity is small.
References
- Smith R. Chemical Process Design and Integration, 2nd edition. Wiley, Chichester, 2016.
- Astrom KJ, Hagglund T. Advanced PID Control. ISA, Research Triangle Park, NC, 2006.
- Modelica Association. Modelica Standard Library, version 4.1.0. Linkoping, 2025.
- International Association for the Properties of Water and Steam. Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam, IAPWS R7-97(2012). 2012.