Aerationcontrol
Introduction and Model¶
The aeration control system aims to maintain a constant dissolved oxygen (DO) concentration in all sludge reactors using real time feedback. This ensures that microorganisms have enough oxygen for metabolism while avoiding over-aeration, which wastes energy.
The implementation of the aeration system is based on Benchmark Simulation Model no.2 (BSM2) by J.Alex et al. (2008), with only a small deviation ensuring better performance. It consists of a sensor, a PID controller and an actuator.
Broken down to the key aspects, the sensor measures the DO concentration, passes it on the PID controller which then sends a control signal to the actuator controlling the airflow into the tank. Both sensor and actuator are modeled as dynamic systems.
Sensor and Actuator¶
Both the sensor and the actuators are modelled as realistic and dynamic systems with a set of equations using a state space system and transfer function. The desired behavior is achieved by transforming the original input signal \(u(t)\) into a delayed input signal \(u_{2}(t)\) with a desired time response \(t_{r}\). This is implemented by using a first-order delay transfer function \(G_{S}(s)\). In order to to simulate the components in a time domain, \(G_{S}(s)\) is transformed into a state space system \(y_1(t)\). Further enhancing the realistic behaviour of the sensor noise is modelled with a constant noise level and added to the delayed sensor signal. The noise signal is white noise with a standard deviation of \(\delta = 1\) multiplied with the noise level \(nl\) and the maximum value of the measurement interval \(y_{max}\). Note, that the actuator is not equipped with a noise model. Lastly, within both components, the signal is checked against a maximum and minimum value resulting in a final output signal \(y\).
PID controller¶
The PID controller consists of a proportional, an integral and a derivative term, as well as an anti-windup-mechanism. The primary control objective is to maintain a constant DO concentration within the reactor, for example by manipulating the KLa parameter. This is achieved by calculating the error as the difference of the measured process value and the desired setpoint. The error is then evaluated with the proportional, integral and derivative component, and subject to limit checking via the anti-windup mechanism - resulting in the final control output \(y(t)\).
Equations¶
Components¶
| \(i\) | Component | Symbol | Unit |
|---|---|---|---|
| 1 | Dissolved oxygen | \(S_O\) | g(O2)\(\cdot\)m-3 |
| 2 | Mass transfer coefficient | \(K_{L}a\) | \(d^{-1}\) |
| 3 | Original input signal | \(u(t)\) | - |
| 4 | Delayed input signal | \(u_{2}(t)\) | - |
| 5 | Response time | \(t_{r}\) | s |
| 6 | Maximum value of the measurement | \(y_{max}\) | - |
| 7 | Noise level | \(nl\) | - |
Sensor and Actuator model¶
| Symbol | Description | Equation |
|---|---|---|
| \(G_{S}(s)\) | Transfer function for class A | \((\frac{1}{1+\tau s})^2\) |
| \(y_1(t)\) | state space function | \(u_{2}(t) + y_{max}\times nl \times n(t)\) |
| \(y(t)\) | state space function | \(y(t) =\begin{cases} y_{max}, & y_1(t) > y_{max} \\ y_1(t), & y_{min} \leq y_1(t) \leq y_{max} \\ y_{min}, & y_1(t) < y_{min} \end{cases}\) |
PID controller model¶
| Symbol | Description | Equation |
|---|---|---|
| \(e\) | Error | \(Z^{setpoint} - Z^{meas}\) |
| \(IAE\) | Integral of Absolute Error | \(\int_{t_{f}}^{t_{0}} ∣e∣ \,dt\) |
| \(ISE\) | Integral of Squared Error | \(\int_{t_{f}}^{t_{0}} e^2 \,dt\) |
| \(Dev^{max}\) | Maximal deviation from set point | \(max(∣e∣)\) |
| \(Var(e)\) | Error variance | \(\overline{e^2} - \left( \overline{e} \right)^2\) |
| \(\overline{e}\) | Mean of \(e\) | \(\frac{\int_{t_{f}}^{t_{0}} e \,dt}{t_{obs}}\) |
| \(\overline{e^2}\) | Mean of \(e^2\) | \(\frac{\int_{t_{f}}^{t_{0}} e^2 \,dt}{t_{obs}}\) |
| \(Var(\Delta u)\) | Variance of manipulated variable (\(u_{i}\)) variations | \(\overline{(\Delta u)^2} - \left( \overline{\Delta u} \right)^2\) |
| \(\Delta u\) | Difference of manipulated variable | \(∣u(t + \Delta t) - u(t)∣\) |
| \(\overline{\Delta u}\) | Mean of \(\delta u\) | \(\frac{1}{t_{\mathrm{obs}}} \int_{t_0}^{t_r} \Delta u \, dt\) |
| \(\overline{(\Delta u)^2}\) | Mean of \((\delta u)^2\) | \(\frac{1}{t_{\mathrm{obs}}} \int_{t_0}^{t_r} (\Delta u)^2 \, dt\) |
Source code documentation¶
-
Benchmarking of Control Strategies for Wastewater Treatment Plants, chap. 4.3 Sensors and Actuators ↩
-
Benchmarking of Control Strategies for Wastewater Treatment Plants, chap. 5.2 BSM2 Controllers ↩
-
Benchmark Simulation Model no. 2 (BSM2), chap. 13 Sensors and Control Handles ↩
-
Benchmark Simulation Model no. 2 (BSM2), chap. 11 Set-up of a default controller ↩