# Performance Evaluation of Real Industrial RTO Systems

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

**Θ**, which are modifiers of $\mathbf{Z}$, so that ${\mathbf{Z}}^{+}=\mathbf{Z}+\mathsf{\Theta}$.

**θ**, and

**ε**is the measurement error vector, being a random variable.

**ε**follows a Gaussian probability density function, such that $E\left[\mathit{\epsilon}\right]=\mathbf{0}$, $Var\left[\mathit{\epsilon}\right]={\mathbf{\sigma}}_{z}^{2}$, and $Cov\left[\mathit{\epsilon}{\mathit{\epsilon}}^{\mathsf{T}}\right]={\mathbf{V}}_{z}$, the problem defined in Equation (9) becomes the weighted least squares (WLS) estimation:

## 3. RTO System Description

- (a)
- Steady-state detection (SSD), which states if the plant is at steady state based on the data gathered from the plant within a time interval;
- (b)
- Monitoring sequence (MON), which is a switching method for executing the RTO iteration based on the information of the unit’s stability, the unit’s load and the RTO system’s status; the switching method triggers the beginning of a new cycle of optimization and commonly depends on a minimal interval between successive RTO iterations, which typically corresponds to 30 min to 2 h for distillation units;
- (c)
- Execution of the optimization layer based on the two-step approach, thus adapting the stationary process model and using it as a constraint for solving a nonlinear programming problem representing an economic index.

- production planning and scheduling, which transfer information to it;
- storage logistics, which has information about the composition of feed tanks;
- Distributed control system (DCS) and database, which deliver measured values.

## 4. Industrial RTO Evaluation

#### 4.1. Steady-State Detection

#### 4.1.1. Tool A

#### 4.1.2. Tool B

#### 4.1.3. Industrial Results

#### 4.2. Adaptation and Optimization

**upd**and

**obj**(i.e., the decision vector

**θ**and the set of variables in the objective function) is almost always based on empirical procedures. It is interesting to note that the premises that support the choices of

**obj**set are hardly observed. According to these premises, an important element is that

**obj**encompasses measured variables, which are non-zero values of $\mathbf{Zw}$, directly affected by the corruption of experimental signals. However, it may be seen in real RTO systems that there is the inclusion of updated parameters and non-measured variables (

**upd**set) in the set

**obj**. A typical occurrence of such a thing is the inclusion of load characterization parameters, which belong to the

**upd**set and are often included in the objective function of the model adaptation problem. In this case, where there is no measured value for $\mathbf{Zw}\left(\mathbf{upd}\right)$, fixed values are arbitrarily chosen for these variables. As a result, this practice limits the variation of these variables around the adopted fixed values. This approach degenerates the estimator and induces the occurrence of bias in estimated variables.

**obj**over the value of the objective function from Equation (21) with real RTO data. Assuming that Equation (10) is valid, measurement errors are independent and the knowledge of the true variance values for each measured variable is available, it is expected that the normalized effects, $ct$, of each variable in

**obj**are similar, as defined in Equation (22). In Tool A, the number of variables in set

**obj**was 49. The time interval between two successful and successive RTO iterations has a probability density with 10th, 50th and 90th percentiles of 0.80, 1.28 and 4.86, respectively. The normalized effects $ct$, as defined in Equation (22), are shown in Figure 6. Results show that it is common that fewer variables within

**obj**have greater effects in objective function ${J}^{id}$ values. For a period of three months, the RTO system from Tool A has been executed 1000 times, but achieved convergence for the reconciliation and model adaptation step in 59.7% of cases. Besides the formulation of the estimation problem, this relatively low convergence rate might be caused by the following reasons: (i) the nonlinearity of the process model that may reach hundreds of thousands of equations, since the optimization of a crude oil atmospheric distillation unit is a large-scale problem governed by complex and nonlinear physicochemical phenomena; (ii) the model is not perfect, and many parameters are assumed to remain constant during the operation, which may not be the real case; such assumptions force estimations of updated parameters to accommodate all uncertainty and may result in great variability in estimates between two consecutive RTO iterations; and (iii) the limitations of the employed optimization technique, along with its parameterization; in most commercial RTO packages, sequential quadratic programming (SQP) is the standard optimization technique, as is the case for Tool A. Regarding convergence, thresholds are empirical choices, generally represented by rules of thumb.

**θ**), i.e., an unmeasured variable. Besides the comparison of the median values, the values for the 10th and 90th percentiles show greater variance in the effects of each variable. As an example, the first two variables in Table 2 vary as high as 100% among percentiles. In fact, some discrepancy among the contribution of variables in

**obj**in objective function values is expected, as long as the variables are expressed with different units, and this effect is not fully compensated by the variances. Nevertheless, this fact does not explain the high variation along the operation, which can be attributed to the violation of one or more hypothesis in different operating scenarios. Even in this case, it is not possible to determine which assumptions do not hold.

- the database might present lagged analyses, given the quality of oil changes with time;
- there might be changes in oil composition due to storage and distribution policies from well to final tank. Commonly, this causes the loss of volatile compounds;
- mixture rules applied to determine the properties of the load might not adequately represent its distillation profile;
- eventually, internal streams of the refinery are blended with the load for reprocessing.

## 5. Conclusions

**θ**. Since the plant will typically not be in the set of models obtained by spanning all of the possible values of the model parameters, the two-step approach will fail in the presence of structural plant-model mismatch. However, we should ask if the structural inability of the two-step approach is the primary source of uncertainty in RTO systems. Indeed, the disagreement in the mathematical model structure is not the only source of plant-model mismatch. The fault in steady-state detection is a source of model uncertainty and not a parametric one. This is due to the fact that derivative terms are ignored. Even for methods that theoretically are not affected by plant-model mismatch, a wrong steady-state detection will negatively impact the method’s result. In addition, given the enormous amount of data and gradient estimations that would be required in a large-scale process, none of such “model-free” methods are viable for application in real industrial processes. This notwithstanding, the aforementioned disagreement may be caused by any factor that impairs information acquisition and processing in real implementations, constituting sources of uncertainty, such as: (i) measurements signals corrupted by noise with an unknown error structure; (ii) variation of the elements of input $\mathbf{I}$, neither measured, nor estimated ($\mathbf{var}\u2288(\mathbf{ms}\cup \mathbf{upd})$); (iii) use of the wrong default values for fixed variables; (iv) use of an inaccurate process model; (v) violation of maximum likelihood assumptions; (vi) imperfect numerical optimization method; and (vii) imperfect steady state detection or gross error filtering.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

RTO | Real-time optimization systems |

MPC | Model predictive control |

MON | Monitoring sequence |

SSD | Steady-state detection |

APC | Advanced process control |

DCS | Distributed control system |

SM | Statistical method |

HM | Heuristic method |

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**Figure 1.**Topology of a real RTO system running in a crude oil distillation unit. MON, monitoring sequence; SSD, steady-state detection; APC, advanded process control; DCS, distributed control system.

**Figure 2.**Percentage of values assumed static as a function of critical values ${C}_{c}$ (the analogous of ${R}_{c}$ as defined by Equation (13)) of the hypothesis test applied to signals of four process variables.

**Figure 3.**Normalized values of flow rate $F2$ (blue line) and the corresponding indication of stationarity according to the RTO system (pink line), where one means steady and zero means unsteady.

**Figure 4.**(

**A**) Normalized values of flow rate $F2$, as shown in Figure 3, in restricted axes values; (

**B**) C-statistic values for the corresponding time interval. The value of ${C}_{c}$ according to the method is 1.64.

**Figure 5.**Evolution of ${s}_{d}^{2}$ and $2{s}^{2}$ along the first 200 min for the signal depicted in Figure 3.

**Figure 6.**Interval between the first and third quartiles of the normalized effect of each variable within

**obj**for the objective function from the model adaptation problem (Equation (21)). The x-axis refer to the relative position of variables in the vector

**obj**.

**Figure 7.**(

**A**) Distribution of estimated values of knots; (

**B**) distribution of the relative deviation between two successive estimations of a knot.

**Figure 8.**Distribution of the relative difference of objective function values under convergence between two successive iterations ($\Delta {J}^{id}$).

**Figure 9.**Distribution of the values for $\Delta {\varphi}_{prev}$ and $\Delta {\varphi}_{verif}$ for the RTO profit. Above the graph, the percentiles P5, P50 and P95 are indicated as [P5 P50 P95].

**Table 1.**Percentage of points deemed static for the set of 8 variables. ${\left.{r}_{EE}\right|}_{SM}$: percentage computed using results obtained by the C-statistic. ${\left.{r}_{EE}\right|}_{RTO}$: percentage computed using real results obtained by the RTO system Tool A. F, flow rate; T, temperature.

Tag | ${\left.{\mathit{r}}_{\mathit{EE}}\right|}_{\mathit{SM}}$ | ${\left.{\mathit{r}}_{\mathit{EE}}\right|}_{\mathit{RTO}}$ |
---|---|---|

$F1$ | 0.0 | 98.3 |

$F2$ | 3.2 | 81.1 |

$T1$ | 0.0 | 97.3 |

$T2$ | 0.0 | 90.9 |

$F3$ | 0.0 | 97.1 |

$F4$ | 4.8 | 99.5 |

$F5$ | 0.0 | 90.1 |

$F6$ | 6.8 | 84.4 |

**Table 2.**Percentile values of normalized effects (%) of the 10 most influential variables in the value of the objective function from the model adaptation problem. The rank is relative to the 50th percentile (P50).

Position in Rank | Tag | $\in \mathbf{ms}$ | P50 | P90 | P10 |
---|---|---|---|---|---|

1 | $T01$ | yes | 21.53 | 54.83 | 1.15 |

2 | $T02$ | yes | 15.99 | 54.33 | 0.19 |

3 | $F01$ | yes | 2.37 | 11.39 | 0.10 |

4 | $T03$ | yes | 1.95 | 7.49 | 0.48 |

5 | $T04$ | yes | 1.68 | 14.21 | 0.35 |

6 | $T05$ | yes | 1.41 | 7.01 | <0.01 |

7 | $T06$ | yes | 1.17 | 5.94 | 0.04 |

8 | $F07$ | yes | 1.12 | 7.39 | <0.01 |

9 | θ(8) | no | 0.96 | 4.08 | 0.09 |

10 | $T07$ | yes | 0.65 | 2.17 | 0.12 |

**Table 3.**Lower and upper bounds on the knots of the model adaptation problem and the percentage number of iterations in which the constraint is active under convergence.

Knot | ${\mathbf{Lim}}_{\mathbf{inf}}$ | ${\mathbf{Lim}}_{\mathbf{sup}}$ | Active Constraint (%) |
---|---|---|---|

1 | 0.2 | 10 | 42.5 |

2 | 0.2 | 10 | 13.4 |

3 | 0.2 | 7.5 | 31.5 |

4 | 0.2 | 7.5 | 81.7 |

5 | 0.2 | 7.5 | 0.7 |

6 | 0.2 | 7.5 | 0.2 |

7 | 0.2 | 7.5 | 92.1 |

8 | 0.2 | 3 | 19.4 |

9 | 0.2 | 3 | 1.2 |

10 | 0.2 | 3 | 14.6 |

11 | 0.2 | 3 | 0.3 |

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Câmara, M.M.; Quelhas, A.D.; Pinto, J.C.
Performance Evaluation of Real Industrial RTO Systems. *Processes* **2016**, *4*, 44.
https://doi.org/10.3390/pr4040044

**AMA Style**

Câmara MM, Quelhas AD, Pinto JC.
Performance Evaluation of Real Industrial RTO Systems. *Processes*. 2016; 4(4):44.
https://doi.org/10.3390/pr4040044

**Chicago/Turabian Style**

Câmara, Maurício M., André D. Quelhas, and José Carlos Pinto.
2016. "Performance Evaluation of Real Industrial RTO Systems" *Processes* 4, no. 4: 44.
https://doi.org/10.3390/pr4040044