Abstract - University of Edinburgh



Dynamic optimization of beer fermentation: sensitivity analysis of attainable performance vs. product flavor constraintsAlistair D. Rodman, Dimitrios I. Gerogiorgis*School of Engineering, University of Edinburgh, The King’s Buildings, Edinburgh, EH9 3FB, UK*Corresponding author: D.Gerogiorgis@ed.ac.uk (+44 131 651 7072)AbstractThe declining alcohol industry in the UK and the concurrent surge in supply and variety of beer products has created extremely competitive environment for breweries, many of which are pursuing the benefits of process intensification and optimisation. To gain insight into the brewing process, an investigation into the influence of by-product threshold levels on obtainable fermentation performance has been performed, by computing optimal operating temperature profiles for a range of constraint levels on by-product concentrations in the final product. The DynOpt software package has been used, converting the continuous control vector optimisation problem into nonlinear programming (NLP) form via collocation on finite elements, which has then been solved with an interior point algorithm. This has been performed for increasing levels of time discretisation, by means of a range of initialising solution profiles, for a wide spectrum of imposed by-product flavour constraints. Each by-product flavour threshold affects process performance in a unique way. Results indicate that the maximum allowable diacetyl concentration in the final product has very strong influence on batch duration, with lower limits requiring considerably longer batches. The maximum allowable ethyl acetate concentration is shown to dictate the attainable ethanol concentration, and lower limits adversely affect the desired high alcohol content in the final product. IntroductionDetermining how a modern industrial production process shall be operated typically involves mathematical optimisation in some form. Often this will include an optimal control problem, where a system of state variables [x] are influenced by an externally manipulatable control variable, u, so the optimal control vector u(t) is sought to maximise an objective, φ, here considering only a terminal payoff (Biegler, 2010; Biegler et al., 2012):minut, tfφ(xtf, tf) (1)s.tdx(t)dt=fxt, ut, xt0=x0(2)hxt, ut=0, gxt, ut≤0(3)hfxt=0gfxtf≤0(4)u(t)L≤ut≤utU, x(t)L≤xt≤x(t)U (5)The ordinary differential equations (ODEs) which dictate the state trajectories (Eq. 2) are influenced at any time by the current control (u) value, while Eq. 3 represents equality and inequality constraints across the entire time horizon, t ∈ t0,tf, with terminal constraints given by Eq. 4. Lastly the state and control boundaries are constrained within permissible bounds by Eq. 5.An investigation into the beer manufacturing industry in the UK has been performed to determine if a strong incentive for process intensification and optimisation exists. The alcohol industry as a whole has been in decline in recent years within the UK as shown in Fig. 1, where annual litres of pure alcohol per capita is the metric used to normalise for beverages of differing alcoholic strength. This a result of several factors: people are drinking from a later age and regular drinkers are turning away from high strength products, towards more costly and lower strength drinks, such as craft beer. Figure 1. Alcohol consumption per capita, for UK adults (15+) (Beer Statistics, 2015).Beer is however one of the few exceptions from the trend of a declining sector. The growing market share fuelled by recent increased demand for high value craft beer products produced on a small scale has led to the beer industry growing both in terms of production volume and market value. 1% year on year growth is predicted over the next 3 years, with the annual production volume in the UK expected to exceed 4.6 billion litres by 2019, compared to 4.2 billion in 2015. Fig. 2 shows the number of breweries in operation over the last 6 years in the UK: it is evident that there is very steady increase which is predicted to continue moving forward.Figure 2. Number of UK breweries in operation by year (Beer Statistics, 2015).Fig. 3 depicts the UK’s alcohol consumption in context vs the rest of Europe. While Scots may have a reputation of being heavy drinkers it is evident that while their per capita consumption is above the average for the rest of the UK, it is still a very typical value within Europe.Figure 3. Alcohol consumption by country (Beer Statistics, 2015).The result of the declining alcohol industry and the surge in supply of beer products has created an extremely competitive environment for producers, many of whom must look towards process intensification if they are to remain profitable, forming the motivation for this study.Within the beer production process the fermentation stage is generally the system bottle neck, with batch times in excess of one week not uncommon. Fermentation progression depends on many variables (Rodman and Gerogiorgis, 2016b), however progression is dominated by the influence of the temperature of the involved substrates. As such, it is necessary to determine the temperature manipulation profile capable of steering the process to competition in an optimal manner.Approaches to process optimisation fall under three areas (Bonvin, 1988):off-line optimisation (open loop optimal control)run-to-run optimisationon-line optimisationThis study is concerned with the former: determining solutions to the off-line optimisation problem to provide optimal open loop trajectories for the manipulated and state variables. These profiles are computed once, off-line, thus feedback elements are not included, and rather an ideal recipe for optimal production is produced. This approach is limited in usefulness as in the presence of disturbances these trajectories lose their optimal character (Canto et al., 2000), however on-line optimisation is not practical: online concentration readings are extremely cumbersome to monitor in many cases. Rather many medium scale breweries elect to take a sample once the prescribed temperature trajectory has completed and determine the residual sugar content based on the product density (a surrogate measurement for total sugar content) via the Plato scale, to confirm if the batch has completed fermentation as expected and desired. This convention renders any attempt to incorporate an online control loop for control of state (concentration) trajectory control non-applicable to this particular problem and is the reason why our study is focused on off-line optimisation. A beer brand or line instead typically has a proprietary temperature manipulation profile (recipe) used for every batch, to ensure product consistency, which fits the scope of this work.Process Description 2.1 Beer FermentationFermentation is an essential step in the manufacture of alcoholic beverages, responsible for the characteristic taste of the final product and its alcohol content (Rodman and Gerogiorgis, 2016c). Upstream processing produces a sugar rich intermediate (wort) from a feedstock starch source (most typically malted barley). Once cooled to an appropriate initial temperature the wort enters stainless steel vessels along with yeast, allowing fermentation to commence. The primary chemical reaction pathway is the conversion of sugars into ethanol and carbon dioxide, which is coupled with biomass (yeast) growth and heat generation from the exothermic reaction. Concurrently, a range of species are formed at low concentrations by a multitude of side reactions, many of which may impact product flavour above threshold concentrations. Fermentation is completed once all consumable sugars have been converted by the yeast into alcohol, following which the solution leaves the fermenter for subsequent downstream processing prior to sale and consumption.2.2 Fermentation modellingSeveral mathematical models for the beer fermentation process have been published (Gee and Ramirez, 1988; de Andres-Toro, 1998, Trelea et al., 2001). Models are reduced order, considering only the key species present due to system complexity (200+ species, Vanderhaegen et al., 2006) rendering exhaustive modelling extremely cumbersome: in fact to date many of the specific chemical interactions in the fermentation process are not understood. The kinetic model of beer fermentation by de Andrés-Toro et al. (1998) has been selected for study due to its direct applicability to the industrial process:Published parameters are derived from a very large array of experiments, resulting in a wide temperature range (8–24 ?C) which ensures high fidelity and applicability. The model includes all prominent by-products which degrade beer product quality in terms of taste and aroma, rendering the model valuable for assessing performance. Predicted profiles indicate the highest fidelity with experimental and pilot-plant data in comparison to other models, due to successful validation against over 200 fermentations. The model considers 7 states (Eqs. 6-12), with trajectories governed by temperature dependant production and consumption factors (Eqs. 13-16). The model structure takes the form shown in Fig. 4. Yeast cells transition from latent to active to dead over time, with only active cells able to promote fermentation (conversion of sugar to ethanol). A more detailed description of the model can be found in its original publication (de Andrés-Toro, 1998), along with the constants for the Arrhenius relationship governing the parameters temperature dependence, as computed from industrial scale fermentation data. The exception is that the rates for diacetyl are taken from a later publication, shown in Table 1 along with the initial state concentrations used for simulation.d[XA](t)dt =μxt,T?[XA](t)-μDT(t,T)?[XA](t)+μL(t,T)?[XL](t) (6)d[XD](t)dt =-μSD(t,T)?[XD](t)+μDT(t,T)?[XA](t) (7)d[S](t)dt =?-μS(t,T)?[XA](t) (8)d[EtOH](t)dt =?ft?μe(t,T)?[XA](t) (9)d[EA](t)dt =YEA(T)?μx(t,T)?[XA](t) (10)d[DY](t)dt =μDY?[S](t)?[XA](t)-μAB?[DY](t)?[EtOH](t) (11)d[XL](t)dt =?-μL(t,T)?[XL](t) (12)μx(t,T) = μx0T?[S](t)0.5?[S]0+[EtOH](t) (13)μSD(t,T) = μSD0(T)?0.5?[S]00.5?[S]0+[EtOH](t) (14)μs(t,T) = μs0T?[S](t)ksT+[S](t) (15)μe(t,T) = μe0T?[S](t)keT+[S](t) (16)f(t) =1-[EtOH](t)0.5?[S]0 (17)Figure 4. Kinetic model for beer fermentation under industrial conditions (de Andres-Toro, 1998).The model considers two by-product species alongside the primary reaction pathway: ethyl acetate (Eq. 12) and diacetyl compounds (Eq. 11). Diacetyl (2,3-butanedione) has a pungent butter-like aroma (Izquierdo-Ferrero et al., 1997), while ethyl acetate is often used as an indicator of all esters present, and is described as having the odour of nail varnish remover (Hanke, S. et al., 2010). Different beer products may contain differing levels of these species, since the specific flavour profile will influence the levels above which these by-products will degrade the flavour. This study has looked into the influence which imposed limits on these compounds concentrations in the final beer product have on the attainable fermentation performance and efficacy, by considering a range of realistic thresholds of each. Table 1. Model parameters for dynamic simulation of beer fermentation.SymbolDescriptionValue Units Diacetyl rates(Carrillo-Ureta et al., 2001)μDY Diacetyl production rate1.27672?10-7 g-1 h-1 LμAB Diacetyl consumption rate1.13864?10-3 g-1 h-1 LInitial simulation conditions[XA]+[XL]+[XD]Biomass inoculum (pitching rate) 4g L-1[S]0 Sugar concentration 130g L-1The primary source of potential modelling error lies in the model parameterization and the validity of these parameters for representing any specific brewery process. While the reduced order model structure has been repeatedly demonstrated to accurately represent the reactions taking place during the industrial fermentation of beer, model parameters are subject to variations due to biological system-specific factors (yeast strain, mutations, species aggregation). In our study the original model parameters have been used (de Andrés-Toro, 1998); these have been reported as computed from industrial data obtained from a fermentation campaign performed at a full scale industrial plant (Cruz Campo Brewery, Madrid, Spain).2.3 Literature reviewNumerous authors have used the de Andrés-Toro (1998) beer fermentation model for optimal control studies. Several have been stochastic approaches, including genetic algorithms (Carrillo-Ureta et al., 2001), ant colony system (Xiao et al., 2003) and simulated annealing (Rodman and Gerogiorgis, 2016a). Additionally, Bosse and Griewank (2014) have used the kinetic model to generate optimal control profiles using a sweeping dynamic optimisation methodology. The process involves guessing a control path and using this to integrate the states forward in time. This allows the costates to be integrated backwards through the process time span: a new control profile is thus deduced by maximising the Hamiltonian for all t ∈ [t0,tf], and the process is repeated until path convergence is attained. The authors were able to compute a more preferable temperature profile using the same objective, compared to a prior stochastic approach (de Andrés-Toro et al., 1997). 2.4 Process targets: objective functionWhen considering what it is desirable to improve in a fermentation process there are two obvious contenders: reduced duration and heightened alcohol content (even if this requires later dilution, it is still desirable to increase yield). In addition to batch time minimisation and alcohol production maximisation, all prior authors have elected to include terms for minimisation of both by-products within their respective optimisation objective functions. However, as is known that within certain beer products the concentrations of both ethyl acetate and diactyl compounds shall be indistinguishable below certain levels, efforts towards further reduction and concentration minimisation are redundant.As such it is deemed more appropriate to consider an objective function only seeking to minimise production time and maximise sugar conversion to ethanol (with variable relative weights) while treating the final concentrations of both ethyl acetate and diacetyl compounds as strict constraints to avoid unnecessary efforts towards further by product reduction.Thus the following objective shall be used in this study:minTt, tfφ(x, tf)=-WE?EtOH- Wt?1t (18)s.t. [EA]t=tf ≤ [EA]max(19)[DY]t=tf ≤ [DY]max(20)Here WE and Wt are the respective weights of the two components in the objective function: while a large range of weight values have been investigated, for conciseness only the case WE = Wt = 50% shall be presented in this paper. 1t is the inverse batch time normalised by division with the maximum value attained from exhaustive simulation (Rodman and Gerogiorgis, 2016a) and [EtOH] is the ethanol concentration normalised in the same way. In doing so the normalised ethanol concentration, [EtOH] ranges from 0.68 when [EtOH]= 42 g L-1 to 1 when [EtOH]= 61.3 g L-1, similarly the normalised inverse batch time, 1t , ranges from 0.62 to 1 when t is 99 hrs and 160 hrs respectively. Given the strong dependence of yeast health on system temperature it is necessary to include an additional constraint such that the control profile (temperature manipulation schedule) remains within acceptable levels. Eq. 21 ensures that the lower temperature limit excludes scenarios in which the system lacks enough energy to promote cell growth while the upper limit ensures bacteria which are present above this temperature cannot thrive, while also preventing the temperature from reaching a level at which undesirably high by-product concentrations are known to be produced.Tt∈[9 °C, 16 °C] for all t ∈ [t0,tf](21)2.5 Temperature profile: Performance insightA prior exhaustive simulation campaign has been performed, considering a finite set (175,000) of piecewise linear temperature manipulation profiles adhering to a low level of discretising with equally spaced time intervals, which also adhere to realistic operability heuristics for control profile formulation in order to reduce the feasible set considered (Rodman & Gerogiorgis, 2016a). To gauge what quantifies as effective fermentation performance the solution set can be examined. Fig. 5 presents the performance of each candidate profile used for simulation, considering four metrics: batch time (x-axis) as well as ethanol, diacetyl and ethyl acetate concentrations on the y-axis such that each point corresponds to the performance of a single candidate profile. Here yellow markers indicate solutions attaining an ethanol concentration above the prescribed minimum for each column, while orange markers represent those which also fall below the base case limit on ethyl acetate concentration, defined as [EA]max = 2.0 ppm. It is shown from the first column of Fig. 5 that profiles producing ethanol concentrations above 60 g L-1 are universally unable to fulfil the tolerable base case ethyl acetate threshold. Decreasing the acceptable ethanol concentration to 59.5 g L-1 in the middle column shows that there is now a subset of profiles which produce by-product concentrations below both [EA]max and [DY]max (0.1 ppm) for the base case, however it is found that these correspond to unfavourably long batch times. Further decreasing the ethanol limit to 59 g L-1 shows in the right most column that there is now a subset of solutions which correspond to short batch times (tf < 120 hr) which attain this ethanol concentration while maintaining by-products below base case thresholds. [EtOH]min = 60 g L-1 [EtOH]min = 59.5 g L-1 [EtOH]min = 59 g L-1Figure 5. Corresponding profile performance for lowering acceptable [EtOH]tf levels. Yellow staining of high [EtOH]tf implies unacceptably high [EA]tf. Orange staining of acceptable [EA]tf indicates [DY]tf satisfaction.Dynamic Optimisation of Beer FermentationA wide range of optimisation methodologies exist for solving optimal control trajectory problems. These include variation methods and finite approximation methods. In the former exploiting Pontryagin’s maximum principle allows the resulting two point boundary value problem to be solved, while the later uses predefined functional forms to represent the control profile (Almeida and Secchi, 2011). Finite formulations may be tackled with simultaneous, sequential or multi shooting strategies which are extensively reviewed in the literature (Biegler et al., 2002). The sequential strategy involves discretisation of the control profile with the ODE system (process model), requiring regular re-integration during the algorithm to compute corresponding state trajectories, an approach effective for problems with few decision variables and constraints (Osorio et al., 2005) which has been widely applied to engineering problems (Farhat et al., 1990; Mujtaba and Macchietto, 1993; S?rensen et al., 1996). In contrast, simultaneous strategies require the ODE system to also be discretised on the time horizon to produce a large scale NLP problem requiring no futher integration of the DEA system, generally using orthogonal collocation techniques. The later offers numerous benefits, being faster to solve and able to handle problems with a greater number of decision variables and constraints (Cervantes et al., 1998; 2000).In this paper we have applied a simultaneous dynamic optimization strategy using collocation on finite elements, which is an established methodology widely used in numerous chemical process optimization studies: the key contributions advancing the state of the art is that we are pursuing dynamic optimization for a combination of two competing objectives, and even more so under explicit constraint level variation. The orthogonal collocation on finite elements is a trusted and robust approach, which has been demonstrated to be applicable for constrained problems such as industrial fermentation optimization, unlike variational methods which are not efficient for solving constrained problems. Discretizing both the state and control variables to form large-scale NLPs allows rapid determination of solution profiles with fewer finite elements than when using sequential methods that apply standard ODE solvers and are unable to properly handle problem instabilities. The simultaneous strategy has further advantages for the treatment of path constraints, useful if we were to extend the problem, for example, to prohibit by-product constraint violation at any time in place of at terminal time only. Limitations of the strategy employed are that efficient large scale NLP solvers are required, while global optimization is cumbersome (indeed, the optimal solution profiles computed here are indeed strongly dependent on the initialization considered, but they also clearly illustrate the multitude of attainable tradeoffs between ethanol maximization and batch duration minimization). 3.1 DynOpt for fermentation optimisation A direct method for dynamic optimisation (simultaneous strategy) has been performed in this study. Orthogonal polynomials on finite elements are used to approximate the control and state trajectories allowing the continuous problem described by Eqs. (6-21) to be converted to NLP form. Implementation has been performed using the DynOpt package for MATLAB (Cizniar et al., 2006). The DAE system is converted to a system of algebraic equations, where decision variables of the derived NLP problem are the coefficients of the linear combinations of these AEs. Precision is known to vary with collocation point locations and step sizes used (Logsdon and Biegler, 1989; Tanartkit and Biegler, 1995). Considering the general problem (Eqs. 1-5) with N elements (i = 1, …, N), each of which with K collocation points (j = 1, …, K). The differential profiles (Eq. 2) can be approximated by:xi=xi-1+?tij=1KΩjt-ti-1 ?tidxdti,j(22)Where ?ti is the length of element i and dx/dti,j is the derivative of the state variable in element i at the jth collocation point. Ωj is a Kth order polynomial satisfying:Ωj0=0, Ω'jρj= δj for j=1,…, K(23)Continuity of the state trajectories is ensured with:xt=xi-1+?tij=1KΩj1dxdti,j(24)while the control profile is approximated by: ut=q=1K?ψjt-ti-1 ?tiui,j(25)Where ?ψj is a Lagrange polynomial of degree K that satisfies?ψjρj= δj for j=1,…, K. It is shown in Fig. 6 how control variables may have discontinuities at element boundaries, while Eq. 24 produces continuity in states at these same boundaries. Figure 6. Collocation method for state and control profiles (based on Biegler, 2006). Applying Eqs. 22-25 to the fermentation problem described by Eqs. 6 – 21 the resulting NLP problem is as follows, where x is a vector containing the 7 model states (Eq. 6-12) and Ti,j is the fermenter temperature in element i at collocation point j:minxi,j, Ti,j, ?ti φx, tf= -WE?EtOHN- Wt?1tf (26) s.t dxdti,j=f(xi,j, Ti,j)(27)xi,j=xi-1+?tij=1KΩj'ρjdxdti,j(28)hxi,j, Ti,j=0, i = 1, …, N, j = 1, …, K(29)xi,j=xi-1+?tij=1KΩj1dxdti,j, i = 1, …, N(30)gfxN≤0 → [EA]N - [EA]max≤0, [DY]N-[DY]max ≤ 0(31)9≤ Ti,j≤16, 0≤xi,j, i = 1, …, N, j = 1, …, K(32)tf=i=1N?ti(33)This large scale NLP problem produced from the DynOpt code has been solved with the fmincon MATLAB function, using the interior point algorithm from the optimisation toolbox, a detailed derivation of which is given by Waltz (2006). Three collocation points have been used for state trajectories, with one collocation point being used for control profiles, resulting in the computation of temperature profiles which are piecewise-constant. The NLP solver has been executed in each case for a fixed number of iterations, rather than setting solution tolerances as stopping criteria: the maximum function evaluations for each discretisation level is given in Table 2.Table 2. Summary of solution conditions, producing 800 cases.Number of valuesRange[EA]max (ppm)5[0.5, 1.0, 1.5, 2.0, 3.0][DY]max (ppm)5[0.05, 0.10. 0.15, 0.20, 0.25]Discretisation level, N8[6, 12, 18, 24, 30, 36, 42, 48] Initializing profile4[A, B, C, D]Max function evaluations for N[2000, 5000, 10000, 15000, 20000, 25000, 30000, 30000]3.2 Initialization Due to the high number of local extrema that exist when discretizing a control vector problem to NLP form, the initializing profile has considerable bearing on the resultant output profile, which cannot be guaranteed as globally optimal. An investigative campaign was performed using five isothermal profiles to initialize the solver (T = 11, 12, 13, 14, 15 °C). As these isothermal profiles do not show particularly suitable performance for industrial beer fermentation, the result was that the outputs in fact did not represent profiles for great performance either, due to confinement to local solutions in the vicinity of the isothermal input. To overcome this limitation, it is desirable to input a profile known to have good performance, such that the algorithm can act to improve on this. We have selected a range of profiles from a prior exhaustive simulation campaign (Rodman & Gerogiorgis 2016a) performed with a low discretization level, N = 6. Figure 7. Promising profiles from exhaustive simulation to be used for initilaization of DynOpt.These corresponding profiles which have been used for initializing the DynOpt code, are shown in Fig. 7, where their position from Fig. 5 is highlighted with stars of corresponding colour in the plot which is cropped around the desirable region. To visualise how these profiles perform, Fig. 8 depicts the position of these points on the performance plots for the entire solution set. The top plot (ethanol vs. batch time) shows that the four profiles taken forward from exhaustive simulation for initialising the simultaneous optimisation procedure all fall towards the more desirable portion of the plot.Figure 8. Performance of four highlighted profiles with respect to by-product concentration thresholds. The vast number of points which correspond to lesser batch time and greater ethanol concentration (top left corner of plot) however suggest that there is significant scope to improve upon these profiles. The lower two plots (by-products vs. batch time) show that profiles A, C & D all universally fulfil both base limits of the by-product species, while prolife B in fact does violate the diacetyl limit. This is of interest to observe how a constraint violation in the initializing solution affects the performance of the algorithm in producing optimal T(t) profile outputs.As these initializing profiles are piecewise linear and DynOpt is computing piecewise constant temperature profiles it is necessary to approximate the profiles in Fig. 7 to a piecewise constant form, which will differ for each discretisation level solved. This approximation is performed by averaging the temperature over N steps of equal duration: this transformation is shown in Fig. 9 for profile D. It is demonstrated that N increases the profile tends to the original piecewise linear form. Figure 9. Piecewise constant approximations of profile D (Fig. 7) for varying discretisation levels. Results & DiscussionTo thoroughly investigate effect which by-product constraint thresholds (Eqs. 19-20) have the attainable fermentation performance, and to access the methodology performance, a large campaign of cases have been solved. Five realistic thresholds for ethyl acetate and diacetyl have been selected producing 25 constraint permutations for which the system will be solved. Each is performed for eight different discretisation levels and initialized with each of the four input profiles (Fig. 7) in turn, meaning a solution set of 6400 control profiles has been produced, summarized in Table 2. A sample of solution profiles are presented here for conciseness. Fig. 10 shows the temperature profiles computed for the base case by-product concentration thresholds, for each initializing profile and a selection of the discretization levels used. Figures 11-13 shows the corresponding profiles for different constraint threshold levels as indicated by the figure captions: low [EA]max & high [DY]max; high [EA]max & low [DY]max; low [EA]max & low [DY]max respectively, such that the Figs. 10 – 13 represent 4 of the 25 total permutations computed. Within the final column of each figure (N = 42) the best and worst performing profiles for each aspect of the bi-criteria objective are highlighted with the corresponding performance metric value within the plot panel. Remarkably several profiles produced have a strong resemblance to the profile form obtained by Bosse and Griewank (2014), showing characteristic dual peaks with a moderate dip in temperature between them. The authors did use the same fermentation model, however described a quite different objective, considering [EA] and [DY] as minimisation criteria rather than constraints as done here, in addition to not considering batch time as a minimisation target.Figure 10. Computed profiles for base case: [EA]max = 2.0 ppm, [DY]max = 0.10 ppm.Figure 11. Computed profiles for low [EA]max = 0.5 ppm, high [DY]max = 0.25 ppm.Figure 12. Computed profiles for high [EA]max = 3.0 ppm, low [DY]max = 0.05 ppm.Figure 13. Computed profiles for low [EA]max = 0.5 ppm, low [DY]max = 0.05 ppm.4.1 Effect of increasing time domain discretisationInspecting horizontal rows from Figs. 10 – 13 in isolation allows the influence of the discretisation level to be observed. It is apparent that regardless of the discretisation level the solution form is similar: in general as N increases the profile becomes refined, following a similar trajectory in a smoother manner. There are however several instances where considerable deviations occur. Firstly from Figure. 10 the upper row (input profile A) shows that depending on the discretisation level, the solution profile has a drastically different initial temperature, T(t=0), ranging from 11 to 16 °C. A similar observation can be made from the third row of the same figure (input profile C). The case presented in Figure 11 also shows considerable differences in the profile as N increases. At lower levels of N the solution produced is very flat, while once a greater degree of control is allowed a much more variable solution is obtained, which corresponds to a significantly improved objective. A further example of the solution being sensitive the discretisation level is shown in Figure 12 row 4 (input profile D). The profiles initial temperature, T(t=0), is considerably lower at high values of N compared to that at lower discretisation levels. These differences indicate the discretisation level can have significant bearing on the specific profile being produced, however in most cases the overall solution form does not differ drastically. There are some cases in which lower discretization levels yield shorter batch times compared to those obtained for higher discretization levels (e.g. Figure 13, cases A and C); these occurences are affected by both the initialisation T(t) profile, but also by the fact that gradual temporal grid refinement favors further [EtOH] maximization at the expense of batch duration (tf) minimization.It is demonstrated that as the number of piecewise constant sections in the profile exceeds 40 the solution is smoothed and tends towards a continuous form, which ultimately removes the implementation issues of using piecewise constant profiles (instantaneous temperature adjustments). It is shown that as N is increased the attainable value for the objective function (Eq. 18) increases, as expected due to the greater level of control possible with a higher number of manipulatible sections in the temperature profile. 4.2 Effect of initialising temperature profileTo observe the influence of the initializing profile it is necessary to inspect columns from Figs. 10 – 13 in isolation. An immediate observation is that solutions do not converge to the same solution, meaning that globally optimal solutions are not being produced, rather the input initialising profile has significant impact on the profile output for any set of conditions when using this methodology. However, there exists a large number of similarities between the solution profiles’ appearance, with significant features present across all solutions even when differences in the duration or magnitude of these features exists. Comparing the performance of solutions which differ only in the initialising profile (each figure column) show very similar values, with the objective value (Eq. 18) only differing by more than 1% in very few cases. Within Figure 11. The profile produced at N = 42 when initialized with profile D is considerably different from those from the alternative input profiles. Its secondary peak is not present in the other profiles which significantly affects performance. A lower ethanol concentration is produced than all three other solutions, however batch time is drastically less such that case D has the most desirable objective. This is likely a result of initializing profile D featuring this late peak, which does correspond to the best performing of the 4 input profiles considered. Furthermore, within Figure 13 the final columns (N = 42) shows two differing solution forms: input profiles B and D produce an output profile with a gradual temperature reduction followed by a later peak, while inputs A and C produce an alternative form with a gradual temperature increase towards this peak. The former solution form corresponds to a greater ethanol production, while the later permits a much shorter batch duration, however once again the overall objective is not drastically different. This emphasises the influence which the input profile has on the resultant solution obtained by the algorithm, given that globally optimality is not being achieved. 4.3 Effect of by-product constraint thresholdsTo discover the influence which the concentration limits of by-product species in the beer product have the attainable fermentation efficiency it is necessary visualise the performance of all 25 threshold permutations simultaneously. Figure 14 presents this for the discretization level N = 30 with the four plots corresponding to the results using the four different initializing profiles. Each point corresponds the performance an output profile; the corresponding by-product concentration limits are represented on the x-y plane with the z axis showing the batch time, tf and the marker colour corresponding to the ethanol product concentration, [EtOH]tf. Figure 14. Performance of output profiles (N = 30) for all constraint permutations and initialising profiles. It is observed that for all input profiles, the resultant performance points are very close, reiterating that while the solutions do differ their performance is highly comparable across the initialling profiles used. The results show a very coherent pattern indicating the manner in which fermentation performance is influenced. It is shown that batch time universally increases as the acceptable threshold on diacetyl, [DY]max, is reduced. Batch time does indeed also increase as [EA]max is reduced, however the relationship is far less significant, with the dependency on the diacetyl threshold much stronger. The marker colours show how it is exclusively the ethyl acetate threshold, [EA]max, which influences the final ethanol yield. In all cases when [EA]max, = 0.5 ppm the product ethanol, [EtOH]tf, is very low (under 56 g L-1), which increases steadily towards 61 g L-1 as this permitted [EA]max, threshold is relaxed towards 3 ppm. These results reveal as to how the two components of the bi-criteria objective are dictated by the two inequality constraints on the by-product concentrations:[DY]max has very strong influence on batch time.[EA]max is shown to dictate the attainable ethanol concentration.Figure 15. Performance plot projection on by-product threshold plane (N = 30): marker size scaled with tf-1.Fig 15. represents the projection of the four plots from Fig 14. on the by-product concentration plane. Here the marker size is scaled relative to inverse batch times: smaller makers show the worst performing solutions (longest batch times) while larger markers show the best performance (shortest batch times). It is shown that a small selection of solutions do not fall directly on the intersect of the two by-product limits imposed for the particular case being solved. There are two factors responsible for this; firstly there are several cases where the constraints are comfortably fulfilled, ie the solution produced for the case [EA] < 1.0 ppm & [DY] < 0.25 ppm in fact has a much lower [DY] of only 0.22 ppm. This could suggest sub optimality in the solution, perhaps with a shorter batch time possible if the concentration of diacetyl were to increase more. Secondly, all performance results presented in this work have been computed after reintegration of the system using the solution control profile. Slight deviations exist between the performance of the profile during the NLP algorithm and later integration of the solution, depending on the accuracy of the piecewise polynomial representation of the continuous state trajectories in the NLP formulation. Deviations have are shown to be non-significant as the algorithm used captures the state trajectories effectively. 4.4 Performance of key output profilesOf the entire solution set computed it is of interest to inspect the profiles which correspond to the extrema in terms of performance. As such 5 cases are presented from the N = 30 solution set:The best performing profile from the base case ([EA]max = 2.0 ppm, [DY]max = 0.10 ppm)The profile corresponding to the longest batch time, tf, from all cases The profile corresponding to greatest ethanol concentration, [EtOH]tf, from all casesThe profile corresponding to the shortest batch time, tf, from all casesThe profile corresponding lowest ethanol concentration, [EtOH]tf, from all casesTo allow visualization of how these compare among the rest of the solutions, they are highlighted in the collated performance plot depicted in Figure 16. The corresponding profiles are shown in Figure. 17, as well as the state trajectories of all species considered in the dynamic model. The performance of these profiles is summarised in Table 3.Figure 16. Collated profile performance (N = 30) for all constraint permutations and initialising profiles.Table 3. Performance of extrema profiles computed. SolutionInitialising profile[EtOH]tf (g L-1)tf (hr)[EA]max (ppm)[DY]max (ppm)ID59.8114.22.00.10IIB59.6142.93.00.05IIID60.5128.53.00.05IVD60.194.93.00.25VB55.4120.30.50.05It is demonstrated that the solution which corresponds to the shortest batch times aligns with the greatest value of [DY]max, and conversely the longest batch corresponds to the maximum value of [DY]max, further confirming the strong correlation between the two parameters. It is shown that the greatest product ethanol concentration occurs when both constraints are fully relaxed, with the lowest ethanol concentration corresponding to the tightest by-product limits, indicating that both species thresholds in fact influence the obtainable ethanol concentration. It is noteworthy that the solutions corresponding to desirable results (I, III, IV) are produced using initializing profile D; this input profile is the scenario which maximises Eq. 18 from the entire exhaustive simulation set (Rodman & Gerogiorgis, 2016a). Similarly the two undesirable solutions representing minimum ethanol and longest batch times (II, V) have been computed from profile B which is the least preferable of the 4 input profiles considered. This demonstrates that a better performing input profile enables the computation of the most preferable output profiles from the DynOpt algorithm, given the solution sensitivity to the input profile.Figure 17. State trajectories corresponding to key temperature profiles computed. ConclusionsA multi-objective dynamic optimisation study has been carried out, investigating the potential for process improvement of industrial scale batch beer fermentation via modifications to the fermentor temperature profile throughout the duration of the process stage. A simultaneous method for direct dynamic optimisation of the temperature profile has been performed for a spectrum of threshold values on by-product concentrations to investigate the effect which these have on the obtainable process performance. The scope and demand for beer fermentation optimisation has been motivated and justified on the basis of a detailed survey of production statistics, and optimal fermentor temperature manipulations (dynamic profiles) towards minimising batch duration and maximising product yield (final ethanol concentration) have been computed via a simultaneous strategy. A key contribution of the present study is the explicit consideration of by-product constraint level variability, as the extensive illustration and comparative evaluation of optimal T(t) profiles for constraint variation (ethyl acetate, [EA] and diacetyl, [DY] thresholds) clearly reveal the pivotal influence that these limits have on processing targets. The maximum allowable ethyl acetate concentration significantly affects the ethanol yield: a relaxation from 0.5 ppm to 3 ppm increases ethanol from 55.5 g L1 to 65.5 g L1.The maximum allowable diacetyl concentration in the final product also has a very strong influence, but mainly on batch duration: a relaxation from 0.05 ppm to 0.25 ppm can reduce batch time by up to 33%. The sequential dynamic optimisation procedure has been performed using the DynOpt package for MATLAB. Discretising the state trajectories in addition to the control vector using orthogonal collocations permits a large scale NLP problem to be solved, here for piecewise constant temperature profiles. This is performed for 25 different pairs of constraint thresholds on ethyl acetate and diacetyl in the beer product to compare the performance of the optimal profiles produced in each case. A range of discretization levels (N) and initializing profiles are used for each scenario producing a total of 6400 unique cases which have been solved. The implemented algorithm does not address global optimisation, exactly because of the clear and prominent tradeoffs between the two objectives (ethanol maximisation and batch duration minimisation), but also due to the sensitivity of optimal T(t) solution profiles which depend on the initialization profile T0(t) considered. It is shown that the best performing input profiles produce the best performing outputs, however the difference in performance is not significant providing an acceptable profile is used for initialisation. It is shown that as N is increased the attainable value for the objective improves, as expected due to the greater level of control possible with a higher number of manipulatible sections in the temperature profile. It is demonstrated that as the number of piecewise constant sections in the profile exceeds 40 the solution is smoothed and tends towards a continuous form, which ultimately removes the implacability of using piecewise constant profiles (instantaneous temperature adjustments). The investigation into the influence of by-product threshold limits on obtainable fermentation performance has revealed new insight into how each by-product uniquely affects process performance. It is found that the permitted diacetyl concentration in the product has very strong influence on batch time, with lower limits requiring considerably longer batches. Ethyl acetate is shown to dictate the attainable ethanol concentration, such that low limits prohibit a reasonable alcohol content in the product.Acknowledgements The authors gratefully acknowledge the financial support of the Eric Birse Charitable Trust for a Birse Doctoral Fellowship awarded to Mr A.D. Rodman, and that of the Engineering and Physical Sciences Research Council (EPSRC) via funding from an Impact Acceleration Account (IAA) administered by Edinburgh Research & Innovation (ERI). Moreover, Dr D.I. Gerogiorgis gratefully acknowledges a Royal Academy of Engineering (RAEng) Industrial Fellowship which he has been awarded (2017). Both authors express thanks to Mrs Hilary Jones, Mr Simon P. Roberts and Mr Udo Zimmermann (WEST Beer) for consistent encouragement and inspiring discussions throughout this research project.Nomenclature Roman symbols WEObjective ethanol weight (%)WtObjective time weight (%)XAActive biomass concentration (g L-1)XDDead biomass concentration (g L-1)XLLatent biomass concentration (g L-1)YEAEthyl acetate production stoichiometric factor (g L-1)keEthanol affinity constant (g L-1)ks Sugar affinity constant (g L-1)kxBiomass affinity constant (g L-1)DY Diacetyl (-)EAEthyl Acetate (-)EtOHEthanol (-)g Inequality constraint hEquality constraintK Number of collocation pointsN Number of elemets in time horizon SSugar (-)TFermenter temperature (K)fFermentation inhibition factor (g L-1)tTime (h)tfBatch time (h)uModel controlxModel stateGreek symbols?tiLength of element iμABDiacetyl consumption rate (g-1 h-1 L)μDTSpecific cell death rate (h-1)μDYDiacetyl growth rate (g-1 h-1 L)μEEthanol production rate (h-1)μLSpecific cell activation rate (h-1)μSSugar consumption rate (h-1)μSDSpecific dead cell settling rate (h-1)μxSpecific cell growth rate (h-1)ΩState approximation polynomialφObjective function (-)ψControl approximation polynomialSubscripts and operators ( )Normalised parameter (-)( )0Initial condition (-)( )LLower bound (-)( )UUpper bound (-)( )iProperty in element i (-)( )jProperty at collocation point j (-)ReferencesAlmeida Nt. E. and Secchi, A. R., 2011. Dynamic optimization of a FCC converter unit: numerical analysis.?Brazilian Journal of Chemical Engineering,?28(1): 117-136.Beer Statistics (2015) – Available online at: uploads/mycms-files/documents/publications/2015/statistics_2015_v3.pdf.Biegler, L. T., 2006. An overview of simultaneous strategies for dynamic optimisation. Chemical Engineering and Processing, 46: 1043-1053.?Biegler, L. T., 2010. Nonlinear programming: concepts, algorithms, and applications to chemical processes. SIAM publishing. Print.Biegler, L. T., Campbell, S. L. and Mehrmann, V., 2012. Control and optimization with differential-algebraic constraints. SIAM publishing. Print.Biegler, L. T., Cervantes, A. M. and W?chter, A., 2002. Advances in simultaneous strategies for dynamic process optimization. Optimization, Chemical Engineering Science, 57: 575-593.?Bonvin, D., 1998. Optimal operation of batch reactors—a personal view. Journal of Process Control,?8(5): 355-368.Bosse, T. and Griewank, A., 2014. Optimal control of beer fermentation processes with Lipschitz‐constraint on the control. Journal of the Institute of Brewing,?120(4): 444-458.Carrillo-Ureta, G., Roberts, P. and Becerra, V., 2001. Genetic algorithms for optimal control of beer fermentation. Proceedings of the IEEE International Symposium on Intelligent Control, 391-396.Cervantes, A. and Biegler, L. T., 1998. Large-scale DAE optimization using a simultaneous NLP formulation. AIChE Journal, 44(5): 1038-1050. Cervantes, A., Wachter, A., Tutuncu, R. H. and Biegler, L. T., 2000. A reduced space interior point strategy for optimization of differential algebraic systems. Computers and Chemical Engineering, 24: 39-51.Cizniar, M., Fikar M, and Latifi, M. A., 2006. MATLAB Dynamic Optimisation Code DYNOPT, User's guide, Technical Report, KIRP FCHPT STU, Bratislava.de Andrés-Toro, B., Giron-Sierra, J., Lopez-Orozco, J. and Fernandez-Conde, C., 1997. Application of genetic algorithms and simulations for the optimization of batch fermentation control. 1997 IEEE International Conference on Systems, Man, and Cybernetics. Computational Cybernetics and Simulation, 392-397.de Andrés-Toro, B., Giron-Sierra, J., Lopez-Orozco, J., and Fernandez-Conde, C., 1998. A kinetic model for beer production under industrial operational conditions. Mathematics and Computers in Simulation, 48(1): 65-74.ence on Decision and Control, Vol. 1, pp. 828–829.Farhat, S., Czernicki, M., Pibouleau, L. and Domenech, S., 1990. Optimization of multiple-fraction batch distillation by nonlinear programming. AIChE Journal. 36: 1349-1360.Gee, D. A. and Ramirez, W. F., 1988. Optimal temperature control for batch beer fermentation. Biotechnology and Bioengineering, 31: 224-234.Hanke, S., Ditz, V., Herrmann, M., Back, W., Becker, T and Krottenthaler, M., 2010. Influence of ethyl acetate, isoamyl acetate and linalool on off-flavour perception in beer. Brewing Science, 63(7): 94-99.Izquierdo-Ferrero, J. M., Fernández-Romero, J. M., Luque de Castro, M. D., 1997. On-line flow injection–pervaporation of beer samples for the determination of diacetyl. The Analyst, 122(2): 119-122.Logsdon, J. S. and Biegler, L. T., 1989. Accurate solution of differential-algebraic optimization problems. Industrial and Engineering Chemistry Research, 28(11): 1628-1639.?Mujtaba, I. M. and Macchietto, S., 1993. Optimal operation of multicomponent batch distillation-multiperiod formulation and solution. Computers and Chemical Engineering. 17(12): 1191-1207.Osorio, D., Pérez-Correa, J. R., Biegler, L. T. and Agosin, E., 2005. Wine distillates: practical operating recipe formulation for stills.?Journal of agricultural and food chemistry,?53(16): 6326-6331.Rodman, A. D. and Gerogiorgis. D. I., 2016a. Multi-objective process optimisation of beer fermentation via dynamic simulation. Food and Bioproducts Processing, 100: 255-274.Rodman, A. D. and Gerogiorgis, D. I., 2016b. Dynamic simulation and visualisation analysis of fermentation: effect of conditions on beer quality, IFAC-PapersOnline (DYCOPS 2016), 49(7): 615-620.Rodman, A. D. and Gerogiorgis, D. I., 2016c. Multi-objective optimisation of flavour and processing time in beer fermentation via dynamic simulation, Computer-Aided Chemical Engineering, 38: 1033-1038.S?rensen, E., Macchietto, S., Stuart, G. and Skogestad, S., 1996. Optimal control and on-line operation of reactive batch distillation. Computers and Chemical Engineering, 20(12): 1491-1498.Tanartkit, P. and Biegler, L. T., 1995. Stable decomposition for dynamic optimization. Industrial and Engineering Chemistry Research, 34(4): 1253-1266.Trelea, I. C., Titica, M., Landaud, S., Latrille, E., Corrieu, G. and Cheruy, A., 2001. Predictive modelling of brewing fermentation: from knowledge-based to black-box models. Mathematics and Computers in Simulation, 56(4): 405-424.Vanderhaegen, B., Neven, H., Verachtert, H. and Derdelinckx, G., 2006. The chemistry of beer aging – a critical review. Food Chemistry, 95(3): 357-381.Waltz, R. A., Morales, J. L., Nocedal, J. and Orban, D., 2006. An interior algorithm for nonlinear optimization that combines line search and trust region steps, Mathematical Programming, 107(3): 391–408.Xiao, J., Zhou, Z., Zhang, G., 2003. Ant colony system algorithm for the optimization of beer fermentation control. Journal of Zhejiang University SCIENCE, 5(12): 1597-1603. ................
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