Dynamic optimization of beer fermentation: sensitivity analysis of attainable

performance vs. product flavor constraints

Alistair 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)

ABSTRACT

The 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.

1. INTRODUCTION

Determining 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):

minu (t ) ,t f

φ(x (t f ) ,t f ) (1)

s.tdx (t)

dt=f ( x (t ) , u (t )) , x ( t0 )=x0 (2)

h ( x (t ) , u (t ) )=0, g ( x ( t ) ,u ( t ) ) ≤0 (3)

h f ( x ( t ) )=0 gf ( x (t f ))≤0 (4)

u(t )L ≤ u (t ) ≤u (t )U , x (t )L≤ x (t )≤ 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 , t f ] , 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).

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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.

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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 optimisation

on-line optimisation

This 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.

2. PROCESS DESCRIPTION 2.1 Beer Fermentation

Fermentation 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

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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 modelling

Several 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 ºC) 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 [ X ¿¿ A ](t )dt

¿

¿ μx (t , T ) ∙ [X A ](t )−μDT (t ,T ) ∙[X ¿¿ A]( t)+μL (t , T ) ∙[ X ¿¿ L](t )¿¿ (6)

d [ XD ](t)dt

¿−μSD( t , T )∙ [X ¿¿D ](t)+μDT (t ,T )∙[X ¿¿ A] (t)¿¿ (7)

d [S ](t)dt

¿−μS(t , T ) ∙[X ¿¿ A]( t)¿ (8)

d [EtOH ]( t)dt

¿ f (t ) ∙ μe (t ,T )∙[ X ¿¿ A] (t)¿ (9)

d [EA ](t)dt

¿Y EA(T ) ∙ μx (t , T ) ∙[ X ¿¿ A]( t)¿ (10)

d [DY ]( t)dt

¿ μDY ∙[S ](t) ∙[ X ¿¿ A]( t)−μAB ∙[DY ]( t)∙ [EtOH ](t )¿ (11)

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d [ X¿¿ L](t)dt

=−μL(t , T ) ∙[ X ¿¿ L](t )¿¿ (12)

μx( t , T )=μx0

(T ) ∙ [S] (t)0.5 ∙[S ]0+[ EtOH ](t)

(13)

μSD (t , T )=μS D 0

(T ) ∙0.5 ∙[S ]00.5 ∙[S ]0+[EtOH ](t )

(14)

μs( t , T )=μs0

(T ) ∙[ S](t )k s (T )+[ S]( t)

(15)

μe (t , T )=μe0

(T ) ∙[S ](t )ke (T )+[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.

Symbol Description Value Units

Diacetyl rates(Carrillo-Ureta et al., 2001)

μDY Diacetyl production rate 1.27672∙10-7 g-1 h-1 LμAB Diacetyl consumption rate 1.13864∙10-3 g-1 h-1 L

Initial simulation conditions[XA]+[XL]+[XD] Biomass inoculum (pitching rate) 4 g L-1

[S ]0 Sugar concentration 130 g L-1

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The 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 review

Numerous 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 ∈ [t 0 ,t f ], 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 function

When 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:

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minT ( t ) ,tf

φ(x , t f)=−W E ∙~

[ EtOH ]−W t ∙~

( 1t ) (18)

s . t . [ EA ]t=tf≤ [EA ]max (19)

[ DY ]t=tf≤ [ DY ]max (20)

Here W E and W t 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 W E = W t = 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.

T (t )∈[9 ° C , 16 °C ] for all t ∈ [t 0 ,t f ] (21)

2.5 Temperature profile: Performance insight

A 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

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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 (t f < 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-1

Figure 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.

3. DYNAMIC OPTIMISATION OF BEER FERMENTATION

A 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,

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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; Sørensen 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

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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:

x i=x i−1+∆ ti∑j=1

K

Ω j( t−ti−1

∆ ti) dx

d t i , j(22)

Where ∆ t i is the length of element i and dx /d ti , j is the derivative of the state variable in element i at

the jth collocation point. Ω j is a Kth order polynomial satisfying:

Ω j (0 )=0 , Ω'j ( ρ j )=δ j for j=1 , …, K (23)

Continuity of the state trajectories is ensured with:

x (t )=x i−1+∆ t i∑j=1

K

Ω j (1 ) dxd ti , j

(24)

while the control profile is approximated by:

u ( t )=∑q=1

K

ψ j( t−t i−1

∆ t i)u i , j (25)

Where ψ j is a Lagrange polynomial of degree K that satisfiesψ j ( ρ j )=δ j 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) andT i , j is the fermenter

temperature in element i at collocation point j:

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minxi , j, T i, j , ∆ ti

φ ( x , t f )=−W E ∙~

[ EtOH ]N−W t ∙~

( 1t f ) (26)

s.t d x

d t i , j=f (xi , j ,T i , j) (27)

x i , j=x i−1+∆ ti∑j=1

K

Ω j ' ( ρ j )d x

d t i , j(28)

h ( x i , j , T i , j )=0, i = 1, …, N, j = 1, …, K (29)

x i , j=x i−1+∆ ti∑j=1

K

Ω j (1 ) d xd t i , j

, i = 1, …, N (30)

gf ( x N ) ≤ 0 → [ EA ]N−[ EA ]max ≤ 0 , [DY ]N−[DY ]max ≤ 0 (31)

9≤ T i , j ≤16 , 0≤ xi , j, i = 1, …, N, j = 1, …, K (32)

t f =∑i=1

N

∆ t i (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 values Range

[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, N 8 [6, 12, 18, 24, 30, 36, 42, 48]

Initializing profile 4 [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

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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.

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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.

4. RESULTS & DISCUSSION

To 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.

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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.

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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.

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Figure 13. Computed profiles for low [EA]max = 0.5 ppm, low [DY]max = 0.05 ppm.

4.1 Effect of increasing time domain discretisation

Inspecting 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

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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 profile

To 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 thresholds

To 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

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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, t f 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.

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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 profiles

Of 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:

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I. The best performing profile from the base case ([EA]max = 2.0 ppm, [DY]max = 0.10 ppm)

II. The profile corresponding to the longest batch time, tf, from all cases

III. The profile corresponding to greatest ethanol concentration, [EtOH]tf, from all cases

IV. The profile corresponding to the shortest batch time, tf, from all cases

V. The profile corresponding lowest ethanol concentration, [EtOH]tf, from all cases

To 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.

Solution Initialising profile [EtOH]tf (g L-1) tf (hr) [EA]max (ppm) [DY]max (ppm)I D 59.8 114.2 2.0 0.10II B 59.6 142.9 3.0 0.05III D 60.5 128.5 3.0 0.05IV D 60.1 94.9 3.0 0.25V B 55.4 120.3 0.5 0.05

It 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

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(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.

5. CONCLUSIONS

A 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.

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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 L -1 to 65.5 g L-1.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.

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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.

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NOMENCLATURE Roman symbols

W E Objective ethanol weight (%)W t Objective time weight (%)X A Active biomass concentration (g L-1)X D Dead biomass concentration (g L-1)X L Latent biomass concentration (g L-1)Y EA Ethyl acetate production stoichiometric factor (g L-1)k e Ethanol affinity constant (g L-1)k s Sugar affinity constant (g L-1)k x Biomass affinity constant (g L-1)DY Diacetyl (-)EA Ethyl Acetate (-)EtOH Ethanol (-)g Inequality constraint h Equality constraintK Number of collocation pointsN Number of elemets in time horizon S Sugar (-)T Fermenter temperature (K)f Fermentation inhibition factor (g L-1)t Time (h)t f Batch time (h)u Model controlx Model state

Greek symbols

∆ t i Length of element iμAB Diacetyl consumption rate (g-1 h-1 L)μDT Specific cell death rate (h-1)μDY Diacetyl growth rate (g-1 h-1 L)μE Ethanol production rate (h-1)μL Specific cell activation rate (h-1)μS Sugar consumption rate (h-1)μSD Specific dead cell settling rate (h-1)μx Specific cell growth rate (h-1)Ω State approximation polynomialφ Objective function (-)ψ Control approximation polynomial

Subscripts and operators

~() Normalised parameter (-)()0 Initial condition (-)

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()L Lower bound (-)()U Upper bound (-)()i Property in element i (-)()j Property at collocation point j (-)

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