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Reserve management and real time optimization for a solar powered Membrane Distillation Bio-Reactor water recycling plant via convex optimization
Renewable Energy 60 (2013) 489e497
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Reserve management and real time optimization for a solar powered Membrane Distillation Bio-Reactor water recycling plant via convex optimization Avinash Vijay a, K.V. Ling a, *, A.G. Fane b a b
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Singapore Membrane Technology Centre, School of Civil and Environmental Engineering, Nanyang Technological University, Singapore
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 November 2012 Accepted 12 May 2013 Available online 17 June 2013
This paper aims to design a system that can enable optimal operation of the solar powered Membrane Distillation Bio-Reactor (MDBR) water recycling plant. Optimal operation is described as maximization of the clean water produced while being a self sufficient entity in terms of energy. During periods of adequate solar radiation, the system will optimize production levels. When faced with inadequate sunlight, the system will focus on reserving sufficient energy for the survival of the essential microorganisms in the MDBR. Hence, uninterrupted plant operation during periods of unfavorable weather entails the management of a back up reserve. These engineering objectives are translated into mathematical functions that can be incorporated into convex optimization problems. The effectiveness of the control framework is demonstrated through simulation. 2013 Elsevier Ltd. All rights reserved.
Keywords: Solar energy Waste water Convex optimization Mathematical programming
1. Introduction Clean water is a scarce resource. This long squandered commodity will be in great demand in the future unless a cost effective means of producing clean water is established. Many advances have been made in the field of water reclamation in an effort to bridge the gap between supply and demand. Such advances have given rise to the creation of the Membrane Distillation Bio-Reactor (MDBR). Apart from 100% (theoretical) rejection of non-volatile organics, salts and biomass, MDBRs also offer lower carbon footprints and lower energy requirements in comparison to conventional distillation and pressure driven processes [1]. Further improvements can be brought about through optimization of the operating conditions. Initial efforts made to realize the full potential of this possibility were presented in Ref. [2], which proposed an optimization framework to obtain the operational schedule of an MDBR plant. Details regarding the MDBR plant have been presented in Section 2. The underlying ideas that motivated the design of the control architecture in Ref. [2] are the same as the ones used by the current
* Corresponding author. Tel.: þ65 6790 5567; fax: þ65 6791 2687. E-mail addresses: [email protected] (A. Vijay), [email protected] (K.V. Ling), [email protected] (A.G. Fane). 0960-1481/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.renene.2013.05.035
work. They are: maximize production during favorable weather and protect micro-organisms during unfavorable weather. The structure of the control architecture is also the same. There are three levels: Scheduling layer: For long term reserve management Supervisory layer: For real time optimization of production Regulatory layer: For disturbance rejection and follow up control This paper reinforces those efforts by enhancing the optimization capabilities of the system. The supervisory layer now requires fewer optimization problems to be solved. Improvements in the scheduling layer decrease the computational load required to determine the reserve management strategy. Additionally, experimental validation has also been provided to justify the modeling principles used. The usage of modeling and optimization techniques with regards to solar powered water treatment plants has been explored in the past. It has been utilized by Salcedoa et al. [3] for assessing cost and environmental impact. A mixed integer non-linear program was used in the analysis to help decision-makers select better technological alternatives. Optimization can also refer to the selection of the best choice of parameters either for design [4,5] or for the best choice of a reference operating point [6,7]. In this work, optimization refers determination of a suitable trajectory of
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operating strategies which are based on both present and future conditions. These trajectories are calculated through the use of mathematical programming techniques, namely, convex optimization. The rest of this work is structured as follows. Section 2 provides an introduction to the operation and the mathematical details used to describe the plant. Section 2.2 pertains to the control framework. It discusses the objectives motivating control and the architecture organized to satisfy them. The main focus of this work is the translation of engineering objectives into convex optimization problems as shown in Sections 3 and 4. Since some relations in these problems are non-convex, the modifications required to solve them using convex optimization algorithms are discussed. Illustrative examples demonstrate the operation of the proposed system. 2. System overview The control system proposed in this paper is based on the Membrane Distillation Bio-Reactor (MDBR) water recycling plant being developed at Nanyang Technological University, Singapore [2]. The plant schematic is shown in Fig. 1. The following sections will detail plant operation and modeling of the significant decision variables. 2.1. Plant operation As the solar radiation increases at dawn, the solar collectors and the solar thermal storage system kick into action. These two subsystems help sustain the thermal and electrical load required
to produce clean water. The MDBR, along with its microorganisms, helps recycle the waste water. The bio-reactor portion of the MDBR makes use of micro-organisms to remove dissolved and suspended organic chemical content via biodegradation. Specifically, the micro-organisms involved are aerobic thermophillic and salt tolerant bacteria. These bacteria can function at relatively high temperatures, such as 50 Ce55 C. The microbial community was developed by gradual acclimatization to the reactor conditions. Further work is planned to optimize the microbial community. The raw feed and the clean permeate streams move parallel to the membrane surface but in opposite directions inside the MDBR. The vapor pressure difference between the two allows the water vapor to diffuse and/or convect across the membrane to the permeate side where it is condensed. The feed and the clean permeate streams also come into contact with each other in the feed tank. The feed tank acts as a heat exchanger where energy is transferred back to the feed from the permeate. To ensure smooth operation, the feed and the permeate streams have to be returned to their original temperature before they enter the MDBR module. The permeate stream is allowed to pass through another heat exchanger to enhance energy recovery and further reduce its temperature. The temperature at which the permeate stream enters the MDBR is controlled through manipulation of the flow rate of the cold stream. The cooling liquid used for this purpose is pumped from the main raw water storage tank. During stable operation, there is a build up of product water in the permeate stream. This build up is removed and stored in the product tank separately.
Fig. 1. Plant schematic [2].
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2.2. Control system and architecture
491
Table 1 Plant Parameters.
The three main objectives that have been considered for the solar powered MDBR plant are: Safety: The plant must remain within the safe operating envelope at all times. Maximizing value: To ensure economic feasibility, the output produced per unit amount of energy available has to be maximized. Self-sustaining operation: The plant must be able to survive without solar radiation during periods of unfavorable weather. Design of a centralized monolithic unit to handle all of the control objectives is usually plausible only when the plant model, dynamics and control objectives are reasonably simple and well defined. Most practical situations involve complex non-linear multi-variable processes which do not come under this category. Hence, the main control problem is usually divided into a set of simpler, hierarchically formulated sub-problems. Each subproblem is assigned a separate layer. These layers or sub problems are characterized by different time scales. The hierarchical control structure is shown in Fig. 2. The scheduling layer uses forecasts to manage back up reserves several days ahead. The supervisory layer manages operation throughout the day while satisfying the reserve targets set by the scheduling layer. It maximizes production in favorable periods and protects the microorganisms in the MDBR during unfavorable periods. Finally, the regulatory layer minimizes set point deviations from the references provided by the supervisory layer. Since this layer has been extensively studied (see Ref. [8] for examples) it is not the focus of this work. The scheduling and supervisory layers are interconnected as shown in Fig. 2. The scheduling layer enforces an upper bound on the power consumption of the supervisory layer. Thus, in case the current solar radiation levels are high but the forecasts indicate bad weather, the scheduling layer will decrease the value of the bound to conserve energy. When the forecasts are bright and favorable, the scheduling layer will allow the plant to increase energy consumption. The selection of measured and manipulated variables shown in Fig. 2 was done through the analysis of cause and effect relationships embedded into the equations that describe the plant.
Cp h
4.2 103 kJ m3 k1 2
0.03 kW m
1
k
2
A
2m
CoP
3
Tin Tfeed
305 K 303 K
Tamb es N
303 K 34 kWh 15
max Fðcirc;feedÞ
10 l min1
min Fðcirc;feedÞ max Php min Php max Tmdbr min Tmdbr
3 l min1
Tsustenance ep
318 K 88 kWh
2 kW 0 kW 353 K 328 K
The main knobs or manipulated parameters for the control framework are: the bounds on consumption P, Q(bound) from the scheduling layer and the optimal values of Ffeed, Fcirc, Php and Tmdbr from the supervisory layer. As for the regulatory layer, it treats the values of Ffeed, Fcirc, Php and Tmdbr from the supervisory layer as control set-points and closes the loop with sensor feedback. The variables which need to be measured are Tin, Tfeed and Tamb. This work assumes knowledge of certain quantities for its operation. These are, solar predictions W over a planning horizon, current or available reserve for the scheduling layer R0 and the current or available reserve for the supervisory layer P, Q(available). The rationale behind such choices will become clearer as the reader goes through the following sections (Table 1). 2.3. Modeling 2.3.1. Temperature dynamics of the re-circulated liquid The inlet temperature of the re-circulated liquid is maintained constant through control of flow rate of cooling liquid in heat exchanger. The outlet temperature of re-circulated liquid can be determined from the equations that describe the rate of heat transfer in the MDBR tank [9], hA
Tout ¼ Tmdbr þ eCp Fcirc ðTin Tmdbr Þ
(1)
where Tmdbr, Tout and Tin denote temperatures of the MDBR tank, outlet and inlet of the re-circulated liquid respectively. The terms Cp, h and A denote the specific heat capacity of water, heat transfer co-efficient of the MDBR tank and surface area of the MDBR tank
Fig. 2. MDBR control architecture.
A. Vijay et al. / Renewable Energy 60 (2013) 489e497
f ðqx þ ð1 qÞyÞ qf ðxÞ þ ð1 qÞf ðyÞ: 2.3.2. Output flux Output flux is the amount of clean water produced by the plant. The driving force required to produce output flux is the difference in temperature between the feed and the permeate side [12]. Although the variation of vapor pressure with respect to the temperature gradient is not a perfect straight line, we have assumed linearity over the operating region considered in this work. Fig. 4 shows experimental validation of this relation. Therefore the management system will use the driving force as a proxy to manipulate the output flux (J):
J ¼ Tmdbr 0:5ðTout þ Tin Þ
(2)
The output flux is a function of the outlet temperature of the recirculated liquid. Hence, the relation in Eq. (2) is also non-convex in terms of Fcirc and Tmdbr. It does not satisfy the conditions required for convexity stated above.
8
7
6
−2 −1
respectively and Fcirc denotes the flow rate of the re-circulated liquid. Experimental validation for this relation is shown in Fig. 3 [10]. Variations in flow rate of re-circulation could not be covered in these experiments due to limitations in lab equipment. The relation in Eq. (1) is non-convex in terms of the manipulated variables Fcirc and Tmdbr. It does not satisfy the definition of a convex function [11]: A function f : Rn /R is convex if dom f is a convex set and if for all x, y ˛ dom f, and q with 0 q 1, we have
Output flux (lm h )
492
5
4
3
2
1
0
0
5
10
15
20 25 Driving force (K)
30
35
40
Fig. 4. MDBR experimental validation: output flux vs driving force.
Thermal power (Q load) requirements considered are that of feed liquid (Q feed), re-circulated liquid (Q circ), MDBR tank loss (Q mdbr tank) and the heat recovered by the heat pump (Q hp). The overall equation is:
Q load ¼ Q feed þ Q circ þ Q mdbrtank Q hp ¼ Ffeed Cp Tmdbr Tfeed þ Fcirc Cp ðTout Tin Þ þ hAðTmdbr Tambient Þ CoP Php
2.3.3. Power consumed Power is consumed by the plant in two different forms: electrical and thermal. Electrical power (Pload) is consumed by two circulation pumps and a heat pump. Since the circulation pumps used are centrifugal in nature, shaft power consumed is proportional to their respective flow rates [13].
where CoP is the co-efficient of performance of the heat pump. The terms Cp, Ffeed, Tmdbr and Cp, Fcirc, Tout are non-convex in terms of the manipulated variables Ffeed, Fcirc and Tmdbr.
Pload ¼ Kp Ffeed þ Kp ðFcirc Þ þ Php
3. Design and evaluation of the supervisory layer
(3)
where Ffeed and Fcirc denote flow rates of feed and re-circulated liquid. Kp is the constant of proportionality and Php is the power supplied to the heat pump.
(a)
The supervisory layer has two modes of operation. It is said to be in production mode during periods of favorable weather and plant output is maximized. It is said to be in sustenance mode during
(b)
311
311 ACTUAL ESTIMATE 310
309
309
308
308
Toutlet (k)
310
T
outlet
(k)
ACTUAL ESTIMATE
307
307
306
306
305
305
304 304
(4)
304.5
305
305.5
306
306.5
307
304 305
310
315
320 T
Tinlet (k)
325 (k)
330
335
mdbr
Fig. 3. MDBR experimental validation: (a) outlet temperature vs inlet temperature (b) outlet temperature vs MDBR temperature [10].
340
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periods of unfavorable weather and the focus is on conserving energy necessary for the micro-organisms. Inclusion of a sustenance mode is not necessary if the system has an auxiliary source of power other than solar radiation. The auxiliary source can support continuous operation during periods of unfavorable weather. Otherwise, if the back up reserves are large enough to act as this auxiliary source, then also, sustenance mode is not required. There will be no penalties on production.
3.1. Problem formulation From a mathematical perspective, the supervisory layer optimization problem is of the form:
minimize
cðx; uÞ
subject to
hðx; uÞ 0
u
(5)
where c(x, u) is the objective function and h(x, u) represents the constraints. The symbols x and u represent the state variables and the manipulated variables respectively. They are:
3 Tamb x ¼ 4 Tfeed 5 Tin
3 Fcirc 6 Ffeed 7 7 u ¼ 6 4 Php 5 Tmdbr 2
2
and
There are multiple ways in which this problem can be formulated. The designer can optimize a weighed sum of the power consumed and the output produced to obtain a stationary operating point. The component operation limits for the manipulated set-point u take the form U u U as shown below:
d1 Jdf þ d2 ðQ load þ Pload Þ
minimize u
UuU
subject to
Q load þ P load
subject to
Jdf Ji ˛½ Jmin ; Jmax
u
UuU
(7)
But, in this case, numerous optimization problems have to be solved before arriving at the operating strategy. Hence, the system would benefit from a formulation that captures all the objectives in a single optimization problem. Maximizing the output flux while imposing bounds on the power consumed would allow the system to do that. The optimization problem is of the form:
maximize u
subject to
J Q load Q bound Pload Pbound UuU
where the bounds on thermal power consumed (Qbound)and electrical power consumed (Pbound) correspond to the available power. When integrated with the scheduling layer, these bounds will also depend on weather forecasts also. The optimization problems presented in this section assume that the plant is operating in production mode. This implies that the power available is equal to or above the minimum value required to support production of water. The minimum value of power required is dependent upon the sizes and the operational limits of the components in the plant. Intuitively, the minimum power required can be determined by assigning the least possible values to all manipulated variables. In mathematical terms this can be stated as u ¼ U. So if the power is enough for the circulation pumps, the heat pump and the MDBR temperature control to operate at their respective minimum capacities the plant can produce water [10]. 3.2. Implementation 3.2.1. Model simulation Before proceeding to the simulation details, let us state the assumptions being made about the MDBR plant’s operation at each sampling period: The temperature of feed liquid Tfeed, ambient temperature Tamb are known from measurements. The heat recovery system maintains the inlet temperature of the re-circulation liquid Tin constant at the prescribed value. The available reserve or the maximum power that can be consumed by the plant during the current period is known. The regulatory layer is capable of enforcing the set points determined by the supervisory layer. The supervisory layer acts as a steady state optimizer, hence the transients or dynamics of the thermal tank are not considered in this work.
(6)
This approach though simple does not take into account the changes in available power. It determines a single operating point that the system has to stick to irrespective of external conditions. In order to be more robust the system needs to adjust the set-points according to environmental changes. Another strategy that can be considered is to remove the output flux from the objective function and include it as a constraint. Thus, the optimization problem will minimize power consumed while forcing the output flux to take different values between its nominal minimum and maximum bounds. The combination which is closest to the available power is chosen as the control set-point. The resulting optimization problem takes the form:
minimize
493
(8)
Taking these assumptions into consideration, the values of output flux, thermal power and electrical power can be determined by simultaneously solving Eqs. (1)e(4). Upon careful examination, one will notice that the model is not explicitly represented as an equality constraint in the optimization problem described in Eq. (8). Rather, they are included in the cost functions and the inequality constraints in this work. This is done since the model of the process can be analytically solved. But many of the expressions presented here are not convex and hence require modification as shown below. 3.2.2. Obtaining convex problems The optimization problem is non-linear and non-convex. It requires modification before it can be fed to a convex optimization solver. Convex relaxations that closely resemble the original supervisory layer problem help the system handle non-convex terms. Hence, the optimization problem from Eq. (8) is modified to give,
maximize u
subject to
J ðrÞ ðrÞ
Q load Q bound Pload Pbound UuU ðrÞ
(9)
where J (r) and Q load are convex alternatives used to replace the ðrÞ original functions. The relaxation Q load must satisfy the constraint ðrÞ from the original problem, Q load Q bound . Otherwise the results obtained through optimization will violate the actual constraints. As long as the relaxations used resemble the original relations satisfactorily, further modifications are not required. The true test for these relaxations lies in their ability to produce viable optimal
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solutions when used with the actual relations during simulation. In this work, we have used simple relaxations of the form:
J ðrÞ ¼ cT1 U þ d1 ðrÞ
Q load ¼ cT2 U þ d2
(10) (11)
The linear relaxations and simulations shown in the next section have produced practically feasible results. 3.3. Simulation example This example provided a demonstration of the proposed supervisory layer control system. Fig. 5(a) shows the solar radiation profiles used in this simulation. They were obtained from the NREL data archive at http://www.nrel.gov/. Given the solar radiation profile, the available power can be computed. At the beginning of the day, the system encounters low solar radiation as shown in Fig. 5(a). It operates in sustenance mode. Output
is not produced as seen in Fig. 5(b). As the day progresses from dawn, the available power exceeds the required threshold. The system now realizes that production of output can commence. Hence, it switches to production mode and production of output is initiated as shown as shown in Fig. 5(b). As the solar radiation increases, the available power increases and the optimizer chooses the decision variables T accordingly. The output produced u ¼ Fcirc ; Ffeed ; Php ; Tmdbr and power consumed increases with an increase in available power as seen in Fig. 5(b) and (c). This increase can be attributed to optimizer choosing a high value for the temperature inside the MDBR (Tmdbr) as seen in Fig. 5(d). Fig. 5(e) shows that as long as the solar radiation is adequate to operate in production mode, the system does not need to expend electrical power through the heat pump. One can understand from Fig. 5(f) that the optimization problem generally selects the maximum value of flow rate for the re-circulated liquid. It is attempting to increase output flux by increasing flow rate of the recirculation loop. The feed liquid on the other hand is assigned minimum flow rate so as to minimize the thermal and electrical power consumed as shown in Fig. 5(g). At the end of the day, the solar radiation declines and the system switches back to sustenance mode. Production of output is stopped. Hence, both mode selection and optimization capabilities of the supervisory layer have been explained in this example. 4. Design and evaluation of the scheduling layer Energy reserves are managed by the scheduling layer. It ensures that the plant can maintain the conditions required for survival of MDBR bacteria even when it is subjected to unfavorable weather conditions. 4.1. Problem formulation
Fig. 5. Supervisory layer e typical solar profile.
The scheduling layer has to decide whether the plant can be allowed to consume all the power that is available or whether some of it has to be reserved for unfavorable periods without sunlight. For this purpose, it needs to be aware of weather variations and requires information regarding the variation of solar radiation in the future. The scheduling layer will depend on the solar radiation prediction system to provide forecasts about the future. The scheduling layer divides the interval between sunrise and sunset into three periods. It looks ahead over a prediction horizon of five days (i.e. 15 periods). Energy consumption targets for these periods are determined based on information available from the prediction system. These targets are revised when the prediction horizon is shifted forward. This is akin to a receding horizon strategy one would use while planning finances. A multi-month expenditure plan is decided based on the information available about a finite section of the income forecast. But only the decision regarding saving and spending for the current month is applied. The expenditure plan is then revised in the next month when new income forecasts are available. A new financial plan is determined. Again, only the current decision is put into practise and the process repeats. For the scheduling layer the consumption targets represent expenditure, the solar predictions represent the income forecast and the back up reserves represent the savings. This receding horizon process shown in Fig. 6 allows the plant to account for uncertainty in the forecasts and optimize its strategy accordingly. One does not need to scrutinize a financial plan if the forecasts indicate a lean period of very low or no income. When the influx of resources ceases to exist, the consumption of available resources must be reduced to the minimum amount possible. Similarly, if the solar predictions indicate that the solar profile will not be able to sustain even the minimum back up reserves required, the scheduling layer will not attempt to solve the optimization problem.
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495
(a) 100
wi (kWh)
80 60 40 20 0
(b)
1
2
3
4
5
100 Max/Min Bounds
80
It will merely cut down expenditure to the bare minimum. This is known as the feasibility condition. During normal periods of sunlight, the system is designed to maximize output flux. Higher values of output flux require more energy consumption. If the system were to minimize the available reserves (r), it would be maximizing energy consumption. Hence, the objective function takes the form:
minimize U
N X
ep ui es
i ¼ 1; .; N
2
3
4
5
6
100 80 60 40 20 0
(14) (15)
The constraint from Eq. (14) makes sure that the available reserve should always be greater than the load required to support plant operation in sustenance mode, es. The constraint from Eq. (15) ensures that the scheduling targets lie between their upper and lower bounds. The upper bound is the maximum consumption possible in production mode, ep. The lower bound should obviously be the consumption level required to sustain plant operation in sustenance mode, es. All functions and constraints are convex. Hence, they do not require any modifications during implementation. Furthermore, the special structure of the optimization problem discussed in this section can be exploited to reduce the complexity and computational effort required to determine the solution as shown in the appendix. 4.2. Simulation example The solar profile wi considered in Fig. 7(a) shows that there are bursts of radiation on days 1, 3 and 4. Days 2 and 5 do not receive sunlight. Using the information available from wi the optimization problem has determined values for consumption targets ui as shown in Fig. 7(b). The consumption remains at the minimum level
1
2
3
4
5
Time(Days)
(13)
where wi represents the predicted energy level and ui represents the consumption target for the ith period. The constraints are:
i ¼ 1; .; N
1
(c)
(12)
where, U is the optimization variable referring to the vector of consumption targets for each period in the planning horizon. N refers to the number of periods in the planning horizon. Expressions for the available reserve levels at the end of the ith period are described as the difference between the energy available and the energy consumed:
ri es
0
Sustenance Bound
i¼1
ri ¼ ri1 þ wi ui
40
0
# ðri Þ
60
20
ri (kWh)
"
ui (kWh)
Fig. 6. Receding horizon approach in the scheduling layer.
Fig. 7. Scheduling layer e intermittent profile.
when there is no sunlight. When there is sunlight, the system is allowed to increase consumption, but never allowed to exhaust everything available. This is evident on day 4, where the consumption is markedly lesser than the available energy in order to tackle unfavorable periods in day 5. The values of forecasts and consumption targets are used to calculate the reserve levels shown in Fig. 7(c). The system intuitively builds up reserves before the dark periods in days 2 and 5. Also, the reserve levels are never allowed to dip beyond a certain minimum value. Thus the scheduling layer is able to make sure that the system can overcome periods of undesirable weather. 5. Conclusion This paper provides an introduction to the formulation and design of a plant-wide optimization and control system for an MDBR water recycling plants. It presents a means to manage both real time optimization and long term reserve management requirements. The simulated examples show that the proposed framework is able to make best use of the available energy to produce clean water while maintaining sufficient reserves required for the sustenance of micro-organisms in the MDBR. Implementation of the proposed control system will greatly reduce the need for operator intervention. The layered structure can accommodate future modifications, improvement and inclusion of better functions can be established without having to incur the costs of overall replacement.
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For future work, the performance of the system can be improved by dealing with some issues. The MDBR is not designed to handle rapid changes in operating conditions. In order to leverage the computation that can be performed in real-time, the control strategies have to be adjusted according to variations in solar radiation. But, in the current set-up, changes in solar radiation are faster than the changes in operating conditions. This is because current industry practice suggests that trends are rather slow in the MDBR due to system capacitance [9]. Hence, further research is required to assess the minimum time interval between changes in operating conditions for MDBRs. The system can also be extended to account for transients in membrane performance due to fouling. It is caused by gradual deposition of particles and macro-solutes on the membrane surface [14]. This effect will be incorporated into the required driving force as a function of time. Fouling can usually be reversed by regular cleaning [9] (as shown in this reference).
Appendix A. Reduced complexity algorithm for the scheduling layer Appendix A.1. Analysis and discussion of problem structure Further analysis of the scheduling layer optimization problem reveals that, it can be solved without the use of an on-line optimization solver. The structure of the problem allows us to define an algorithm of reduced complexity to determine the optimal solution. For this purpose, one must begin with an assessment of the optimization problem from Section 4.1: The objective function from Eq. (12) is a linear function of the values of reserves at different intervals,
" minimize U
N X
# ðri Þ
i¼1
Since the reserve is known to be a linear function of the optimization variable U from Eq. (13),
ri ¼ ri1 þ wi ui the original objective function can be equivalently formulated as maximization of a linear function of the optimization variable:
minimize cT U þ d
(A.1)
U
It is observed that Eq. (14):
ri es ci ¼ 1; .; N
By re-arranging the terms and multiplying both sides of the above equation by L1 we get,
L1 ðR0 Es Þ þ W U
(A.2)
The relation from Eq. (A.2) makes up a key element of the solution algorithm. All the elements on the left hand side of the equation are constants and represent the maximum values that components of the vector u can take. Hence by replacing the inequality by an equality sign and solving the set of simultaneous equations, one can obtain a preliminary estimate of the solution of the optimization problem without the use of an optimization solver. This solution is referred to as the preliminary estimate since it does not take into account the upper and lower bounds imposed upon the optimization variable by Eq. (15). In other words, this estimate can contain values that may either be higher than the plant’s designed capacity or may be lower than the value required for operation in sustenance mode. Hence, some adjustment is required to ensure that the values of the optimization variables do not violate these constraints. This is done in two stages. In the first stage of adjustment, the variables ui are analyzed starting from u1 to uN. If at any variable, there is a surplus or excess above the maximum bound, this surplus is shifted to the subsequent interval. This process removes any violations of the constraint related to the upper bound. In the second stage of adjustment, the variables ui are analyzed from uN to u2. If at any variable, there is a deficit or gap below the minimum bound, this deficit is borrowed from the interval preceding the current interval. Performing these steps for all variables up till u2 will remove any violations of the lower bound. Notice that the second stage of adjustment is not allowed to modify u1. This is done to check whether the problem is feasible. If the second stage of adjustments results in a value of u1 that is lesser than the minimum bound, then it is obvious that the predicted values are not sufficient to sustain the minimum reserve levels required during all periods. In such cases, the problem is deemed to be infeasible. The rationale behind the operation of this algorithm can be understood through the operation of the plant itself. If radiation available is more than what the plant can use, any surplus will obviously be transferred to the reserve to be used the next day. This is the principle behind the first stage of surplus adjustment. As for the second stage, if the consumption is below the minimum required at any interval, it is an indication that additional reserves have to be saved up before that point of time. Hence, by borrowing the deficit from the preceding intervals, the second adjustment accomplishes that goal. Thus the algorithm is reduced to a series of simple mathematical operations as shown in the next section.
can be written as,
Appendix A.2. Algorithm sequence
R0 þ LW LU Es
The Matlab code for the algorithm described in this Appendix is as follows:
where, T
R0 ¼ ½ r0 r0 / r0 W ¼ ½ w1 w2 / wN T U ¼ ½ u1 u2 / uN T Es ¼ 2½ es es / es T3 2 1 0 0 / 0 1 0 0 / 61 1 0 / 07 6 1 1 0 1 6 7 6 1 6 7 L ¼ 6 6 1 1 1 1 0 7and hence; L ¼ 6 0 1 1 1 4« 4 « 1 1 1 1 05 1 1 1 / 1 0 / 0 1
3 0 07 7 «7 7 05 1
u_alg ¼ inv (L)*((r0 es) * ones(n, 1)) þ W; %if there is surplus in the current period, % store it as reserve and use in the next period for i ¼ 1:n if u_alg (i) > ep u_alg (i þ 1) ¼ u_alg (i þ 1) þ (u_alg (i) ep); u_alg (i) ¼ ep; end end %if there is deficit in the current period, % use less in the previous period
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for i ¼ n: 1:2 if u_alg (i) < es u_alg (i 1) ¼ u_alg (i 1) (es u_alg (i)); u_alg (i) ¼ es; end end if u_alg (1) < es, disp (‘Problem Infeasible!’), end In relative terms, first statement of the algorithm requires more computational effort since it involves matrix inversion, L1. Since L is constant and has a special structure, L1 can be easily determined. References [1] Phattaranawik J, Fane A, Pasquier A, Bing W. A novel membrane bioreactor based on membrane distillation. Desalination 2008;223(1e3):386e95. [2] Vijay A, Ling KV, Fane AG. Applications of convex optimization in plant-wide control of membrane distillation bio-reactor (MDBR) water recycling plant. In: Proceedings of the 11th International Conference on Control, Automation, Robotics and Vision (ICARCV 2010) 2010. p. 246e51. [3] Salcedo R, Antipova E, Boer D, Jiménez L, Guillén-Gosálbez G. Multi-objective optimization of solar Rankine cycles coupled with reverse osmosis desalination considering economic and life cycle environmental concerns. Desalination 2011;286:358e71.
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[4] Bourouni K, Ben MBarek T, Al Taee A. Design and optimization of desalination reverse osmosis plants driven by renewable energies using genetic algorithms. Renewable Energy 2011;36(3):936e50. [5] Khayet M, Cojocaru C, García-Payo M. Experimental design and optimization of asymmetric flat-sheet membranes prepared for direct contact membrane distillation. Journal of Membrane Science 2010;351(1e2):234e45. [6] Bui VA, Vu LTT, Nguyen MH. Simulation and optimisation of direct contact membrane distillation for energy efficiency. Desalination 2010;259(1e3): 29e37. [7] Chang H, Wang GB, Chen YH, Li CC, Chang CL. Modeling and optimization of a solar driven membrane distillation desalination system. Renewable Energy 2010;35(12):2714e22. [8] Ogunnaike B, Ray W. Process dynamics, modeling, and control. USA: Oxford University Press; 1994. [9] Khaing TH, Li J, Li Y, Wai N, Wong FS. Feasibility study on petrochemical wastewater treatment and reuse using a novel submerged membrane distillation bioreactor. Separation and Purification Technology 2010;74(1):138e43. [10] Vijay A. Applications of convex optimization in plant-wide control of a solar powered MDBR water recycling system. Master’s thesis. Singapore: Nanyang Technological University; 2012. [11] Boyd S, Vandenberghe L. Convex optimization. Cambridge Univ Pr; 2004. [12] Lawson K, Lloyd D. Membrane distillation. Journal of Membrane Science 1997;124(1):1e25. [13] Palgrave R. Troubleshooting centrifugal pumps and their systems. Elsevier Science Ltd; 2003. [14] Zhang J, Chua H, Zhou J, Fane A. Factors affecting the membrane performance in submerged membrane bioreactors. Journal of Membrane Science 2006;284(1e2):54e66.
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Reserve management and real time optimization for a solar powered Membrane Distillation Bio-Reactor water recycling plant via convex optimization Avinash Vijay a, K.V. Ling a, *, A.G. Fane b a b
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Singapore Membrane Technology Centre, School of Civil and Environmental Engineering, Nanyang Technological University, Singapore
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 November 2012 Accepted 12 May 2013 Available online 17 June 2013
This paper aims to design a system that can enable optimal operation of the solar powered Membrane Distillation Bio-Reactor (MDBR) water recycling plant. Optimal operation is described as maximization of the clean water produced while being a self sufficient entity in terms of energy. During periods of adequate solar radiation, the system will optimize production levels. When faced with inadequate sunlight, the system will focus on reserving sufficient energy for the survival of the essential microorganisms in the MDBR. Hence, uninterrupted plant operation during periods of unfavorable weather entails the management of a back up reserve. These engineering objectives are translated into mathematical functions that can be incorporated into convex optimization problems. The effectiveness of the control framework is demonstrated through simulation. 2013 Elsevier Ltd. All rights reserved.
Keywords: Solar energy Waste water Convex optimization Mathematical programming
1. Introduction Clean water is a scarce resource. This long squandered commodity will be in great demand in the future unless a cost effective means of producing clean water is established. Many advances have been made in the field of water reclamation in an effort to bridge the gap between supply and demand. Such advances have given rise to the creation of the Membrane Distillation Bio-Reactor (MDBR). Apart from 100% (theoretical) rejection of non-volatile organics, salts and biomass, MDBRs also offer lower carbon footprints and lower energy requirements in comparison to conventional distillation and pressure driven processes [1]. Further improvements can be brought about through optimization of the operating conditions. Initial efforts made to realize the full potential of this possibility were presented in Ref. [2], which proposed an optimization framework to obtain the operational schedule of an MDBR plant. Details regarding the MDBR plant have been presented in Section 2. The underlying ideas that motivated the design of the control architecture in Ref. [2] are the same as the ones used by the current
* Corresponding author. Tel.: þ65 6790 5567; fax: þ65 6791 2687. E-mail addresses: [email protected] (A. Vijay), [email protected] (K.V. Ling), [email protected] (A.G. Fane). 0960-1481/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.renene.2013.05.035
work. They are: maximize production during favorable weather and protect micro-organisms during unfavorable weather. The structure of the control architecture is also the same. There are three levels: Scheduling layer: For long term reserve management Supervisory layer: For real time optimization of production Regulatory layer: For disturbance rejection and follow up control This paper reinforces those efforts by enhancing the optimization capabilities of the system. The supervisory layer now requires fewer optimization problems to be solved. Improvements in the scheduling layer decrease the computational load required to determine the reserve management strategy. Additionally, experimental validation has also been provided to justify the modeling principles used. The usage of modeling and optimization techniques with regards to solar powered water treatment plants has been explored in the past. It has been utilized by Salcedoa et al. [3] for assessing cost and environmental impact. A mixed integer non-linear program was used in the analysis to help decision-makers select better technological alternatives. Optimization can also refer to the selection of the best choice of parameters either for design [4,5] or for the best choice of a reference operating point [6,7]. In this work, optimization refers determination of a suitable trajectory of
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operating strategies which are based on both present and future conditions. These trajectories are calculated through the use of mathematical programming techniques, namely, convex optimization. The rest of this work is structured as follows. Section 2 provides an introduction to the operation and the mathematical details used to describe the plant. Section 2.2 pertains to the control framework. It discusses the objectives motivating control and the architecture organized to satisfy them. The main focus of this work is the translation of engineering objectives into convex optimization problems as shown in Sections 3 and 4. Since some relations in these problems are non-convex, the modifications required to solve them using convex optimization algorithms are discussed. Illustrative examples demonstrate the operation of the proposed system. 2. System overview The control system proposed in this paper is based on the Membrane Distillation Bio-Reactor (MDBR) water recycling plant being developed at Nanyang Technological University, Singapore [2]. The plant schematic is shown in Fig. 1. The following sections will detail plant operation and modeling of the significant decision variables. 2.1. Plant operation As the solar radiation increases at dawn, the solar collectors and the solar thermal storage system kick into action. These two subsystems help sustain the thermal and electrical load required
to produce clean water. The MDBR, along with its microorganisms, helps recycle the waste water. The bio-reactor portion of the MDBR makes use of micro-organisms to remove dissolved and suspended organic chemical content via biodegradation. Specifically, the micro-organisms involved are aerobic thermophillic and salt tolerant bacteria. These bacteria can function at relatively high temperatures, such as 50 Ce55 C. The microbial community was developed by gradual acclimatization to the reactor conditions. Further work is planned to optimize the microbial community. The raw feed and the clean permeate streams move parallel to the membrane surface but in opposite directions inside the MDBR. The vapor pressure difference between the two allows the water vapor to diffuse and/or convect across the membrane to the permeate side where it is condensed. The feed and the clean permeate streams also come into contact with each other in the feed tank. The feed tank acts as a heat exchanger where energy is transferred back to the feed from the permeate. To ensure smooth operation, the feed and the permeate streams have to be returned to their original temperature before they enter the MDBR module. The permeate stream is allowed to pass through another heat exchanger to enhance energy recovery and further reduce its temperature. The temperature at which the permeate stream enters the MDBR is controlled through manipulation of the flow rate of the cold stream. The cooling liquid used for this purpose is pumped from the main raw water storage tank. During stable operation, there is a build up of product water in the permeate stream. This build up is removed and stored in the product tank separately.
Fig. 1. Plant schematic [2].
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2.2. Control system and architecture
491
Table 1 Plant Parameters.
The three main objectives that have been considered for the solar powered MDBR plant are: Safety: The plant must remain within the safe operating envelope at all times. Maximizing value: To ensure economic feasibility, the output produced per unit amount of energy available has to be maximized. Self-sustaining operation: The plant must be able to survive without solar radiation during periods of unfavorable weather. Design of a centralized monolithic unit to handle all of the control objectives is usually plausible only when the plant model, dynamics and control objectives are reasonably simple and well defined. Most practical situations involve complex non-linear multi-variable processes which do not come under this category. Hence, the main control problem is usually divided into a set of simpler, hierarchically formulated sub-problems. Each subproblem is assigned a separate layer. These layers or sub problems are characterized by different time scales. The hierarchical control structure is shown in Fig. 2. The scheduling layer uses forecasts to manage back up reserves several days ahead. The supervisory layer manages operation throughout the day while satisfying the reserve targets set by the scheduling layer. It maximizes production in favorable periods and protects the microorganisms in the MDBR during unfavorable periods. Finally, the regulatory layer minimizes set point deviations from the references provided by the supervisory layer. Since this layer has been extensively studied (see Ref. [8] for examples) it is not the focus of this work. The scheduling and supervisory layers are interconnected as shown in Fig. 2. The scheduling layer enforces an upper bound on the power consumption of the supervisory layer. Thus, in case the current solar radiation levels are high but the forecasts indicate bad weather, the scheduling layer will decrease the value of the bound to conserve energy. When the forecasts are bright and favorable, the scheduling layer will allow the plant to increase energy consumption. The selection of measured and manipulated variables shown in Fig. 2 was done through the analysis of cause and effect relationships embedded into the equations that describe the plant.
Cp h
4.2 103 kJ m3 k1 2
0.03 kW m
1
k
2
A
2m
CoP
3
Tin Tfeed
305 K 303 K
Tamb es N
303 K 34 kWh 15
max Fðcirc;feedÞ
10 l min1
min Fðcirc;feedÞ max Php min Php max Tmdbr min Tmdbr
3 l min1
Tsustenance ep
318 K 88 kWh
2 kW 0 kW 353 K 328 K
The main knobs or manipulated parameters for the control framework are: the bounds on consumption P, Q(bound) from the scheduling layer and the optimal values of Ffeed, Fcirc, Php and Tmdbr from the supervisory layer. As for the regulatory layer, it treats the values of Ffeed, Fcirc, Php and Tmdbr from the supervisory layer as control set-points and closes the loop with sensor feedback. The variables which need to be measured are Tin, Tfeed and Tamb. This work assumes knowledge of certain quantities for its operation. These are, solar predictions W over a planning horizon, current or available reserve for the scheduling layer R0 and the current or available reserve for the supervisory layer P, Q(available). The rationale behind such choices will become clearer as the reader goes through the following sections (Table 1). 2.3. Modeling 2.3.1. Temperature dynamics of the re-circulated liquid The inlet temperature of the re-circulated liquid is maintained constant through control of flow rate of cooling liquid in heat exchanger. The outlet temperature of re-circulated liquid can be determined from the equations that describe the rate of heat transfer in the MDBR tank [9], hA
Tout ¼ Tmdbr þ eCp Fcirc ðTin Tmdbr Þ
(1)
where Tmdbr, Tout and Tin denote temperatures of the MDBR tank, outlet and inlet of the re-circulated liquid respectively. The terms Cp, h and A denote the specific heat capacity of water, heat transfer co-efficient of the MDBR tank and surface area of the MDBR tank
Fig. 2. MDBR control architecture.
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f ðqx þ ð1 qÞyÞ qf ðxÞ þ ð1 qÞf ðyÞ: 2.3.2. Output flux Output flux is the amount of clean water produced by the plant. The driving force required to produce output flux is the difference in temperature between the feed and the permeate side [12]. Although the variation of vapor pressure with respect to the temperature gradient is not a perfect straight line, we have assumed linearity over the operating region considered in this work. Fig. 4 shows experimental validation of this relation. Therefore the management system will use the driving force as a proxy to manipulate the output flux (J):
J ¼ Tmdbr 0:5ðTout þ Tin Þ
(2)
The output flux is a function of the outlet temperature of the recirculated liquid. Hence, the relation in Eq. (2) is also non-convex in terms of Fcirc and Tmdbr. It does not satisfy the conditions required for convexity stated above.
8
7
6
−2 −1
respectively and Fcirc denotes the flow rate of the re-circulated liquid. Experimental validation for this relation is shown in Fig. 3 [10]. Variations in flow rate of re-circulation could not be covered in these experiments due to limitations in lab equipment. The relation in Eq. (1) is non-convex in terms of the manipulated variables Fcirc and Tmdbr. It does not satisfy the definition of a convex function [11]: A function f : Rn /R is convex if dom f is a convex set and if for all x, y ˛ dom f, and q with 0 q 1, we have
Output flux (lm h )
492
5
4
3
2
1
0
0
5
10
15
20 25 Driving force (K)
30
35
40
Fig. 4. MDBR experimental validation: output flux vs driving force.
Thermal power (Q load) requirements considered are that of feed liquid (Q feed), re-circulated liquid (Q circ), MDBR tank loss (Q mdbr tank) and the heat recovered by the heat pump (Q hp). The overall equation is:
Q load ¼ Q feed þ Q circ þ Q mdbrtank Q hp ¼ Ffeed Cp Tmdbr Tfeed þ Fcirc Cp ðTout Tin Þ þ hAðTmdbr Tambient Þ CoP Php
2.3.3. Power consumed Power is consumed by the plant in two different forms: electrical and thermal. Electrical power (Pload) is consumed by two circulation pumps and a heat pump. Since the circulation pumps used are centrifugal in nature, shaft power consumed is proportional to their respective flow rates [13].
where CoP is the co-efficient of performance of the heat pump. The terms Cp, Ffeed, Tmdbr and Cp, Fcirc, Tout are non-convex in terms of the manipulated variables Ffeed, Fcirc and Tmdbr.
Pload ¼ Kp Ffeed þ Kp ðFcirc Þ þ Php
3. Design and evaluation of the supervisory layer
(3)
where Ffeed and Fcirc denote flow rates of feed and re-circulated liquid. Kp is the constant of proportionality and Php is the power supplied to the heat pump.
(a)
The supervisory layer has two modes of operation. It is said to be in production mode during periods of favorable weather and plant output is maximized. It is said to be in sustenance mode during
(b)
311
311 ACTUAL ESTIMATE 310
309
309
308
308
Toutlet (k)
310
T
outlet
(k)
ACTUAL ESTIMATE
307
307
306
306
305
305
304 304
(4)
304.5
305
305.5
306
306.5
307
304 305
310
315
320 T
Tinlet (k)
325 (k)
330
335
mdbr
Fig. 3. MDBR experimental validation: (a) outlet temperature vs inlet temperature (b) outlet temperature vs MDBR temperature [10].
340
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periods of unfavorable weather and the focus is on conserving energy necessary for the micro-organisms. Inclusion of a sustenance mode is not necessary if the system has an auxiliary source of power other than solar radiation. The auxiliary source can support continuous operation during periods of unfavorable weather. Otherwise, if the back up reserves are large enough to act as this auxiliary source, then also, sustenance mode is not required. There will be no penalties on production.
3.1. Problem formulation From a mathematical perspective, the supervisory layer optimization problem is of the form:
minimize
cðx; uÞ
subject to
hðx; uÞ 0
u
(5)
where c(x, u) is the objective function and h(x, u) represents the constraints. The symbols x and u represent the state variables and the manipulated variables respectively. They are:
3 Tamb x ¼ 4 Tfeed 5 Tin
3 Fcirc 6 Ffeed 7 7 u ¼ 6 4 Php 5 Tmdbr 2
2
and
There are multiple ways in which this problem can be formulated. The designer can optimize a weighed sum of the power consumed and the output produced to obtain a stationary operating point. The component operation limits for the manipulated set-point u take the form U u U as shown below:
d1 Jdf þ d2 ðQ load þ Pload Þ
minimize u
UuU
subject to
Q load þ P load
subject to
Jdf Ji ˛½ Jmin ; Jmax
u
UuU
(7)
But, in this case, numerous optimization problems have to be solved before arriving at the operating strategy. Hence, the system would benefit from a formulation that captures all the objectives in a single optimization problem. Maximizing the output flux while imposing bounds on the power consumed would allow the system to do that. The optimization problem is of the form:
maximize u
subject to
J Q load Q bound Pload Pbound UuU
where the bounds on thermal power consumed (Qbound)and electrical power consumed (Pbound) correspond to the available power. When integrated with the scheduling layer, these bounds will also depend on weather forecasts also. The optimization problems presented in this section assume that the plant is operating in production mode. This implies that the power available is equal to or above the minimum value required to support production of water. The minimum value of power required is dependent upon the sizes and the operational limits of the components in the plant. Intuitively, the minimum power required can be determined by assigning the least possible values to all manipulated variables. In mathematical terms this can be stated as u ¼ U. So if the power is enough for the circulation pumps, the heat pump and the MDBR temperature control to operate at their respective minimum capacities the plant can produce water [10]. 3.2. Implementation 3.2.1. Model simulation Before proceeding to the simulation details, let us state the assumptions being made about the MDBR plant’s operation at each sampling period: The temperature of feed liquid Tfeed, ambient temperature Tamb are known from measurements. The heat recovery system maintains the inlet temperature of the re-circulation liquid Tin constant at the prescribed value. The available reserve or the maximum power that can be consumed by the plant during the current period is known. The regulatory layer is capable of enforcing the set points determined by the supervisory layer. The supervisory layer acts as a steady state optimizer, hence the transients or dynamics of the thermal tank are not considered in this work.
(6)
This approach though simple does not take into account the changes in available power. It determines a single operating point that the system has to stick to irrespective of external conditions. In order to be more robust the system needs to adjust the set-points according to environmental changes. Another strategy that can be considered is to remove the output flux from the objective function and include it as a constraint. Thus, the optimization problem will minimize power consumed while forcing the output flux to take different values between its nominal minimum and maximum bounds. The combination which is closest to the available power is chosen as the control set-point. The resulting optimization problem takes the form:
minimize
493
(8)
Taking these assumptions into consideration, the values of output flux, thermal power and electrical power can be determined by simultaneously solving Eqs. (1)e(4). Upon careful examination, one will notice that the model is not explicitly represented as an equality constraint in the optimization problem described in Eq. (8). Rather, they are included in the cost functions and the inequality constraints in this work. This is done since the model of the process can be analytically solved. But many of the expressions presented here are not convex and hence require modification as shown below. 3.2.2. Obtaining convex problems The optimization problem is non-linear and non-convex. It requires modification before it can be fed to a convex optimization solver. Convex relaxations that closely resemble the original supervisory layer problem help the system handle non-convex terms. Hence, the optimization problem from Eq. (8) is modified to give,
maximize u
subject to
J ðrÞ ðrÞ
Q load Q bound Pload Pbound UuU ðrÞ
(9)
where J (r) and Q load are convex alternatives used to replace the ðrÞ original functions. The relaxation Q load must satisfy the constraint ðrÞ from the original problem, Q load Q bound . Otherwise the results obtained through optimization will violate the actual constraints. As long as the relaxations used resemble the original relations satisfactorily, further modifications are not required. The true test for these relaxations lies in their ability to produce viable optimal
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solutions when used with the actual relations during simulation. In this work, we have used simple relaxations of the form:
J ðrÞ ¼ cT1 U þ d1 ðrÞ
Q load ¼ cT2 U þ d2
(10) (11)
The linear relaxations and simulations shown in the next section have produced practically feasible results. 3.3. Simulation example This example provided a demonstration of the proposed supervisory layer control system. Fig. 5(a) shows the solar radiation profiles used in this simulation. They were obtained from the NREL data archive at http://www.nrel.gov/. Given the solar radiation profile, the available power can be computed. At the beginning of the day, the system encounters low solar radiation as shown in Fig. 5(a). It operates in sustenance mode. Output
is not produced as seen in Fig. 5(b). As the day progresses from dawn, the available power exceeds the required threshold. The system now realizes that production of output can commence. Hence, it switches to production mode and production of output is initiated as shown as shown in Fig. 5(b). As the solar radiation increases, the available power increases and the optimizer chooses the decision variables T accordingly. The output produced u ¼ Fcirc ; Ffeed ; Php ; Tmdbr and power consumed increases with an increase in available power as seen in Fig. 5(b) and (c). This increase can be attributed to optimizer choosing a high value for the temperature inside the MDBR (Tmdbr) as seen in Fig. 5(d). Fig. 5(e) shows that as long as the solar radiation is adequate to operate in production mode, the system does not need to expend electrical power through the heat pump. One can understand from Fig. 5(f) that the optimization problem generally selects the maximum value of flow rate for the re-circulated liquid. It is attempting to increase output flux by increasing flow rate of the recirculation loop. The feed liquid on the other hand is assigned minimum flow rate so as to minimize the thermal and electrical power consumed as shown in Fig. 5(g). At the end of the day, the solar radiation declines and the system switches back to sustenance mode. Production of output is stopped. Hence, both mode selection and optimization capabilities of the supervisory layer have been explained in this example. 4. Design and evaluation of the scheduling layer Energy reserves are managed by the scheduling layer. It ensures that the plant can maintain the conditions required for survival of MDBR bacteria even when it is subjected to unfavorable weather conditions. 4.1. Problem formulation
Fig. 5. Supervisory layer e typical solar profile.
The scheduling layer has to decide whether the plant can be allowed to consume all the power that is available or whether some of it has to be reserved for unfavorable periods without sunlight. For this purpose, it needs to be aware of weather variations and requires information regarding the variation of solar radiation in the future. The scheduling layer will depend on the solar radiation prediction system to provide forecasts about the future. The scheduling layer divides the interval between sunrise and sunset into three periods. It looks ahead over a prediction horizon of five days (i.e. 15 periods). Energy consumption targets for these periods are determined based on information available from the prediction system. These targets are revised when the prediction horizon is shifted forward. This is akin to a receding horizon strategy one would use while planning finances. A multi-month expenditure plan is decided based on the information available about a finite section of the income forecast. But only the decision regarding saving and spending for the current month is applied. The expenditure plan is then revised in the next month when new income forecasts are available. A new financial plan is determined. Again, only the current decision is put into practise and the process repeats. For the scheduling layer the consumption targets represent expenditure, the solar predictions represent the income forecast and the back up reserves represent the savings. This receding horizon process shown in Fig. 6 allows the plant to account for uncertainty in the forecasts and optimize its strategy accordingly. One does not need to scrutinize a financial plan if the forecasts indicate a lean period of very low or no income. When the influx of resources ceases to exist, the consumption of available resources must be reduced to the minimum amount possible. Similarly, if the solar predictions indicate that the solar profile will not be able to sustain even the minimum back up reserves required, the scheduling layer will not attempt to solve the optimization problem.
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495
(a) 100
wi (kWh)
80 60 40 20 0
(b)
1
2
3
4
5
100 Max/Min Bounds
80
It will merely cut down expenditure to the bare minimum. This is known as the feasibility condition. During normal periods of sunlight, the system is designed to maximize output flux. Higher values of output flux require more energy consumption. If the system were to minimize the available reserves (r), it would be maximizing energy consumption. Hence, the objective function takes the form:
minimize U
N X
ep ui es
i ¼ 1; .; N
2
3
4
5
6
100 80 60 40 20 0
(14) (15)
The constraint from Eq. (14) makes sure that the available reserve should always be greater than the load required to support plant operation in sustenance mode, es. The constraint from Eq. (15) ensures that the scheduling targets lie between their upper and lower bounds. The upper bound is the maximum consumption possible in production mode, ep. The lower bound should obviously be the consumption level required to sustain plant operation in sustenance mode, es. All functions and constraints are convex. Hence, they do not require any modifications during implementation. Furthermore, the special structure of the optimization problem discussed in this section can be exploited to reduce the complexity and computational effort required to determine the solution as shown in the appendix. 4.2. Simulation example The solar profile wi considered in Fig. 7(a) shows that there are bursts of radiation on days 1, 3 and 4. Days 2 and 5 do not receive sunlight. Using the information available from wi the optimization problem has determined values for consumption targets ui as shown in Fig. 7(b). The consumption remains at the minimum level
1
2
3
4
5
Time(Days)
(13)
where wi represents the predicted energy level and ui represents the consumption target for the ith period. The constraints are:
i ¼ 1; .; N
1
(c)
(12)
where, U is the optimization variable referring to the vector of consumption targets for each period in the planning horizon. N refers to the number of periods in the planning horizon. Expressions for the available reserve levels at the end of the ith period are described as the difference between the energy available and the energy consumed:
ri es
0
Sustenance Bound
i¼1
ri ¼ ri1 þ wi ui
40
0
# ðri Þ
60
20
ri (kWh)
"
ui (kWh)
Fig. 6. Receding horizon approach in the scheduling layer.
Fig. 7. Scheduling layer e intermittent profile.
when there is no sunlight. When there is sunlight, the system is allowed to increase consumption, but never allowed to exhaust everything available. This is evident on day 4, where the consumption is markedly lesser than the available energy in order to tackle unfavorable periods in day 5. The values of forecasts and consumption targets are used to calculate the reserve levels shown in Fig. 7(c). The system intuitively builds up reserves before the dark periods in days 2 and 5. Also, the reserve levels are never allowed to dip beyond a certain minimum value. Thus the scheduling layer is able to make sure that the system can overcome periods of undesirable weather. 5. Conclusion This paper provides an introduction to the formulation and design of a plant-wide optimization and control system for an MDBR water recycling plants. It presents a means to manage both real time optimization and long term reserve management requirements. The simulated examples show that the proposed framework is able to make best use of the available energy to produce clean water while maintaining sufficient reserves required for the sustenance of micro-organisms in the MDBR. Implementation of the proposed control system will greatly reduce the need for operator intervention. The layered structure can accommodate future modifications, improvement and inclusion of better functions can be established without having to incur the costs of overall replacement.
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For future work, the performance of the system can be improved by dealing with some issues. The MDBR is not designed to handle rapid changes in operating conditions. In order to leverage the computation that can be performed in real-time, the control strategies have to be adjusted according to variations in solar radiation. But, in the current set-up, changes in solar radiation are faster than the changes in operating conditions. This is because current industry practice suggests that trends are rather slow in the MDBR due to system capacitance [9]. Hence, further research is required to assess the minimum time interval between changes in operating conditions for MDBRs. The system can also be extended to account for transients in membrane performance due to fouling. It is caused by gradual deposition of particles and macro-solutes on the membrane surface [14]. This effect will be incorporated into the required driving force as a function of time. Fouling can usually be reversed by regular cleaning [9] (as shown in this reference).
Appendix A. Reduced complexity algorithm for the scheduling layer Appendix A.1. Analysis and discussion of problem structure Further analysis of the scheduling layer optimization problem reveals that, it can be solved without the use of an on-line optimization solver. The structure of the problem allows us to define an algorithm of reduced complexity to determine the optimal solution. For this purpose, one must begin with an assessment of the optimization problem from Section 4.1: The objective function from Eq. (12) is a linear function of the values of reserves at different intervals,
" minimize U
N X
# ðri Þ
i¼1
Since the reserve is known to be a linear function of the optimization variable U from Eq. (13),
ri ¼ ri1 þ wi ui the original objective function can be equivalently formulated as maximization of a linear function of the optimization variable:
minimize cT U þ d
(A.1)
U
It is observed that Eq. (14):
ri es ci ¼ 1; .; N
By re-arranging the terms and multiplying both sides of the above equation by L1 we get,
L1 ðR0 Es Þ þ W U
(A.2)
The relation from Eq. (A.2) makes up a key element of the solution algorithm. All the elements on the left hand side of the equation are constants and represent the maximum values that components of the vector u can take. Hence by replacing the inequality by an equality sign and solving the set of simultaneous equations, one can obtain a preliminary estimate of the solution of the optimization problem without the use of an optimization solver. This solution is referred to as the preliminary estimate since it does not take into account the upper and lower bounds imposed upon the optimization variable by Eq. (15). In other words, this estimate can contain values that may either be higher than the plant’s designed capacity or may be lower than the value required for operation in sustenance mode. Hence, some adjustment is required to ensure that the values of the optimization variables do not violate these constraints. This is done in two stages. In the first stage of adjustment, the variables ui are analyzed starting from u1 to uN. If at any variable, there is a surplus or excess above the maximum bound, this surplus is shifted to the subsequent interval. This process removes any violations of the constraint related to the upper bound. In the second stage of adjustment, the variables ui are analyzed from uN to u2. If at any variable, there is a deficit or gap below the minimum bound, this deficit is borrowed from the interval preceding the current interval. Performing these steps for all variables up till u2 will remove any violations of the lower bound. Notice that the second stage of adjustment is not allowed to modify u1. This is done to check whether the problem is feasible. If the second stage of adjustments results in a value of u1 that is lesser than the minimum bound, then it is obvious that the predicted values are not sufficient to sustain the minimum reserve levels required during all periods. In such cases, the problem is deemed to be infeasible. The rationale behind the operation of this algorithm can be understood through the operation of the plant itself. If radiation available is more than what the plant can use, any surplus will obviously be transferred to the reserve to be used the next day. This is the principle behind the first stage of surplus adjustment. As for the second stage, if the consumption is below the minimum required at any interval, it is an indication that additional reserves have to be saved up before that point of time. Hence, by borrowing the deficit from the preceding intervals, the second adjustment accomplishes that goal. Thus the algorithm is reduced to a series of simple mathematical operations as shown in the next section.
can be written as,
Appendix A.2. Algorithm sequence
R0 þ LW LU Es
The Matlab code for the algorithm described in this Appendix is as follows:
where, T
R0 ¼ ½ r0 r0 / r0 W ¼ ½ w1 w2 / wN T U ¼ ½ u1 u2 / uN T Es ¼ 2½ es es / es T3 2 1 0 0 / 0 1 0 0 / 61 1 0 / 07 6 1 1 0 1 6 7 6 1 6 7 L ¼ 6 6 1 1 1 1 0 7and hence; L ¼ 6 0 1 1 1 4« 4 « 1 1 1 1 05 1 1 1 / 1 0 / 0 1
3 0 07 7 «7 7 05 1
u_alg ¼ inv (L)*((r0 es) * ones(n, 1)) þ W; %if there is surplus in the current period, % store it as reserve and use in the next period for i ¼ 1:n if u_alg (i) > ep u_alg (i þ 1) ¼ u_alg (i þ 1) þ (u_alg (i) ep); u_alg (i) ¼ ep; end end %if there is deficit in the current period, % use less in the previous period
A. Vijay et al. / Renewable Energy 60 (2013) 489e497
for i ¼ n: 1:2 if u_alg (i) < es u_alg (i 1) ¼ u_alg (i 1) (es u_alg (i)); u_alg (i) ¼ es; end end if u_alg (1) < es, disp (‘Problem Infeasible!’), end In relative terms, first statement of the algorithm requires more computational effort since it involves matrix inversion, L1. Since L is constant and has a special structure, L1 can be easily determined. References [1] Phattaranawik J, Fane A, Pasquier A, Bing W. A novel membrane bioreactor based on membrane distillation. Desalination 2008;223(1e3):386e95. [2] Vijay A, Ling KV, Fane AG. Applications of convex optimization in plant-wide control of membrane distillation bio-reactor (MDBR) water recycling plant. In: Proceedings of the 11th International Conference on Control, Automation, Robotics and Vision (ICARCV 2010) 2010. p. 246e51. [3] Salcedo R, Antipova E, Boer D, Jiménez L, Guillén-Gosálbez G. Multi-objective optimization of solar Rankine cycles coupled with reverse osmosis desalination considering economic and life cycle environmental concerns. Desalination 2011;286:358e71.
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