continuous stirred tank reactor and a 'dynamic mass balance' over the reach written as ... estimation; in particular, the EKF does not provide a very systematic ...
Water quality models for waste water management
Paul G. Whitehead
Abstract. A methodology is proposed for evaluating the impact of effluent on t h e aquatic environment and for the development of management strategies for the planning and control of waste water systems. The particular approach considers the dynamic behaviour of t h e river system and relates time varying upstream influents to downstream water quality using simple, low order but robust models of water quality. Identification and estimation problems are investigated using recursive algorithms of time series analysis, and practical applications are considered for an artificial aeration scheme on the River Cam and a probabilistic assessment of the impact of effluent from the new town of Milton Keynes in the United Kingdom. Modèles de qualité pour la gestion des eaux usées Résumé. L'auteur propose une méthodologie pour évaluer l'impact des effluents sur le milieu aquatique et pour la mise en oeuvre de stratégies de gestion dans la planification et le contrôle des systèmes d'eaux usées. Cette façon de procéder prend en compte le comportement dynamique du système 'rivière' et établit des relations entre les variations dans le temps sur les parties amont, y compris les effluents, et la qualité de l'eau à l'aval, ceci au moyen de modèles de qualité des eaux peu sophistiqués mais solides. Les problèmes d'identification et d'estimation sont abordés par des procédés de calculs récurrents de l'analyse des séries chronologiques; la méthode est appliquée à un projet d'aération artificielle de la rivière Cam, en Angleterre, et à une évaluation probabiliste de l'influence des effluents de la nouvelle ville de Milton Keynes en Royaume Uni.
INTRODUCTION In the management and planning of water resource systems one of the major needs is for adequate forecasting of possible future conditions. The complex interactions between water quality variables, biological components and the physical environment ensure that forecasting is a difficult procedure, while systems analysis is further complicated by the problems of obtaining representative samples of water and the management errors associated with laboratory analysis. In such an 'ill-defined' environment a statistically based methodology is required to analyse water quality data. At the same time descriptions of water quality systems should be based on the day to day behaviour or the dynamic aspects of river pollution, since it is the transient violations of water quality standards that create particular problems. In line with these requirements a study of the Bedford Ouse River System in Eastern England was initiated in 1972 by the Great Ouse River Division of the Anglian Water Authority and the Department of the Environment, in association with the Control Division of the Engineering Department, University of Cambridge. The objective of the study was to develop and utilize water quality models in the planning and control of water resource systems on the Ouse and, in particular, to investigate the impact of effluent from the new city of Milton Keynes being developed 55 km upstream of the Bedford Abstraction Plant. The research has been directed towards obtaining robust, low order models of water quality; models that are of practical utility and describe both the dynamic and stochastic aspects of the aquatic environment. 421
422
Paul G. Whitehead Effluent
1
Upstream input
Downstream outout
VOLUME V
(a) Effluent
Upstream input
w
Q
v, e„
downstream output
C
(b) FIGURE 1.
(a) River reach, (b) continuous stirred tank representation.
DYNAMIC WATER QUALITY MODELS The dynamic water quality models developed for the Bedford Ouse have been based on the conservation of mass principle. As shown in Fig.l, the reach is represented as a continuous stirred tank reactor and a 'dynamic mass balance' over the reach written as follows: dCg
it
9£i V
V
(1)
where V is the reach volume [m^], Q is the volumetric flow rate [m 3 /day], q i s the concentration at the upstream input point [mg/1.], c 0 is the concentration at the downstream output point [mg/1.], c and c • are respectively, additional sources and sinks acting on the system [mg/1./day]. This basic mass balance equation for a reach may be modified to suit the nature of the problem and the additional sources and sinks represent such factors as effluent entering the reach or a chemical decay producing a loss of material down the reach. Each water quality variable or state of interest is described by a single equation and the interaction between states represented by cross coupling terms. Model identification The identification of the causal relationships in such water quality models is an important aspect of analytical water quality studies, and during the early stages of the Bedford Ouse study, the extended Kalman filter (Jazwinski, 1970) proved to be a flexible tool with which to identify suitable dynamic water quality model structures. An initial model was formulated, based on some heuristic feeling or physico-chemical understanding of the system, and the EKF was then used to test the efficacy of these models and, where
Water quality models for waste water management
423
Observed Downstream Water Q u a l i t y
input information
Forecast downstream
RIVER MODEL
water quality
/* \^ Model
error
Corrected model parameters.
RECURSIVE ESTIHATION
Parameter Value 1.2 0.9 k
0.6
v
"
8
'
/ *
i~
o. : 1
45
1
92
1
138
— I
184
1
230 TiiT.e d a y s
lb) FIGURE 2. (a) Schematic diagram of recursive model estimation, (b) Estimation of parameters fc8 and &g *.
necessary, to supplement the basic model with additional terms to account for disturbances. Whilst the mathematical details of the extended Kalman filter (EKF) may appear complex the principle of recursive estimation employed is relatively simple and is of particular relevance to the analysis of time varying water resource systems. The parameters of the model are updated step by step through the data using the estimation algorithm and, as shown in Fig.2, the errors between the model forecast and the observed water quality state is used as a corrective factor. In addition to automatically tuning the model, the technique provides information on the time varying nature of the system, since parameter variations will reflect structural changes in the system. It is often necessary at this early identification stage to alter the structure of the model to allow for nonstationarity of the system. A complete statistical analysis of a water quality system is given by Beck and Young (1976), and the approach has been used extensively in the Bedford Ouse study to identify model structure.
424
Paul G. Whitehead
Model estimation Unfortunately there are certain limitations of the LKF as a tool tor final parameter estimation; in particular, the EKF does not provide a very systematic approach to parameter estimation in those situations where large quantities of data a r e available for several reaches of the river. I n these situations, there is a need for a m o r e rigid estimation procedure where the model structure is assumed known a priori and where it is desirable to have some form of repeated iteration through the data, in o r d e r to 'refine' the estimates of the parameters and so obtain better statistical efficiency (lower variance estimates). Such a technique is available in the iterative instrumental variable approximate maximum likelihood (IVAML) (Young, 1974) approach utilized during the rainfallrunoff time series modelling of the Bedford Ouse (Whitehead and Young, 1977). The extension of this technique to the multivariable situation in which the m o d e l is composed of a coupled set of dynamic equations has been restricted to the following discrete-time state equation: xk = Axk-i
+ Buk~i
(2)
where x and u represent the states (e.g. water quality variables at time k ) and the inputs (e g sunlight etc.) respectively, while A and B represent coefficient matrices which include unknown parameters to be estimated by reference to the available data. However transforming the differential equation model of water quality into the discrete-time form tends to destroy some of the simplicity of the model. This simplicity may be retained, however, by writing the model directly in discrete form while retaining all the salient features identified in the previous EKF studies. Discrete water quality models developed for the Bedford Ouse are in the form of equation (2) with the multivariable model describing the relationships between biochemical oxygen demand (BOD) and dissolved oxygen (DO) at the o u t p u t of a single reach in a nontidal river system. This model takes the form of two coupled first-order equations and can be written as: BOD: xlk DO: x2k
= ki
s (~ — y k _ 1
= kvcifi-l +
*l,*-l
+
fc2"U-l
+
k
^k-^l
y + k5 ç^j*2,k~l
(3) + k(,u2,k~i
+^fc-l
ks,CSlk-i
where xi is the BOD at the output of the reach, which can be considered as an aggregate or macro measure of the oxygen absorbing potential of substances in the stream (such as decaying organic material from effluent discharges), and is defined as the oxygen absorbed in mg l . _ 1 over a five day period by a sample of river water in the absence of light at a constant temperature of 20 C; x2 is the DO at the output of the reach [mg 1. ~ ] ; Q is the volumetric flow rate in the stream [m 3 day- 1 ] ; Vm is the mean volumetric hold-up in the reach [m 3 ] ; u\ is the input BOD from the previous upstream reach in the river system [mg lr1}; M2'is the input DO from the previous upstream reach [mg \:1] ; S is the term dependent upon sunlight hours and chlorophyll A level to account for photosynthetic effects such as algal growth and decay; Cs is the saturation concentration of DO [mg l.- 1 ] ; and kh k2, •.. , k8 are coefficients or parameters which will be either constant or slowly time-variable depending on the period of observation.
Water quality models for waste water management DO mg/L
Summer
Autumn
425
Winter
12.0 8.8
l
flfof
y
• • i V f ; -V. .A
*
\ /
f'V . '»-
7.2 4.8 2.4 0.0 82
BOD mg/L
12.û
138
104 m
230 o«rs
f
8.8 7.2 4.8
/v^¥
2.4 0.0 48
82
138
184
230
CTI D«YS
FIGURE 3. Forecast DO and BOD on the Bedford Ouse. . observed quality state, - forecast quality state, forecast DO over winter prior to inclusion of a reaeration term.
The basic estimation problem is then to use the observations on the states to determine consistent estimates of the parameters and the particular approach utilized is a multivariable version of the instrumental variable approximate maximum likelihood technique discussed in detail elsewhere (Young and Whitehead, 1977). Time varying parameters
In the same manner as the EKF, the recursive estimation algorithm may be utilized to track time varying parameters. During the analysis of water quality data a satisfactory model was obtained for the summer and autumn low flow conditions, as shown in Fig.3 but a large discrepancy occurs over the winter period with the model output well below the observed DO. The discrepancy is probably due to the additional reaeration in winter, which is not included explicitly in the model at this stage and which can be investigated by examining the recursive estimates of the 'reaeration' parameter k% on the assumption that this may be varying over the data period. The parameter is reasonably constant over the first 140 days, as shown in Fig.2, but moves rapidly to a higher value over the winter period when the flow increases are encountered. The variation of reaeration rate as a function of flow has been described in the
426
Paul G. Whitehead
literature (Owens et al, 1964) as
D
" '[%)
where Q is the volumetric flow rate, d is the depth and D is the reaeration rate, and may be introduced into the model by defining the parameter fcg in the form *8 = ^
k8*
(4)
An estimate of the new parameter kg* is shown in Fig.2 as a dotted line: it is clear that, by incorporating the information on flow via equation (4), it has been possible to obtain a description in terms of a predominantly constant estimated parameter, k%* and as shown in Fig,3, obtain an improved model forecast.
OPERATIONAL CONTROL USING DYNAMIC WATER QUALITY MODELS Up to this point in the Bedford Ouse study we have necessarily concentrated on the identification and estimation of stochastic-dynamic models since little was known about the mechanisms of transient change and it was necessary to establish some fundamental appreciation for the nature of the dynamic system. But now that we have been able to achieve some success in these preliminary modelling studies, it makes good sense to consider the possible applications of the kind of models that have been developed. Operational management Dynamic models have been used widely in the process industries for the design and evaluation of engineering systems from the operational management viewpoint. Often complex chemical reactions have to be closely controlled in order to meet product specifications and, for example, it is possible by continuously monitoring a quality state downstream of a reactor, to detect any deviation from the specified level and take whatever remedial action is required to correct the situation. The idea of monitoring, detecting an error from a reference level and then taking remedial action is termed closed loop control; the loop is closed either by the operational managers, who take a decision to change certain appropriate variables in the system or by automatic controllers which directly adjust the input control variables. One control problem of particular interest to the Bedford Ouse study is the maintenance of dissolved oxygen levels in rivers immediately downstream of sewage outfalls. Young and Beck (1974) have investigated two methods of maintaining the DO level in this reach of the Cam using either a variable effluent quality from the treatment plant or a variable effluent discharge rate into the river. For both control systems it was assumed that BOD information was not available and that only the dissolved oxygen was continuously monitored at the downstream point. Control by artificial instream reaeration One low cost alternative approach to the maintenance of dissolved oxygen levels in a river system is by artificial aeration using mechanical aerators. However, artificial aerators should always be viewed as a temporary solution since they cannot cope with a continually worsening effluent from an overloaded sewage works; indeed this may well tend to mask a slowly deteriorating situation. Artificial aeration of a polluted river has been investigated both experimentally and theoretically (Whipple et al, 1970)
Water quality models for waste water management Feedforward loop
CONTROLLER
427
Feedback loop
Effluent
A
Upstream input
FIGURE 4.
£
AERATORS
TTT^
D" F r o b "
Downstream output
Aerator and control scheme for the River Cam.
and it has been used for some time in practical situations as a method of alleviating pollution problems. An instream aeration control study on the River Cam During the past few years the dissolved oxygen levels in the River Cam downstream of Cambridge have steadily deteriorated, resulting in fish mortalities on several occasions. This has been attributed to a combination of sewage pumping station overflows, unsatisfactory effluent from Cambridge sewage works and large diurnal variations induced by photosynthesis and respiration of aquatic plants. A model of this stretch of the River Cam has been used to assess the performance of the aerator scheme. The model parameters have been estimated using the dissolved oxygen concentrations at 7.00 a.m., the minimum levels, producing the following model:
Xfc
0.32 -0.41
0 0.22
0.14 0.21
Sk-1
Xfc-l
+
0.41 0
0 0.15
0 0.73
Q,k~ 1
u\,k~l U2, k-l
(5)
On the basis of this model a feedback—feedforward control scheme has been developed. The main features of this scheme are shown in block diagram form in Fig.4. A dissolved oxygen probe located at the minimum DO sag point downstream of the discharge feeds back DO measurements to a controller. At the same time feedforward signals from an upstream probe provide information on the upstream water quality and a combination of feedforward and feedback signals determine the control signal to the aerators and hence the degree of oxygenation. Details of the control system design are given elsewhere (Whitehead, 1976b). An additional criterion has been incorporated such that the control only operates when the dissolved oxygen level falls below the reference level of 5 mg 1 . _ 1 ; an obvious practical requirement since it is unnecessary to inject further oxygen when high levels exist. In order to test the control scheme an additional load on the system, equivalent to a 5 mg. I." 1 increase in the BOD in the river was simulated and as shown in Fig.5, the DO level prior to control is well below the reference level except during the sunny period from day 32 to day 45. The controlled response, by comparison, is reasonably constant at 6 mg. I.""1 with the aerators providing an input of oxygen over the entire period
428
Paul G. Whitehead C o n t r o l s i g n a l DO mg/1 6
3 +
^ujUMlllillllllUMUilll 16
32
48
64
80 Time days
-3
+
Downstream DO mg/1
9.6
1
* controlled ** uncontrolled
7.2 4.8 2.4
16
32
64
80 Time days
FIGURE 5. loading.
Controlled downstream DO for the River Cam given additional effluent
increasing to 3 mg 1. - 1 after day 46. The control input during the sunny period is reduced since photosynthetic and respiratory effects provide a natural net input of oxygen. The analysis indicates that during poor quality conditions up to 3 mg 1. l of additional oxygen may be required, which, for the equipment being contemplated, implies that three units will be required. STOCHASTIC MODELS IN THE PLANNING ROLE The stochastic aspects of water quality models provide an estimate of the inherent uncertainty in the system and this information may be utilized via the technique of stochastic or Monte Carlo simulation to provide forecasts directly in terms of probability distributions rather than exact values. The stochastic simulation approach is extremely useful where analytical solutions are difficult or even impossible to obtain, as is often the case with reasonably complicated dynamic systems. The system calculations (usually simulations) are performed a large number of times, each time with the values for the stochastic inputs or uncertain parameters selected at random from their assumed (i.e. estimated) parent probability distributions. Each such random experiment or simulation yields a different result for any variable of interest and when all these results are taken together the required probability distribution can be ascertained to any required degree of accuracy from the sample statistics. The degree of accuracy of the probability distribution function estimated in this manner is ,of course, a function of the number of random simulations used to calculate the sample statistics, but it is possible to quantify the degree of uncertainty on the distribution using non-parametric statistical tests such as the Kolmogorov-Reyii statistics (see Spear, 1970). Monte Carlo simulation is a flexible, albeit computationally expensive tool with which
Water quality models for waste water management
429
cumulative p r o b a b i l i t y
1.000
0.000
12.0
15.0
BOD m g / 1 .
cumulative p r o b a b i l i t y
1.000 0.800
1 1/
0.600
w
0.400
It
11 '
0.200 0.000 3.0
6.0
9.C
12.0
15.0 DO m g / 1 .
FIGURE 6. Distributions for differing effluent treatment levels. G observed distribution at Bedford forecast distribution given 0.2 m /s of effluent, forecast distribution given 0.4 m /s of effluent, forecast distribution given 1.0 m /s of effluent.
to investigate certain design problems. For example, the water quality standards proposed in a previous report (Taylor and Fawcett, 1974) are presented in terms of probability distribution functions, and therefore, provide a reference against which the water quality can be tested. It would be possible to perform Monte Carlo simulation analysis using the water quality models developed for the study section of the Bedford Ouse and making various assumptions about future levels of effluent input. The outcome of such an analysis would be probability density functions for the water quality states which could be compared directly with the designed water quality standards. Such information would be extremely useful in assessing the impact of influent to the system and determining the degree of treatment necessary at Milton Keynes in order to ensure a satisfactory water quality at the abstraction point. Statistical water quality forecasts An initial assessment of the impact of Milton Keynes effluent on the aquatic environment may now be obtained using the Monte Carlo simulation, details of which are given by Whitehead (1976a). Altogether three effluent conditions were considered with discharge rates of 0.2, 0.4 and 1 m 3 / s and with BOD levels of 5, 10 and 10 m g / 1 . respectively. It was assumed that the effluent has no dissolved oxygen present; a condition that represents the worst situation but which is not unrealistic as the effluent is to be pumped direct from the treatment works via a 4-km pipe into the river. Effluent BOD levels fluctuate in practice and a stochastic component defined by a noise signal of
430
Paul G. Whitehead
variance 1,4 and 4 mg/1. respectively was added to the three BOD levels. The distributions of BOD and DO at Bedford given these three effluent conditions are compared with present situations in Fig.6. At low discharge conditions there is relatively little effect on the aquatic environment. At the 1 m 3 /s condition, however, the mean BOD level has risen to 4.5 mg/1. and mean DO level has fallen to 6.5 mg/1. and to DO distribution ranges from 4.5 to 9 mg/1. These distributions represent only an initial assessment of the impact of Milton Keynes effluent and an updated forecast based on a re-estimated model in two years time may indicate an improved situation. On the other hand, the DO levels may be adversely affected by the changing biological nature of the river and some form of control action may be necessary to improve the DO distribution. CONCLUSIONS In this paper we have outlined the major aspects of the study of short-term water quality variations in a non-tidal river system. We have been able to develop a complete dynamicstochastic model for flow BOD-DO in a 55-km stretch of the Bedford Ouse River System between the site of the new town of Milton Keynes and the Bedford Water Boarc Abstraction Plant. We believe that this model is one of the first examples of a dynamicstochastic water quality model for a long stretch of river to be satisfactorily identified and statistically validated by reference to field data.
REFERENCES Beck M B. and Young, P.C. (1976) Systematic identification of DO-BOD model structure. /. ' Environ EngngDiv., Proc. Amer. Soc. Civ. Engrs 102, no. EE5, October, 909-927. Jazwinski A. H. (1970) Stochastic Processes and Filtering Theory: Academic Press, New York. Owens, M., Edwards, R. W. and Gibbs, J. W. (1964) Some reaeration studies in streams. Int. J. Air Water Pollution 8, 469-486. Spear, R. C. (1970) The application of Kolmogorov-Renyi statistics to problems of parameter ' uncertainty in systems design. Int. J. Control 11, no. 5. 771-778. Taylor N. and Fawcett, A. (1974) Report on the Bedford Ouse Steady State Model, Great Ouse River Division, Anglian Water Authority. Whipple W. M., Goughlan, F. P. and Yu, S. L. (1970) Instream aeration for polluted rivers. J. Samt. EngngDiv Proc. Amer. Soc. Civ. Engrs, October. Whitehead P G (1976a) Dynamic models in the planning role. In System Simulation in Water Resources (edited by G. C. Vansteenkiste): North Holland/American Elsevier, Amsterdam/ New York. . Whitehead P G (1976b) A control and systems analysis of artificial instream aeration. In Modelling and Optimization in the Service of Man. (Proc. 7th IFIP Conference) (edited by G. Goos and J Hartmanis): Springer-Verlag Berlin, Heidelburg, New York. Whitehead, P. G. and Young, P. C. (1977) A systems model of the Bedford Ouse River - Part 1. Streamflow modelling, submitted to J. Hydrol. Young P. C. (1974) Recursive approaches to time series analysis. Bull. Inst. Maths and its Applications 10, nos. 5 and 6,209-224. Young, P. C. and Beck M. B. (1974) The modelling and control of water quality in a river system. Automatica 10,5,455-468. Young P. C. and Whitehead, P. G. (1977) Recursive approaches to time series analysis for multivariable systems. Int. J. Control.
