Abstract
We present an analytical dynamic mathematical model and a design optimization of a residential scale combined heat and power system. The mathematical model features a detailed description of the internal combustion engine based on a mean value approach, and simplified submodels for the throttle valve, the intake and exhaust manifolds, and the external circuit. The validated zerodimensional dynamic mathematical model of the system is implemented in gPROMS\(^{\textregistered }\), and used for simulation and optimization studies. The objective of the design optimization is to estimate the optimum displacement volume of the internal combustion engine that minimizes the operational costs while satisfying the electrical and heating demand of a residential 10house district. The simulation results show that the mathematical model can accurately predict the behavior of the actual system while the design optimization will later be the basis for advanced control studies.
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Notes
 1.
Assuming 4 occupants per residence and a daily need of water equal to 80 gallons per day per capita.
 2.
Refer to the Appendix for more information.
Abbreviations
 \(A\) :

Area \((\mathrm{m}^{2})\)
 \(B\) :

Cylinder bore (m)
 \(c_{d}\) :

Valve discharge coefficient
 \(c_{p}\) :

Mass specific heat capacity (J/kg K)
 \(cpf\) :

Pressure–flow coefficient
 \(CR\) :

Engine compression ratio
 \(D\) :

Diameter (m)
 \(E\) :

Internal energy (J)
 \(FI\) :

Flywheel inertia
 \(H\) :

Height (m)
 \(h\) :

Mass specific enthalpy (J/kg)
 \(H_{l}\) :

Lower heating value (J/kg)
 \(\Delta H_{c}\) :

Enthalpy change of combustion (J/kg)
 \(L\) :

Length (m)
 \(l\) :

Latency
 \(m\) :

Mass (kg)
 \(NoC\) :

Number of engine cylinders
 \(P\) :

Pressure (Pa)
 \(Pec\) :

Electric power (Watt)
 \(Q\) :

Heat (J)
 \(R_{\beta }\) :

Ideal gas constant (J/kg K)
 \(S\) :

Stroke (m)
 \(SR\) :

Stroke to bore ratio
 \(T\) :

Temperature (K)
 \(TC\) :

Heat transfer rate coefficient
 \(To\) :

Torque (N m)
 \(u\) :

Control signal
 \(V\) :

Volume \((\mathrm{m}^{3})\)
 \(W\) :

Work (J)
 \(Wi\) :

Width (m)
 \(WS\) :

Wetting surface (%)
 \(x\) :

Mass fraction (kg/kg)
 \(\beta ,\gamma , \nu \) :

Engine efficiency coefficients
 \(\eta \) :

Efficiency
 \(\lambda \) :

Excessive air to fuel ratio (kg/kg)
 \(\rho \) :

Mass density \((\mathrm{kg}/\mathrm{m}^{3})\)
 \(\sigma _{0}\) :

Stoichiometric air to fuel ratio (kg/kg)
 \(\phi \) :

Angle (rad)
 \(\omega \) :

Angular velocity (rad/s)
 0:

Initial setting
 \(\dot{a}\) :

Rate of size \(a\) [(units of \(a\))/s]
 \(ab\) :

Ambient environment
 \(air\) :

Atmospheric air
 \(by\) :

Bypass
 \(c\) :

Compression
 \(cont\) :

Continuous value
 \(cg\) :

Cylinder gasket
 \(cl\) :

Flywheel
 \(co\) :

Coolant
 \(cyl\) :

Cylinder or cylindrical
 \(cw\) :

Cylinder walls
 \(d\) :

Displacement
 \(eb\) :

Engine block
 \(ec\) :

Electric
 \(en\) :

Engine block
 \(ex\) :

Exhaust gases
 \(f\) :

Friction
 \(meb\) :

Mean effective break
 \(mef\) :

Corresponding to engine friction losses
 \(mepg\) :

Corresponding to gas pump losses
 \(me\varphi \) :

Corresponding to fuel combustion losses
 \(mn\) :

Intake manifold
 \(OTC\) :

External circuit
 \(pr\) :

External circuit interaction
 \(rec\) :

Rectangular
 \(ss\) :

Steady state
 \(td\) :

Thermodynamic
 \(th\) :

Throttle valve
 \(vl\) :

Volumetric
 \(water\) :

Utility water
 \(\varphi \) :

Fuel
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Appendix: Optimization in gPROMS with gOPT
Appendix: Optimization in gPROMS with gOPT
In general, the CVP approach assumes that the timevarying control variables are piecewiseconstant (or piecewiselinear) functions of time over a specified number of control intervals. The precise values of the controls over each interval, as well as the duration of the latter, are generally determined by the optimization algorithm. As the number of control variables is usually a small fraction of the total number of variables in the problem, the optimization algorithm has to deal only with a relatively small number of decisions, which makes the CVP approach applicable to large problems.
Specifically for this design optimization, two control variables need to be determined over 864 intervals. The first control variable is the engine size. The displacement volume of the engine is a piecewiseconstant design variable that remains steady for the entire optimization horizon. The second control variable is the amount of power that needs to be obtained from the grid to cover any power production deficit. The power that needs to be obtained is treated as a piecewiseconstant function of time. The objective of the optimization is the minimization of the daily cost of the CHP operation while the demand in both electrical and thermal output is covered.
The solver DASOLV (Jarvis and Pantelides 1992) was implemented for integrations of the model equations at each iteration of the optimization. DASOLV is based on backward differentiation formulae methods for integrating the DAEs.
The SRQPD nonlinear programming code (Chen and Macchietto 1989) implements a reduced sequential quadratic programming algorithm. The NLP algorithm requires the gradients of the constraints and the objective function with respect to the optimization decision variables. The solution of the optimization problem is a twostage iterative procedure:

Integration of the model differential–algebraic equations to determine the objective function and constraints, given initial estimates for the control variables and

Implementation of SRQPD to determine new estimates for the control variables.
The optimization ends when the objective function has converged to the minimum. The optimization tolerance was set to 0.0010. More specifically, the criterion for convergence was set as follows:
where
 \(\varepsilon _{0}\) :

optimization tolerance
 \(f\) :

objective function
 \(c\) :

constraint vector
 meq :

number of equality constraints
 \(n\) :

size of the variable vector \(x\)
 \(x_{j}^{L}\) and \(x_{j}^{U}\) :

lower and upper bound of the variable \(x\), respectively
 \(d\) :

vector of corrections to \(x\)
 \(\lambda _{i}\) :

Lagrange multipliers corresponding to the equality constraints
 \(\mu _{i}\) :

Lagrange multipliers corresponding to the inequality constraints.
where \(q_{j}\) is the jth optimisation variable and \(\widetilde{q}_j \) the corresponding scaled variable. \(c_{j}\) and \(d_{j}\) vary according to the scaling procedure. The latter can be based on the range of the optimisation variables, the values of their initial guesses or the value of the gradient of the objective function at the initial guess.
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Diangelakis, N.A., Panos, C. & Pistikopoulos, E.N. Design optimization of an internal combustion engine powered CHP system for residential scale application. Comput Manag Sci 11, 237–266 (2014). https://doi.org/10.1007/s102870140212z
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Keywords
 Combined heat power
 Mathematical modeling
 Design optimization