Optimal Design of Flexible Power Cycles through

Optimal Design of Flexible Power Cycles through Kriging-based Surrogate Models

Luca Riboldi a,*, Lars O. Nord a a

Department of Energy and Process Engineering, Norwegian University of Science and Technology - NTNU, Trondheim, Norway

Abstract The paper presents a novel technique to define the optimal design of a power cycle considering design and off-design performance. A Kriging-based surrogate model is developed in order to simulate the power cycle at design conditions, while decreasing the computational effort. For each design considered, the performance at relevant off-design points is additionally evaluated by means of specific off-design models, developed for the main components of the system. The combination of design and off-design models allows the optimization process to take into account the performance at a selected set of operating conditions. The resulting optimal design will, thus, be characterized by a high degree of flexibility, intended as the ability to work efficiently in the several modes of operations to which the plant will be subjected to. The presented technique was tested on a case study. The optimal design of an offshore combined cycle was evaluated by using a multi-objective approach, where the two objective functions to minimize were the cumulative CO2 emissions and the weight of the bottoming cycle. The resulting designs showed to outperform those defined by a standard optimization procedure, demonstrating the effectiveness of the novel design technique.

Nomenclature GT load ṁsteam ṁcw pcond pcond,out psteam PWnet Tcond,in Tsteam UAECO1 UAECO2 UAOTB UASH WOTSG WST WGEN WCOND ΔpECO1

Gas turbine load Steam mass flow rate, kg/s Mass flow rate of cooling water, kg/s Condenser pressure, bar Pressure at the condenser outlet, bar Steam evaporation pressure, bar Net cycle power output, MW Temperature at the condenser inlet, °C Superheated steam temperature, °C UA coefficient of the 1st economizer, kW/K UA coefficient of the 2nd economizer, kW/K UA coefficient of the evaporator, kW/K UA coefficient of the superheater, kW/K Weight of the OTSG, ton Weight of the steam turbine, ton Weight of the generator, ton Weight of the condenser (wet), ton Pressure drop in the 1st economizer, bar

ΔpECO2 ΔpOTB ΔpSH ΔTcw ΔTOTSG ηcycle ηT

Pressure drop in the 2nd economizer, bar Pressure drop in the evaporator, bar Pressure drop in the superheater, bar Cooling water temperature difference, °C Pinch point difference in the OTSG, °C Net cycle efficiency Isentropic steam turbine efficiency

Introduction The optimization of power cycles’ design is becoming more challenging as a number of additional requirements are emerging. In particular, the power cycles are more and more requested to perform efficiently at varying operating conditions and not only at design condition [1]. This applies to different fields characterized by variable operation profiles, like shipping [2] and offshore oil and gas extraction [3]. A demand for efficient flexible operation is also growing in the power generation sector, due to the increasing penetration of intermittent renewable energy sources [4]. Additionally, multi-objective approaches are becoming common where different and, possibly, conflicting requirements have to be considered in the definition of the optimal design [5]. Within such context, innovative optimization techniques are needed in order to keep up with the increasing complexity of design optimization [2]. Whilst providing a reliable representation of power cycles, high-fidelity models are too computationally intensive to use directly in advanced optimization procedures. Surrogate models are able to reduce the computational effort to simulate complex systems while ensuring a sufficient degree of accuracy. Different approaches have been proposed and showed to be viable options to define a surrogate model: input-output analysis [6]; data interpolation; statistical regression [7]; and artificial neural networks [8]. This paper presents a novel technique to define the optimal design of a power cycle considering design and off-design performance. A Kriging-based surrogate model is used to simulate the behavior of the power cycle at design. Kriging (Gaussian process interpolation) is a surrogate modeling procedure to approximate deterministic data [9]. Kriging was selected after a screening of potential surrogate modeling techniques as it has been successfully used in similar research areas. A regression function first fits the sampling data, and, subsequently, a Gaussian process is constructed through the residuals. The regression function captures the largest variance in the data (the general trend), while the Gaussian process takes care of small details and the interpolation of the data. For each design evaluated through the Kriging model, the cycle performance is calculated at several off-design conditions. Off-design models are developed to simulate the off-design operations of the main components of the power cycle. These models are based on correlations for the off-design performance. Such correlations have been successfully applied in other studies for off-design performance prediction [10]. The combination of the design and off-design models allows taking into account all the expected modes of operations the plant will be subjected to, rather than only specific conditions (i.e. design conditions). The optimal design can be identified as that returning the overall best performance, according to the objective functions selected. The methodology outlined was applied to a case study in order to be validated and to demonstrate its effectiveness. The optimal design of a combined cycle for power generation in an offshore oil and gas installation was investigated, taking into account the varying power requirements characterizing the different periods of the field’s lifetime. A multi-objective approach was considered, where the objective functions were selected to be the cumulative CO2 emissions and the total weight of the bottoming cycle. A genetic algorithm was used to solve the optimization problem by identifying the Pareto front of optimal solutions. A set of optimal designs was defined, accounting for design and off-design performance.

Kriging-based surrogate model The Kriging method is used to construct the surrogate model of the power cycle under investigation. If developed successfully, a Kriging-based surrogate model allows to analyze a much larger number of designs than a computationally expensive high-fidelity model, with obvious potentials in terms of design optimization. Kriging can be defined as a locally weighted regression method. The Kriging approximation of an unknown function is composed by two parts: a regression function f(x) and a Gaussian process Z(x). y  x  f  x  Z  x

(1)

The regression function f(x) can be a known constant (simple Kriging), an unknown constant (ordinary Kriging) or a polynomial (universal Kriging). The regression function is determined by generalized least squares (GLS). Z(x) is a stochastic Gaussian process with mean zero that represents the fluctuations around the trend and ensures the interpolation of the training data. The covariance of Z(x) is given by a process variance and a correlation, which depends on the distance between the points under consideration. More detailed information on Kriging can be found in [11]. The following steps are followed in the development of a Kriging-based surrogate model: (i) highfidelity model development; (ii) input variables sampling; (iii) surrogate model fitting; (iv) surrogate model verification. High-fidelity model development The high-fidelity model is developed in Thermoflex (Thermoflow Inc.) [12], a fully-flexible program for design and off-design simulation of thermal systems. Thermoflex iteratively solves the mass and energy balances at the nodes of a network of pre-defined or user-defined components. It is widely recognized as a reliable tool for the simulation of thermal power plants. Input variables sampling The sampling data set defines the range of points used to train the surrogate model. Once established the input variables influencing the behavior of the system, the choice of a good set of sampling data is of paramount importance in the development of a surrogate model. The first step consists of defining the space within which the model needs to be able to provide reliable predictions. That is determined by assigning bounds to the input variables. The choice of the bounds is a difficult task which requires expertise on the system to model. A combination of deterministic and near-randomized sampling techniques is used for the definition of the sample points. The Box-Behnken and central composite designs specify points on edges, corners, and faces of the sample bounds. The Latin hypercube sampling (LHS) ensures that all portions of the defined space are explored. While the number of deterministic points is a function of the number of input variables selected, the number of near-randomized points has to be specified for the LHS. A large number of points ensures an effective training of the surrogate model, though it is time-consuming to carry out many simulations with the high-fidelity model. A compromise needs to be reached. Surrogate model fitting Rigorous simulations are carried out in accordance with the points defined in the sampling data set. The related simulation outputs are used to fit the Kriging models. A model is built for each output variable selected. A MATLAB Kriging toolbox, called ooDACE toolbox, is used to implement the Kriging

method [13]. Ordinary Kriging is adopted, which assumes a constant regression function, as it has been reported that a constant term is in many cases sufficient [14]. When a simulation does not converge, the related points are discarded. Surrogate model verification The accuracy of the surrogate model needs to be verified before its utilization in the optimization procedure. A test set of points is defined (different than the sampling set of points) and a correspondent number of rigorous simulations are carried out. The outputs of the surrogate model are then checked against the outputs of the high-fidelity model. If the error in the prediction of the selected output variables is sufficiently small, the surrogate model is ready to be utilized for further analyses.

Off-design models The performance of heat exchangers, turbines and pumps is affected by changes in operating conditions. Simple off-design models are defined to simulate the off-design operations. The outputs of the surrogate model are used as starting point to evaluate off-design performance of the single component and of the overall power cycle. Input variables Data-defined models are used to simulate the part-load performance of gas turbines (GTs). The models are developed from data provided by the manufacturers covering the entire operating range of the engines (10–100%). The models encapsulate manufacturer specifications and correction curves into functions dependent on site conditions, inlet and exhaust losses, fuel type and GT load. For a given set of operating conditions, the models return power output, heat rate, temperature and mass flow of the exhaust gases. Turbines The turbines performance at off-design is modelled according to the Stodola’s cone law, which defines a relationship between the mass flow rate (ṁ), the turbine inlet temperature (Tin) and pressure (pin) and the turbine outlet pressure (pout): CS 

m Tin 2 pin2  pout

(2)

where CS is the constant flow coefficient. The isentropic efficiency of the turbine at off-design (ηT) is evaluated through the relation proposed by Schobeiri [15]: hT ,is , d hT ,is , d T 2  T , d hT ,is hT ,is

(3)

where the subscript d indicates the value at design conditions and ΔhT,is is the isentropic enthalpy difference due to the expansion in the turbine. The influence of the rotational speed on the turbine isentropic efficiency is neglected Heat exchangers

The off-design performance of the heat exchangers is based on the relations proposed by Incropera et al. [16]. In particular, the product of the overall heat transfer coefficient (U) and the heat transfer area (A) is assumed to vary according to the following equation:  m  UA  UAd    md 



(4)

where ṁ is a mass flow rate and γ is an exponent depending on the relationship between mass flow rate and heat transfer coefficient [17]. The mass flow rate to consider and the value of the exponent γ are defined according to the side dominating the heat transfer processes. In the case of shell-and-tube heat exchangers, it has to be evaluated whether the heat transfer coefficient inside or outside the tube is the dominant term. For boilers it is common to assume that the gas side heat transfer coefficient is dominant. The gas mass flow rate is then considered and a value of 0.6 is assigned to the exponent γ [18]. However, some works [10] suggested that for the superheating section of a boiler the water side dominates the heat transfer process. Therefore, the water mass flow rate is to be used and a value of 0.8 is assigned to the exponent γ. The pressure drop (Δp) in the heat exchangers is estimated assuming a quadratic dependence with the mass flow rate [19]:  m  p   p d    md 

2

(5)

Condensers A model of the condenser at off-design is not developed, rather the condenser is considered to operate at constant pressure throughout all the operating conditions simulated. The simplification assumes large availability of water as cooling medium, allowing for a control strategy retaining an approximately fixed value of the condensing pressure. Pumps Different correlations can be found in the literature to estimate the variation of pumps isentropic efficiencies at off-design conditions [18,19]. The equation selected was taken from [20] and calculates the isentropic efficiency of the pump (ηpump) as a function of the pump volumetric flow rate:  pump V  V   0.029265    0.14086    pump , d V  d  Vd  3

3

2

V  0.3096    0.86387  Vd 

(6)

The influence of the rotational speed on the pump isentropic efficiency is neglected. The equation reported is used for all the pumps in the cycle (working fluid pumps and cooling water pump of the condenser section). Electric generators The off-design generator efficiency can be estimated through the following equation [21]:

 gen 

load  gen , d

load  gen , d  1   gen , d  1  FCU   FCU load 2 

(7)

where load is the mechanical load, ηgen is the generator efficiency and FCU is a term representing the copper losses (produced in the winding of the stator). The term FCU was set equal to 0.43, in accordance with the suggestion from [21].

Optimization procedure for the design of flexible power cycles The goal of the optimization procedure is to find the optimal design(s) of a power cycle by considering the performance at a range of selected operating conditions. The following iterative steps are implemented: (i) a design is defined by assigning a set of values to the decision variables; (ii) the given design is characterized by means of the Kriging-based model; (iii) the performances at selected operating conditions are evaluated by means of the off-design models; (iv) the objective functions are calculated. The optimization algorithm used to research the optimal solutions is a genetic algorithm (GA), available within the MATLAB Global Optimization Toolbox [22]. The GA is chosen because of the characteristic of the optimization problem. The Kriging models are not directly accessible from the optimization solver, thus they become black box models from the solver perspective. The GA does not need derivative information in its implementation and is a common choice for black-box type optimization problems. Further, GA algorithm in MATLAB is able to handle multi-objective optimization problems in a multiobjective approach.

Application to a case study In the following section, the presented design technique is applied to a case study. Each step is described and the results of the design optimization are reported and discussed. Case study definition The selected case study refers to a combined cycle for power generation in an offshore oil and gas installation. The same offshore installation was assessed in a previous publication [23], where, among other concepts, a combined cycle was studied. The model previously developed constituted the starting point for this work. Some simplifications were introduced in the cycle in order to ease the first implementation of the methodology presented. In particular, the waste heat recovery unit was not considered and the cycle was assumed to supply only power to the plant. The annual variability of the power demand throughout the lifetime of the installation was considered, resulting in a set of off-design operating conditions. Other deviations from average conditions (e.g. different ambient conditions, changing fuel composition, fouling and ageing effects) were not considered in this case study. The operation between 2016 and 2034 was accounted for by average annual power requirements, as shown in Figure 1.

Figure 1. Power demand profile throughout the lifetime of the installation.

Surrogate model of the combined cycle The input parameters considered to have a significant impact on the design of the power cycle are listed in Table 1, together with their bounds. A sampling data set of 913 points was defined (113 deterministic and 800 near-randomized points) and used to train the Kriging models of the selected output parameters. The list of the output parameters is provided in Table 2. Those were the parameters selected to characterize a design. Table 1. Input parameters to the Kriging model and related bounds. The same parameters and bounds were also used as decision variables in the optimization problem. Input parameters GT load

Lower bound Upper bound 0.70

0.95

psteam (bar)

15

40

Tsteam (°C)

400

515

ΔTOTSG (°C)

10

30

pcond (bar)

0.03

0.12

ΔTcw (°C)

3

10

Table 2. Output parameters from the Kriging models. The mean average error (MAE) of the validation process is also indicated. Output parameters MAE

MAE

ηcycle

0.05%

ΔpOTB

0.60%

PWnet

0.08%

ΔpSH

0.00%

ṁsteam

0.14%

Tcond,in

0.00%

ηT

0.47%

pcond,out

0.00%

UAECO1

0.72%

ṁcw

0.42%

UAECO2

0.32%

WOTSG

0.63%

UAOTB

0.17%

WST

0.24%

UASH

1.20%

WGEN

0.14%

ΔpECO1

0.00%

WCOND

1.82%

ΔpECO2

0.04%

In order to verify the accuracy of the Kriging models, a test data set of 30 points was used. The test data set was defined by selecting random values of the input parameters within the established bounds. The output values obtained by the surrogate models were compared to those obtained by the high-fidelity model. Table 2 reports the computed mean average errors (MAE). Figures 2, 3, 4 and 5 show the parity plots of selected output parameters.

Figure 2. Parity plot of net cycle efficiency. On the horizontal axis the high-fidelity model outputs, on the vertical axis the surrogate model outputs.

Figure 3. Parity plot of total weight of the bottoming cycle. On the horizontal axis the high-fidelity model outputs, on the vertical axis the surrogate model outputs.

Figure 4. Parity plot of steam flow rate. On the horizontal axis the high-fidelity model outputs, on the vertical axis the surrogate model outputs.

Figure 5. Parity plot of UA coefficient of the evaporator. On the horizontal axis the high-fidelity model outputs, on the vertical axis the surrogate model outputs.

The mean average error (MAE) obtained was lower than 1% for almost all the output parameters considered (except UASH and WCOND that returned a MEA of 1.20% and 1.82%, respectively). Thus, the surrogate model was deemed as ready to be used for the following analyses. Off-design model of the power cycle Off-design models of the components constituting the power cycle were developed in accordance with the guidelines provided in the specific section of the paper. The off-design models were implemented in MATLAB, where their integration allowed the simulation of the whole power cycle at the different offdesign conditions selected. In order to simulate the off-design operation, a control strategy had to be developed. The plant load was modified primarily through changes in the gas turbine load. A slidingpressure control mode was implemented for the regulation of the pressure upstream of the steam turbine. The live-steam pressure level was kept within a maximum threshold, defined as a percentage of the design value, by means of a steam turbine control valve. The temperature of the steam at the entrance of the once-through heat recovery steam generator (OTSG) section defined as superheater was controlled by means of variations in the feedwater flow to the OTSG. The effectiveness of the off-design model of the power cycle was tested before starting the design optimization procedure. Given a randomly selected design (i.e. a set of input parameters), 30 off-design operating conditions were simulated both with the off-design model and with the high-fidelity model. The off-design points tested were chosen to represent operating conditions that could actually occur in the case study under investigation. Figures 6, 7 and 8 show the parity plots of selected output parameters. Some inaccuracies were noted with regard to the steam turbine operation at off-design and, in particular, to the turbine power output at low loads. The main reason was identified to be the increasing inaccuracy introduced by the evaluation of the UA coefficients through equation (4) when the conditions were

further and further away from design. However, the effect on the overall performance was limited, as the predicted net cycle efficiency showed to closely follow the values obtained by the rigorous simulations throughout all the range of operating conditions tested - the MAE measured was 0.09%.

Figure 6. Parity plot of net cycle efficiency. On the horizontal axis the high-fidelity model outputs, on the vertical axis the off-design model outputs.

Figure 7. Parity plot of steam pressure. On the horizontal axis the high-fidelity model outputs, on the vertical axis the offdesign model outputs.

Figure 8. Parity plot of steam turbine power. On the horizontal axis the high-fidelity model outputs, on the vertical axis the off-design model outputs.

Design optimization A constrained multi-objective optimization problem was defined. The decision variables were selected to be the same as the input variables to the surrogate model and were constrained within the same bounds (Table 1 reports those bounds). The utilization of the same constraints ensured that designs were not searched in spaces where the surrogate model was not able to provide reliable results. Two objective functions were set to be minimized: the cumulative CO2 emissions and the total weight of the bottoming

cycle. Decreasing the CO2 emissions throughout the lifetime of the plant implies advantages in terms of environmental impact, energy efficiency and fuel gas utilization. Limiting the weight of the power cycle is a primary requirement for offshore applications where the weight constraints are often particularly strict. The novel optimization technique entailed an overall evaluation of the objective functions by including the performance at several off-design conditions (in this case the annual operating conditions of the installations dictated by the variable power requirements shown in Figure 1). Therefore, the assessment of a single design involved 11 cycle calculations (one for each year until the power demand stabilizes in the tail-years). A population size of 500 was set for the iterations of the GA. A maximum number of generation of 50 was also set. The average change in the spread of the Pareto front was monitored during the optimization process and the algorithm was stopped when the related indicator was lower than a tolerance factor over a number of stall generations. Otherwise, the algorithm stopped when the maximum number of generations was reached. Results and discussion The utilization of a surrogate modeling technique reduced the computational time of the optimization procedure. The computer used in this work has an Intel Core processor of 2.60 GHz and 16.0 GB of RAM. The Kriging-based model was able to characterize a design in 0.25 s. The high-fidelity model would take on average 10.05 s to run a simulation and define a design (almost 41 times more). For the simulation of a single off-design point, the surrogate model took on average 1.37 s. Assuming that the commercial software used for the rigorous simulations could be automatized to switch to an off-design mode (which actually needs to be done manually), an off-design simulation would take on average 27.95 s, further increasing the computational time difference. The Pareto front of optimal designs obtained through the outlined methodology, taking into account offdesign performance, is shown in Figure 9. A second Pareto front is also reported (again Figure 9), showing the designs obtained through a standard optimization procedure. This second procedure determined the optimal designs at a specific operating point (i.e. the design point), which was selected to represent the plant operation at peak power demand (i.e. 40 MW). The cumulative CO 2 emissions related to the Pareto front of Standard Designs were much lower since a single point (i.e. the year at peak power demand) was simulated rather than a number of operating conditions like with the Novel Designs.

Figure 9. Pareto fronts obtained by the novel (i.e. Novel Designs) and standard (i.e. Standard Designs) optimization procedures. The full red dots indicate the two designs selected for the comparative analysis.

The effect of the novel optimization technique can be noted by observing the output values of the decision variables obtained by the two optimization processes (see Figures 10, 11, 12 and 13).

Figure 10. Gas turbine loads of the Pareto optimal solutions obtained by the novel (i.e. Novel Designs) and standard (i.e. Standard Designs) optimization procedure.

Figure 11. Steam pressures of the Pareto optimal solutions obtained by the novel (i.e. Novel Designs) and standard (i.e. Standard Designs) optimization procedure.

Figure 12. Steam temperatures of the Pareto optimal solutions obtained by the novel (i.e. Novel Designs) and standard (i.e. Standard Designs) optimization procedure.

Figure 13. Condenser pressures of the Pareto optimal solutions obtained by the novel (i.e. Novel Designs) and standard (i.e. Standard Designs) optimization procedure.

The inclusion of the GT load as one of the decision variables of the optimization problem allowed to implicitly consider different design strategies. The variation of the GT load set the thermal energy available in the exhaust gas at design and, consequently, influence the allocation of weights between the components. By letting the GT load to vary in the design definition, it was possible to assess designs targeting optimal performance at different working points. This was believed to be beneficial in case studies whose plant’s lifetime is characterized by a high variability of operating conditions. For instance, it was demonstrated in a previous paper how designing offshore combined power and heat plants at endlife conditions rather than at peak conditions could be advantageous [3]. Such analysis was, to a certain extent, confirmed by the results obtained (see Figure 10). When the design procedure took into account only the performance at design (i.e. Standard Designs considering operation at peak power demand of 40 MW), the Pareto optimal solutions displayed high GT loads. In other words, the cycles were optimized to perform at the only operating point considered (i.e. the design point). This was expected and can be seen as an evidence that the optimization procedure was properly implemented. Conversely, considering the performance at several off-design points in the evaluation of the optimal design (i.e. Novel Designs) led to a different outcome. In that case, when the overall weight of the cycle needed to be minimized, designs at end-life conditions (characterized by lower GT loads) were defined as the optimal ones. When, instead, the high efficiencies were sought, the optimal designs gradually became those at peak conditions (characterized by higher GT loads), thus similar to those obtained with the standard optimization procedure. It must be noted that the GT loads of the Novel Designs sit on the lower bound of the variable, at least for the designs involving limited weight of the cycle. This could suggest that the problem was under-constrained. On the other hand, it could also be due to the fact that the lower bound of the GT load was selected to represent the cycle’s operation at the lowest power demand typical of the long last stage of plant’s life. The optimal designs could have been identified as those performing efficiently at those operating conditions and, thus, defined around the lower bound of the GT load. The distribution of the other optimized decision variables for the two design methods can be checked in Figures 11, 12 and 13. Two design patterns were possible. When high efficiencies at peak conditions were targeted, the high power demand required to operate the GT at high loads making available an increased thermal energy in the exhaust gas. The optimization process returned designs that extensively exploited such large amount of energy, leading to large heat exchange surfaces in the OTSG and relatively low values of live-steam temperature and pressure. These design features allowed to maximize the steam generation. On the other hand, the multi-objective approach implied limitations in the overall weight of the bottoming cycle. Given that a significant weight was allocated for the OTSG, the other components had to be constrained, translating mainly in a higher value of the condenser pressure. The second pattern involved designs optimized at conditions characterized by lower power demands. The diminished thermal energy made available by the exhaust gas led to decreased heat exchange surfaces

in the OTSG. Then, an increased share of weight could be allocated to the steam expansion process by having a higher live-steam temperature and an increased expansion ratio. In other words, since the steam generation was limited by the reduced thermal energy available, a better efficiency was sought by increasing the power that can be extracted by the expansion of the decreased amount of steam. As expected, the first pattern characterized most of the designs defined with the standard optimization method (i.e. Standard Designs). Only those with rather low weights differed given that to obtain so light cycles it was necessary to limit the weight of the OTSG. The Novel Designs, taking into account the performance at different operating conditions (including peak and low power demand instances), had to reach a compromise. Overall, the design features typical of an optimization at low power demand resulted prevalent, due to the larger number of years of operation at diminished plant load rather than at peak load. However, when high efficiencies were sought, irrespective of the weight of the cycle, the Novel Designs started to share the same features of the Standard Designs, as the relaxed weight limitations allowed to use large heat exchange surfaces. The discussed trends can be observed in Figures 11, 12 and 13, where the Novel Designs are generally characterized by relatively higher live-steam temperatures and pressures, and by relatively lower condenser pressures compared to the Standard Designs. An example is provided, where two designs, obtained with the two different procedures and characterized by the same weight of the bottoming cycle, are compared (see full dots in Figure 9). Table 3 showed the values of the decision variables for the two designs. It can be noted that the optimal steam temperature and condenser pressure were respectively higher and lower when the novel optimization technique was applied. The next step was to demonstrate that the Novel Design was actually better. For this purpose, the performance of the two designs was calculated throughout the entire plant’s life. Figure 14 shows the results obtained. In years characterized by peak power demand (years 2019 and 2020), the Standard Design returned a better performance. This was expected as such design was optimized for those conditions. Conversely, the Novel Design had under-dimensioned heat exchange surfaces at peak conditions, while the live-steam pressure level could not be increased over a maximum threshold. It was, thus, made necessary for a fraction of the steam to bypass the expansion in the turbine, resulting in a reduced efficiency. However, the Novel Design outperformed the Standard Design in the remaining years, where the plant power demand is lower. In those conditions, the large heat exchange surfaces of the Standard Design became redundant, while the effect of the more effective expansion path in the Novel Design prevailed. In terms of overall performance, the Novel Design succeeded to cut the cumulative CO2 emissions of 17.4 kton (0.8% reduction compared to the Standard Design), entailing benefits in terms of both environmental impact and economic performance. Figure 15 shows the trend of the CO2 emissions avoided throughout the plant’s lifetime thanks to the Novel Design. The cumulative amount of CO2 emissions avoided increased in the years characterized by a better energy performance, thus in every year apart those of peak oil production (2019 and 2020). Table 3. Breakdown of the two designs selected for the comparative analysis. Input parameters

Novel Design

Standard Design

72.2 %

94.0 %

psteam (bar)

21.3

18.6

Tsteam (°C)

492.2

463.0

ΔTOTSG (°C)

26.9

25.2

pcond (bar)

0.06

0.09

GT load

ΔTcw (°C)

9.5

9.4

Figure 14. Performance of the selected designs throughout the lifetime of the installation. The power demand profile is also showed.

Figure 15. Cumulative CO2 emissions avoided by the Novel Design.

Summing up, the methodology developed was successfully applied and demonstrated to be advantageous in terms of overall performance. One can argue that the benefits achieved through the utilization of the presented technique were rather small. However, this was a very first implementation. As far as the authors are concerned, a refinement of the methodology is expected to lead to improved results. Moreover, additional applications other than design optimization are envisaged to be of interest, e.g. the evaluation of control strategies for the efficient operation of flexible power cycles. The offdesign simulations carried out in this paper included the control of the temperature at the inlet of the superheater. Such simplified choice was made in order to ease the first implementation of the new methodology by ensuring that the evaporation process entirely took place within the proper section of the OTSG. On the other hand, the live-steam temperature at the outlet of the OTSG was not controlled. Its variation at the different operating conditions is shown in Figure 16, where also the design value is reported. The Standard Design appeared to be more critical, as the obtained temperatures consistently overpassed the design value in the last years of plant’s operation. A steam attemperation would probably need to be implemented, reducing the cycle efficiency and further increasing the performance gap with the Novel Design, where such control measure would probably not be necessary.

Figure 16. Superheated steam temperature of the selected designs throughout the lifetime of the installation. The dashed lines shows the design value at which the cycle was optimized.

Conclusions The paper presents a technique for the design of flexible power cycles through Kriging-based surrogate models. The application of Kriging to construct a surrogate model of the power cycle allows the development of complex optimization procedures that would be hardly feasible with high-fidelity models, due to their high computational effort (the Kriging model decreased the computational time for the characterization of a design of almost 41 times). In particular, the developed methodology enables to consider the performance of the cycle at a set of selected off-design conditions in the definition of the optimal design. The performance of each design at the different operating conditions is evaluated through off-design models of the various components of the system. The application of the technique on the design of an offshore combined cycle sparked some interesting considerations. For systems characterized by high variability of operating conditions, the choice of the working point at which to target maximum efficiency is not straightforward. The analysis demonstrated that, for the case study investigated, in many instances it was convenient to design the plant at end-life conditions rather than at peak conditions. The resulting optimal values of cycle’s parameters were determined as those guaranteeing good performance over the considered range of off-design points and respecting the system constraints. One of the novel designs was tested over the entire plant’s lifetime. It showed a better performance, reducing the cumulative CO2 emissions by 17.4 kton, thus validating the optimization procedure developed. Overall, the importance of considering off-design operation in the design of power cycles was demonstrated. The presented technique showed to be a powerful tool with regard to this. The optimization procedure developed allows to identify effective designs for applications characterized by irregular operating conditions.

References [1] [2]

[3] [4]

Montañés RM, Korpås M, Nord LO, Jaehnert S. Identifying operational requirements for flexible CCS power plant in future energy systems. Energy Procedia, vol. 86, 2016, p. 22–31. Baldi F, Larsen U, Gabrielii C. Comparison of different procedures for the optimisation of a combined Diesel engine and organic Rankine cycle system based on ship operational profile. Ocean Eng 2015;110:85–93. Riboldi L, Nord LO. Lifetime Assessment of Combined Cycles for Cogeneration of Power and Heat in Offshore Oil and Gas Installations. Energies 2017;10:744. Domenichini R, Mancuso L, Ferrari N, Davison J. Operating Flexibility of Power Plants with

[5]

[6]

[7]

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19]

[20] [21] [22] [23]

Carbon Capture and Storage (CCS). Energy Procedia 2013;37:2727–37. Kalikatzarakis M, Frangopoulos CA. Multi-criteria selection and thermo-economic optimization of Organic Rankine Cycle system for a marine application. Int J Thermodyn 2015;18:133. Keshavarzian S, Gardumi F, Rocco M V., Colombo E. Off-design modeling of Natural Gas Combined Cycle power plants: An order reduction by means of Thermoeconomic Input-Output Analysis. Entropy 2016;18. Kang CA, Brandt AR, Durlofsky LJ. A new carbon capture proxy model for optimizing the design and time-varying operation of a coal-natural gas power station. Int J Greenh Gas Control 2016;48:234–52. Kesgin U, Heperkan H. Simulation of thermodynamic systems using soft computing techniques. Int J Energy Res 2005;29:581–611. Couckuyt I, Forrester A, Gorissen D, De Turck F, Dhaene T. Blind Kriging: Implementation and performance analysis. Adv Eng Softw 2012;49:1–13. Pierobon L, Benato A, Scolari E, Haglind F, Stoppato A. Waste heat recovery technologies for offshore platforms. Appl Energy 2014;136:228–41. Sacks J, Welch WJ, Mitchell TJ, Wynn HP. Design and Analysis of Computer Experiments. Stat Sci 1989;4:409–23. Thermoflow Inc. Thermoflex Version 26.0 2016. Couckuyt I, Dhaene T, Demeester P. ooDACE Toolbox: A Flexible Object-Oriented Kriging Implementation. J Mach Learn Res 2014;15:3183–6. Quirante N, Javaloyes J, Caballero J. Rigorous Design of Distillation Columns Using Surrogate Models Based on Kriging Interpolation. AIChE J 2015;61:2169–87. Schobeiri M. Turbomachinery flow physics and dynamic performance. Berlin: Springer Berlin; 2005. Incropera FP, DeWitt DP, Bergman TL, Lavine AS. Fundamentals of Heat and Mass Transfer. vol. 6th. John Wiley & Sons; 2007. Andreasen J, Meroni A, Haglind F. A Comparison of Organic and Steam Rankine Cycle Power Systems for Waste Heat Recovery on Large Ships. Energies 2017;10:547. Larsen U, Pierobon L, Baldi F, Haglind F, Ivarsson A. Development of a model for the prediction of the fuel consumption and nitrogen oxides emission trade-off for large ships. Energy 2015;80:545–55. Lecompte S, Huisseune H, van den Broek M, De Schampheleire S, De Paepe M. Part load based thermo-economic optimization of the Organic Rankine Cycle (ORC) applied to a combined heat and power (CHP) system. Appl Energy 2013;111:871–81. Veres JP. Centrifugal and axial pump design and off-design performance prediction. NASA Techincal Memo 106745 1994:1–24. Haglind F, Elmegaard B. Methodologies for predicting the part-load performance of aeroderivative gas turbines. Energy 2009;34:1484–92. MathWorks. Global optimization toolbox. 2016. Riboldi L, Nord LO. Concepts for lifetime efficient supply of power and heat to offshore installations in the North Sea. Energy Convers Manag 2017;148:860–75.