Linear Programming (LP) Model for Cogeneration and Trigeneration – raising energy utilization efficiency

September 3rd, 2011 Posted in Trigeneration and Cogeneration

Linear Programming (LP) Model for Cogeneration and Trigeneration – raising energy utilization efficiency

With the world running out of cheap fossil fuels (crude oil, coal, natural gas), and with continued rise of their prices in the international market, the world is exhausting all possible means of minimizing the impact of diminished supply and expensive cost of fossil fuels.

Among the response of the world’s leading economies as well as emerging economies is to raise energy utilization efficiency in power generation using cogeneration and trigeneration. When the simultaneous provision of electricity and process heat as well as cooling or refrigeration is desirable, the use of cogeneration and trigeneration will raise efficiency from the usual 33% of rankine thermal power systems (oil, gas, coal, biomass thermal) to 56% for cogeneration power systems (oil, gas, biomass) up to 80-90% with trigeneration.

Linear Programming (LP) and setting up LP models to optimize a number of processes and business models is an expertise close to the heart of the energy technology expert. As the LP model custodian of the Petron Bataan Refinery, he was involved in the re-structuring of the refinery’s cost minimization LP model to a profit maximization LP model that ensured that expensive intermediate process streams are blended to the higher valued products which a cost minimization model could not distinguish from.

The same expertise has been applied into practical problems such as transportation planning (tanker fleet assignment for delivering refinery and imported petroleum products to the various bulk plants and distribution networks) and dispatching of process units or power plants.

Recently, he went into energy, power and fuel technology selection and business development. Thus, he prepared an LP model for the optimal load dispatch of various power plants to minimize incremental costs (fuel and operating costs) for the Luzon, Visayas andMindanao grids.

As part of his continuing research, he prepared an easy-to-use LP model for the optimal dispatch of power, process heat and refrigeration & cooling.

Given the selling price of power ($/kWh), process heat ($/kWh or $/MMBtu or $/GJ) and refrigeration & colling ($/kWh or $/MMBtu or $/GJ or $/RTh), the cost of purchased power ($/kWh) and cost of cogeneration fuel ($/kWh or $/MMBtu or $/GJ) and auxiliary boiler fuel ($/kWh or $/MMBtu or $/GJ), the energy efficiency of the cogeneration unit, the efficiency of the auxiliary boiler unit, the coefficient of performance of the electric centrifugal or vapor compression chiller and the coefficient of performance of the absorption chiller, as well as the various material balances at mixing points, the LP model will calculate the operating modes of the various process units in the system.

The LP model in mathematical terms is shown below:


Table 5 – Linear Programming (LP) Model  
  Objective function: minimize variable cost
1) CH = pfc·Fc + pfa·Fa + pep·Ep – pes·Es + rql·Ql
  Subject to the following constraints:  
  Capacity limits  
2) CM: Wc ≤ Wc nom  
3) AB: Qa ≤ Qa nom  
4) AC: Rq ≤ Rq nom  
5) EC: Re ≤ Re nom  
  Production restrictions  
6) CMw: αw·Fc – Wc = 0  
7) CMq: αq·Fc – Qc = 0  
8) AB: ηq·Fa – Qa = 0  
9) AC: COPq·Qr – Rq = 0  
10) EC: COPe·Er – Re = 0  
  Balance equations  
11) S: Wc – Wcc – Es = 0  
12) P: Wcc + Ep – Ed – Er = 0  
13) L: Qc – Qcc – Ql = 0  
14) Q: Qcc + Qa – Qd – Qr = 0  
15) R: Rq + Re – Rd = 0  
  Demand constraints  
16) ED: Ed = 400 kW  
17) QD: Qd = 400 kW  
18) RD: Rd = 400 kW  


The input data of the model are shown in the following tables:

Table 1 – Technical Parameters        
Unit Description Efficiency Coefficient   Nominal Capacity  
CM Cogeneration Module αw ≡ Wc/Fc = 0.35 0.350 Wc nom = 350  
    αq ≡ Qc/Fc = 0.40 0.400    
AB Auxiliary Boiler ηq ≡ Qa/Fa = 0.80 0.800 Qa nom = 400  
AC Absorption Chiller COPq ≡ Rq/Qr = 0.625 0.625 Rq nom = 250  
EC Vapor Compression (Electric) Chiller COPe ≡ Re/Er = 5.0 5.000 Re nom = 250  


Table 2 – Energy Prices        
  Description Euro/kWh Euro/MMBtu US$/kWh $/MMBtu
pep purchased power electricity 0.100   0.1440  
pes export sales electricity 0.080   0.1152  
pfc purchased fuel cogeneration module 0.025 7.327 0.0360 10.551
pfa purchased fuel auxiliary boiler 0.020 5.861 0.0288 8.440
rql cost of waste heat lost 0.000 0.000 0.0000 0.000
  1 kWh = 3600 kJ 3,600      
  1 Btu = 1.05506 kJ 1.05506 3412.12822 Btu/kWh  
  1 MMBtu = 10^6 Btu 1,000,000 12000 Btu/RTh  
  1 Euro = 1.44 US$ (Aug 31, 2011) 1.44      


Table 3 – Demand for Power, Heat and Refrigeration
  Description Demand (kW)
Ed Electricity demand 400.000
Qd Heat demand 400.000
Rd Refrigeration demand 400.000

The LP model is then solved using the “SOLVER” function in Excel or other third party LP optimizers such as “WHAT’S BEST”, etc.

The list of symbols of the LP model and the results of the LP model and the marginal prices or the cost of producing the next 1 unit of demand are shown below:


Table 4 – Nomenclature Current Values Marginal Price
    kWh/h Euro/kWh
c Internal cost [€/kWh]    
CH Cost [€/h]


Ed Demanded electricity [kW]



Ep Purchased electricity [kW]


Er Work input to electric chiller [kW]


Es Sold electricity [kW]


Fa Auxiliary Boiler fuel [kW]


Fc Cogeneration module fuel [kW]


pfa Price of boiler fuel [€/kWh]


pfc Price of engine fuel [€/kWh]


pep Price of purchased electricity [€/kWh]


pes Price of sold electricity [€/kWh]


rql Cost of waste heat lost [€/kWh]


Qa Heat from auxiliary boiler [kW]


Qc Cogenerated heat [kW]


Qcc Consumed cogenerated heat [kW]


Qd Demanded heat [kW]



Ql Waste heat lost [kW]


Qr Heat to absorption chiller [kW]


Rd Cooling demand [kW]



Re Cooling from electric chiller [kW]


Rq Cooling from absorption chiller [kW]


Wc Cogenerated work [kW]


Wcc Consumed cogenerated work [kW]




If you are interested in the model, please email:


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