Optimal Energy Management For A Microgrid Pierre Martinon CAS - - PowerPoint PPT Presentation

Optimal Energy Management For A Microgrid

Pierre Martinon CAS seminar - January 13, 2015

Outline

• Context: microgrid
• Optimal control problem
• Numerical simulations
• Ongoing work

Part of a France-Chile collaboration Inria Saclay / CMAP Polytechnique and Univ. Limoges (F. Bonnans, P.Martinon, B. Heymann, F. Silva) CMM and Centro di Energia, Santiago (A. Joffre, G. Jimenez, F. Lanas)

Huatacondo village, Chile (Tarapac´ a) Population: 53 Prototype of electric microgrid for renewable energy management (project ESUSCON)

Overview of the microgrid

Power sources

• diesel generator*
• photovoltaic panels
• battery storage

(- wind turbines) Power consumption

• domestic power load
• battery storage

(- water pump) → Satisfy power demand with minimal operating cost Operating cost: diesel consumption [ + battery cost ] (* diesel generator sufficient to fulfill the demand)

Optimal control formulation (1/3)

State variables:

• Battery State of Charge: SOC ∈ [SOC min, SOC max]

Control variables:

• Diesel power: PDiesel ∈ {0} ∪ [Pmin

Diesel, Pmax Diesel]

• Charge/discharge power: Pin ∈ [0, Pmax

in

], Pout ∈ [0, Pmax

• ut ]
• Slack power (excess/default): Pexcess, Pdefault

External data:

• Solar power Psolar and power load Pload

Dynamics: ˙ SOC(t) = 1 Qbat

• Pin(t)ρinρinv

in − Pout(t)ρoutρinv

• ut

Optimal control formulation (2/3)

Objective function

• Fuel consumption for diesel generator
• Consumption depending on power output
• Switching cost when generator is turned on

Cdiesel T C(t)dt + NswitchCswitch

• 20

40 60 80 100 120 5 10 15 20 25 30 35 40 CONSUMPTION FOR DIESEL GENERATOR POWER (kW) CONSUMPTION (l/h) ACTUAL DATA LINEAR MODEL: C = P / 3 C = 0.471426 pow (P+1E−4,0.9)

• Price of default power (external source)

Cdefault T Pdefault(t)dt

• Cost of the battery aging (SOH ∈ [0, 1] the State of Health)

Cbattery(SOH(T) − SOH(0))

Optimal control formulation (3/3)

Constraints

• Power equilibrium at all times

Psolar + Pdiesel + Pout + Pdefault = Pload + Pin + Pexcess

• SOC periodicity

SOC(0) = SOC(T)

• Maximal charging power

if SOC < SOC1 : Pin < Pmax

in

if SOC > SOC1 : Pin < Pmax

in

(SOC−SOC1)2 (SOC1−1)2

Summary

• Model with nonlinearities and non-convex objective
• Control, state and mixed control-state constraints
• Switching cost for diesel generator (boolean variable)
• Estimate battery aging under non-nominal conditions
• Stochastic input data (solar power, power load)
• Requires fast computation times (rolling horizon)

Current approach: mixed integer* linear programming

* for the switching cost of the diesel generator Linearization of the model then solving with CPLEX Prediction (neural network) for solar power and domestic demand

New approach: continuous optimal control framework

• No need to linearize the equations
• Local or global optimization methods
• Optimality conditions: insight on the solutions
• Possible expansion to the stochastic case

Optimal control methods

• 1. Direct transcription approach
• 2. Dynamic programming with switchings

→ BOCOP toolbox (www.bocop.org)

Direct transcription

Sometimes referred to as ’Discretize then optimize’

• Discretization in time: state/control variables → vectors
• Rewrite objective and constraints
• Dynamics discretized as additional constraints

(any generalized RK formula, implicit or explicit)

• Solve the resulting NLP problem (SQP, interior point)

Advantages and disadvantages + Easy to implement + Good precision attainable with higher CPU/RAM cost

• Local method (reasonably good convergence)
• No easy way to include the switching cost

Dynamic programming

Based on Bellmann’s optimality principle

• Discretization in both time and state/control
• Compute the Value Function over the whole time × space grid

(backwards in time, using interpolation over the grid) V (tk, xi) := min

u∈U

• h0l(u, xi) + ˜

V (tk+1, xi + h0f (u, x))

• Reconstruct the trajectory for given initial conditions

Advantages and disadvantages + Global method (no iterations, known complexity) + Does not require differentiability, can handle switchings + Can be extended to the stochastic case

• Precision limited by full discretization (time,state, control)
• high CPU/RAM cost, curse of dimension

(Indirect method)

Sometimes referred to as ’Optimize then discretize’

• Apply Pontryagin’s Maximum Principle on continuous OCP
• Solve the resulting BVP problem (shooting, collocation)

→ leads to solving a nonlinear system Advantages and disadvantages + Very high precision for small CPU times

• Local method, starting point can be tricky (adjoint variables)
• Applying PMP is sometimes complicated

(state constraints, singular arcs)

Numerical simulations

• Timeframe: 48h, time step: 15 minutes
• Starting and final SOC set to 0.7
• Interpolated historical data for solar power and power load
• CPU times: a few seconds (similar to MILP)

Test case: winter and summer sample data

5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 SUMMER TEST CASE WINTER TEST CASE 5 10 15 20 25 30 35 40 45 50 4 6 8 10 12 14 16 18 20 SUMMER TEST CASE WINTER TEST CASE

Solar power Power load

Direct method solutions (winter / summer)

5 10 15 20 25 30 35 40 45 50 −20 20 40 60 80 100 TIME (H) SOC * 100 SLACK INVERTER

• INV. LIMIT

DEMAND SOLAR DIESEL 5 10 15 20 25 30 35 40 45 50 −20 −10 10 20 30 40 50 60 70 80 TIME (H) SOC * 100 SLACK INVERTER

• INV. LIMIT

DEMAND SOLAR DIESEL

• Diesel is on at consumption peaks, with lots of switchings
• Battery discharge during the night when diesel is off
• Solar power excess is used to charge battery
• In winter maximal charging limit can be saturated
• Quadratic regularization to improve numerical behaviour
• Still no switching cost or minimal diesel power...

Dynamic programming solutions (winter / summer)

5 10 15 20 25 30 35 40 45 50 −20 20 40 60 80 100 TIME (H) SOC * 100 SLACK INVERTER

• INV. LIMIT

DEMAND SOLAR DIESEL 5 10 15 20 25 30 35 40 45 50 −20 20 40 60 80 100 TIME (H) SOC * 100 SLACK INVERTER

• INV. LIMIT

DEMAND SOLAR DIESEL

• Solutions are close in terms of objective
• Less switchings for the diesel generator
• Diesel is also used to charge the battery
• Can be improved with information from Pontryagin’s Principle

(use the optimality conditions on the control)

Battery aging (1/2)

Overall operating cost includes battery system maintenance Historical data at Huatacondo over 20 years: Cost of replacing the batteries ≈ Diesel cost ! → Estimate battery aging and add it in objective function Main difficulties

• BESS operates at irregular charge / discharge cycles
• Strong restrictions on the model
• few available variables (SOC, power)
• no temperature, intensity or voltage data
• no identified ’cycles’
• must be fast in terms of computation

Battery aging (2/2): weighted Ah throughput

Simple model from Svoboda et al, relies solely on SOC

• Battery State of health based on total

throughput of the battery over its life

• Nominal case weighted by a certain

severity factor depending on SOC ∆SOH = 1 Ahbattery T SF(t)Iout(t)dt

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 SOC SF

+ easy to implement in the optimization

• very abstract model, is it meaningful ?

Effect of the battery aging in the cost

Winter test case Battery aging Diesel cost Battery cost Total cost no battery aging 29409.1 (13216.6) (42625.7) weighted Ah model 30236.2 9945.4 40181.6 Summer test case Battery aging Diesel cost Battery cost Total cost no battery aging 32115.5 (10752.5) (42868.0) weighted Ah model 32639.6 5650.9 38290.1 Slight increase in diesel consumption Significant decrease of the estimated battery cost → lower global cost (according to battery aging model)

Solution with battery aging cost

Summer test case

5 10 15 20 25 30 35 40 45 50 −20 20 40 60 80 100 TIME (H) SOC * 100 DIESEL SLACK INVERTER

• INV. LIMIT

SOLAR 5 10 15 20 25 30 35 40 45 50 20 30 40 50 60 70 80 90 100 SOC x 100 NO BATT. AGING AGING MODEL 2

• SOC is overall higher, to benefit from lower severity factor
• Diesel is (almost) not used to charge battery anymore

Ongoing work (1/2)

More refined battery aging model, with effect on characteristics Requires a more complex model and expensive experimental data

Ongoing work (2/2)

Stochastic model for power load and solar power Sample data over 1 week

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2 4 6 8 10 12 14 16 18 20 FVPower − April,16 to 20, 2012 Hours KWat

day 1 day 2 day 3 day 4 day 5 mean

Load April, 16 to 20, 2012 Hours KWat

day 1 day 2 day 3 day 4 day 5 mean

Thanks !