*A new method is being tested at the 1,450 MW Ita plant in Brazil to provide long-term production optimization of hydropower plants, using setpoints provided by the country's independent system operator.*

**By Marcelo Marcel Cordova, Erlon Cristian Finardi, Fernando Antonio Camargo Ribas, and Felipe Azevedo Brown do Coutto**

This article has been evaluated and edited in accordance with reviews conducted by two or more professionals who have relevant expertise. These peer reviewers judge manuscripts for technical accuracy, usefulness, and overall importance within the hydroelectric industry. |

Hydroelectric plants provide about 80% of all electricity production in the Brazilian National Interconnected System. Because of the uncertainties, mathematical complexities and large size of this system (installed capacity of 115 GW as of August 2011), operational planning must be performed for different time horizons.^{1,2} In Brazil, an independent system operator (ISO) has optimization models consolidated for short-term and medium-term problems.^{3,4} But for long-term scheduling, the ISO does not have a model that considers the inherent nonlinearities and non-convexities.^{5} Hence, the long-term schedule is based on operational guidelines from the short-term model and aims to provide, for each half hour of the following day, a target for hydro plant generation.

To maintain operational coordination, it is important that hydro generators seek not to violate the ISO's target while dispatching units appropriately. The goal is to use the minimum amount of water to meet the generating target provided. The key (and most difficult) aspect is to determine precisely the discharge of each generating unit.

This article describes a system for real-time performance evaluation and optimization of energy production used at Tractebel Energia's 1,450 MW Ita plant on the Uruguay River in southern Brazil. Using data collected from level, pressure and turbine discharge sensors, the system applies optimization algorithms to provide points with the lowest water consumption for different operating conditions.

**Hydropower function**

Power production of a unit can be described as:1

Equation 1

## gp_{i} = 10^{-6} x g x σ x q_{i} x nh_{i} x η_{i}^{ht} – tml_{i} - ggl_{i}

where:

— *gp _{i}* is power output at generator terminals of unit

*i*, in MW;

— 10^{-6} is a factor to convert watts to MW;

— *g* is acceleration of gravity, in m/sec^{2};

— σ is density of water, in kg/m^{3};

— *q _{i}* is discharged outflow of unit

*i*, in m

^{3}/sec;

— *nh ^{i}* is net head of unit

*i*, in m of water column;

— *η _{i}^{ht}* is hydraulic efficiency of turbine

*i*;

— *tml ^{i}* is mechanical losses of unit

*i*, in MW; and

— *ggl ^{i}* is generator global losses of unit

*i*, in MW.

The gravitational acceleration is calculated based on power plant latitude and elevation relative to sea level.^{6} The density of water is a function of its average temperature and plant elevation.^{6}

Net head is defined as:

Equation 2

## nh^{i} = fbl – trl – k_{i}^{0}q_{i}^{2} - hll_{atm}

where:

— *fbl *is forebay level, in m;

— *trl* is tailrace level, in m, which is a function of the plant discharged outflow, composed of the total turbine outflow plus the spillage, in m^{3}/sec;

— *k _{i}^{0}* is the coefficient of hydraulic load losses of unit

*i*, in sec

^{2}/m

^{5}; and

— *hll _{atm}* is the hydraulic loss due to the difference of atmospheric pressure between the forebay level and tailrace level, in meters of water column.

Obtained from a level sensor, *fbl* is considered a constant, given that in real-time operation it has small variation throughout a day. On the other hand, *trl* is obtained from a level sensor in real-time operation, and in the module it is modeled by a set of polynomial functions. The third term in Equation 2 represents a mathematical approximation to the effect of hydraulic load losses. Inclusion of these losses is a demanding task.

Finally, the losses *tml _{i} *and

*ggl*must be defined. The turbine mechanical losses are obtained through field tests and may be divided in three portions: losses due to the shaft seals, thrust bearing, and mechanical friction in the guide bearings. The portion due to shaft seals is considered constant. The third one is modeled as a function of

_{i}*gp*.

_{i}The losses due to the thrust bearing are obtained in a field test that provides a curve relating the losses to *gp _{i}*. These losses are divided into two portions, one referent to the turbine and one to the generator.

Generator global losses consist of electric losses, friction in the guide bearings and the already cited portion of thrust bearing loss. Electric losses are obtained through field testing and depend on generator electric apparent power output. Losses due to friction in the guide bearings may be modeled as a function of *gp _{i}* or may be included in electric losses.

Determination of *tml _{i}* and

*ggl*are different in real-time evaluation and optimization. In real time, cubic spline interpolations are made using the current operating point. But in the optimization module, polynomial functions must be fitted to represent these losses. For modeling, in the module the turbine and generator losses are each modeled by one polynomial.

_{i}## Determination of real-time parameters

One of the modules of the implemented system, real-time processing, aims to determine many variables related to plant energy production for the current operating point. A key feature of real-time processing is the data obtained from a variety of meters and sensors, including level at the trashracks, water temperature, pressure at the spiral casing, slide valve percentage, active and apparent power output, ultrasonic turbine outflow, Winter-Kennedy turbine outflow and tailrace level.

The presence of all these sensors is ideal to adequate visualization of plant operating condition, but not all of them are essential. For example, the level at the trashracks and pressure at the spiral casing are used to determine some hydraulic load losses. However, one can use only theoretical hydraulic load losses.

The discharged outflow of a unit can be obtained from ultrasonic sensors, calculated based on pressure measurements using the Winter-Kennedy method or calculated using a hill chart. In the implemented system, all three methods can be used, but the Winter-Kennedy method is not used due to its low accuracy.

For the method with ultrasonic sensors, the equipment gives the discharged outflow directly. For the hill chart method, outflow is determined in conjunction with other variables. Considering these aspects, the only essential measurements to the operation of the system are *fbl*, *trl*, *gp _{i}*,

*gap*. Details on the steps involved in the real-time evaluation are available.

_{i}^{7}

*Net head and hydraulic load losses*

The total hydraulic load losses represent the sum of load losses in each part of the hydraulic circuit, including the canal intake, trashracks, penstock and draft tube. Hydraulic losses can be determined in three ways:

— Theoretical: all coefficients are theoretical and the the losses related to trash on the trashrack are neglected;

— Measured: hydraulic load losses are calculated based on measured pressure at the spiral casing. All illustrated losses can be determined; and

— Mixed: losses in the trashracks and due to trash are calculated using the measurement of level at the trashracks, and losses in the penstock are theoretical.

Hydraulic load losses due to difference of atmospheric pressure are available.^{8}

Using a pressure meter at the inlet of the spiral casing, it is possible to determine the hydraulic load losses of the entire hydraulic circuit, *hll ^{med}_{adi}*, which includes the canal intake, trashrack, trash and penstock. Using a level sensor at the water intake, one can determine the hydraulic load loss in the canal intake, trashrack and due to trash. The sum of hydraulic load losses in the canal intake, trashrack (due to trash), penstock and draft tube results in the hydraulic load losses of the unit,

*hll*. The coefficient of hydraulic load losses of the generating unit,

_{gu,i}*k*is given by:

^{0}_{i}Equation 3

Each hydraulic load loss has a power loss, determined by:

Equation 4

*phll*_{i} = 10^{-6} x g x σ x hll_{i} x q_{i}

_{i}= 10

^{-6}x g x σ x hll

_{i}x q

_{i}

where:

— *phll _{i}* is power loss due to a hydraulic load loss, in MW; and

— *hll _{i}* is one of the hydraulic load losses presented, in m.

*Hydraulic efficiency of the turbine*

The turbine hydraulic efficiency represents the ratio between its output and input power. When measuring discharged outflow using a sensor, this value is obtained directly. If a sensor is not available, it is necessary to use the manufacturer-provided hill chart for the unit, which is composed by a set of triplets relating output power and net head with hydraulic efficiency. For the Ita plant, the available curve has output power data from 90 to 330 MW, with an interval of 1 MW, and net head data from 80 to 110 m, with an interval of 1 m. A cubic interpolation is performed to increase resolution of the curve.

In real-time evaluation, linear interpolations are performed on the already interpolated curve, providing a reasonable approximation to the current operating point with low computational cost.

Another point to note is that the provided curve refers to a prototype, in which the temperature is kept constant at 25.2 degrees Celsius. In real-time operation, a transposition is performed to the current temperature.^{6}

## Optimization of energy production

The real-time optimization of energy production is also made by the system. For the Ita plant, the ISO provides a generation setpoint for each unit. Thus, it is interesting for the generating agent to know how far the setpoint provided is from the best operating point for the current state of the plant.

Having this information, the real-time optimization model implemented is given by a nonlinear mixed integer optimization problem.^{7}

Minimizing discharged outflow for the generation determined occurs through the distribution of power between the units. The ratio between the discharged outflow and generating power is known as the specific consumption, and its minimization is used as the objective function.

For the Ita plant, in which there are five units, the state space generated by the combination of on and off units is small. Thus, the problem is solved in an enumerative way, and the combination with the lowest consumption is chosen.

Input data for the optimization module consists of: forebay level, tailrace level, total plant outflow (discharged outflows plus spillage), density of water, generator mean power factor, hydraulic loss due to the difference of atmospheric pressure between the forebay and tailrace levels, and coefficients of hydraulic load losses of units. These data are obtained from the real-time module.

*Turbine hydraulic efficiency*

In the hill chart provided by the turbine manufacturer, hydraulic efficiency is a function of its power output. However, the optimization model has the generating units discharge as decision variables. Thus, the hill chart must be transformed, point by point, in a curve dependent of the net head and the unit discharge.

Figure 1, shows the hill chart obtained using this procedure. The forbidden zone (which represents limits of discharge) is delimited by dashed lines. To prevent an operating point lying outside the feasible region, the dashed lines are fitted by polynomials, which are modeled in the optimization problem as turbine outflow bounds.

In the optimization, it is necessary to represent this curve by polynomial functions, usually a second-order polynomial.^{9} However, due to the nonlinearities of the original surface, the nonlinear regression for this polynomial gives a mean relative error of 0.464% and maximum relative error of 3.054% in relation to the original curve. For real-time optimization, these values are considered too high. Using a third-order polynomial still gives high errors — mean and maximum relative errors are 0.335% and 2.057%, respectively.

To minimize the regression errors, the original curve is divided in many segments, and the nonlinear regressions are performed individually, i.e., each segment has its own polynomial.

Segmentation for Ita is performed as:

— Determine the limits of each segment, considering that its size must be 5% of the total interval of net head and 5% of the total interval of unit discharge;

— Classify the triplets in each segment;

— Join neighbor segments if one has less than 20 triplets, aiming to reduce the occurrence of ill-conditioned polynomials;

— Perform nonlinear regressions for each segment, with third-order polynomials.

With this approach, 352 segments are created for the Ita hill chart, each with its own polynomial. The mean relative error is 0.006% and the maximum relative error is 0.168%. It is important to note that segmentation is performed only once, when the discretization of the hill chart is done.

*Tailrace level*

In the real-time processing, the tailrace level is obtained directly from a level sensor. However, in the optimization module the tailrace level must be modeled as a function of the total plant discharge, which consists of the sum of unit discharges and the spillage.

The Foz do Chapeco plant is downstream of the Ita plant, so the forebay level in Foz do Chapeco affects the tailrace level in Ita. Thus, two tailrace curves were determined through field tests: one in which the forebay level in Foz do Chapeco is 264 m and another for 265 m.

For each curve, two polynomials are fit: one for the interval 0 to 2,000 m^{3}/sec, and another for the points above, thus reducing regression errors.

Before executing the optimization algorithm, the system obtains, from real-time operation, the total water discharge and measured tailrace level. With these data, a new tailrace curve is created, through the weighing between 264 and 265 m curves (or an extrapolation, if the obtained point is below 264 m curve or above 265 m curve).

*Optimization levels*

A variation of the optimization model consists of eliminating the constraint of meeting the demand provided by the ISO. Thus, the optimal operating points for each number of units can be determined. With these points, it is possible to determine an interval of generation such that the specific consumption is not below a given optimization level. This ensures that even if the generating units are not in its optimal point, they will be within a range below that optimum. The optimization model to determine this interval is given.^{7}

**System of production optimization**

The system architecture was based on virtual instrumentation technology from National Instruments. The programming language NI LabVIEW was chosen due to its high productivity in instrumentation systems and data acquisition and because it provided a nonlinear programming solver. A centralized Ethernet network was implemented to concentrate the communication of different equipment that provides information to the system.

The software was developed using a three-tier architecture using Web Service technology, aiming to separate the presentation and application layers.

Considering all three methods for hydraulic losses, system variables are determined for six different combinations.

The real-time evaluation module is executed every 10 minutes, using moving averages with a 5-minute window for the measurements. Thus, instantaneous oscillations in the values are avoided. Real-time processing runs if the unit has an output power of 195 to 293 MW. Theoretical limits (allowed operating range) of the generators are 200 and 290 MW. This slack is used so that, in real operation, conditions slightly below the minimum or above the maximum are not discarded. In total, 100 variables related to real-time operation are determined (for each combination of turbine discharge/hydraulic losses).

The real-time optimization module is executed after real-time processing. The optimization routines related to optimization levels, which are more expensive computationally, are executed once an hour. Considering that the only variable that may have some variation with regard to optimization without the constraint of meeting the demand provided by the ISO is the forebay level, and considering that this variation is usually small within a range of minutes, it is not necessary to run that problem as frequently as the original optimization model.

Via the web interface, users can view the operating condition of the plant and generating units, data of sensors, results of real-time optimization and optimization levels, as well as generate system reports. Some reports can be used to monitor plants or generating units' variables through time. Others offer histograms of turbine and generator efficiencies, which can be used in studies of repowering of generating units, for example.

**Numerical results**

On Dec. 21, 2012, at 14:27 p.m., there was one unit in operation. Forebay and tailrace level were 365.75 m and 264.49 m, respectively. Output power was 199.98 MW. Water temperature was 16.7 C. Level at the trashracks was 365.02 m. Pressure at the spiral casing was 9.73 Bar. Turbine discharge measured by ultrasonic flow meter was 220.83 m^{3}/sec.

The system calculates the real-time processing variables for six cases — two methods of measurement of discharged outflow and three for hydraulic losses.

Considering the case with turbine discharge calculated by hill chart and theoretical hydraulic losses, the turbine hydraulic losses correspond to about 71% of total losses. Efficiencies were: 91.17% globally; 92.22%, generating unit; 98.52%, generator; 93.61%, turbine; 93.7%, hydraulic components; and 99.9%, mechanical components.

With respect to hydraulic losses, Figure 2, shows the percentage division considering theoretical losses. The penstock contributes to more than half of the losses, and there are no losses due to trash in the trashracks, as these are not calculated when the hydraulic losses are obtained by a model, not by measurement.

Using the hill chart method, the calculated discharged outflow for theoretical hydraulic load losses was 221.45 m^{3}/sec. Taking as reference the outflow measured by flowmeters, the relative error obtained was 0.28%. Considering that the measurement error of the ultrasonic sensor is on the order of 0.5%, it can be concluded that the results are within the margin of error. This shows that, with a detailed modeling of the hydropower function, one can obtain results consistent with ultrasonic sensors, known for their high accuracy.

*Real-time optimization*

The first case for analysis of optimization routines corresponds to Dec. 20, 2012, 10:54 a.m., when the ISO determined that Units 1 and 2 dispatched 290 MW and Unit 4 dispatched 200 MW. Unit 3 should operate as a synchronous compensator, and Unit 5 was unavailable. Forebay and tailrace level were, respectively, 365.7 m and 264.78 m. Total water release was 889.59 m^{3}/sec. Hydraulic loss due to the difference of atmospheric pressure between the forebay level and tailrace level was 0.119 m. The coefficients of hydraulic losses of the generating units were 1,8382 x 10^{-5}, 1,9013 x 10^{-5} and 2,0718 x 10^{-5} for Units 1, 2 and 4, respectively.

Table 1, shows the current dispatch of the plant and the optimal dispatch to meet that power.

For the same power, the optimal total turbine discharge is 26.42 m^{3}/sec lower than the current point. This difference is reflected in the specific consumptions: 1.1499 [m³/(sec·MW)] at the current point and 1.1158 [m³/(sec·MW)] at the optimal point. It is also observed that the difference in coefficients of hydraulic load loss of the generating units have a small impact on the optimization because the difference between highest and lowest power is 0.62 MW.

The second case examined refers to Dec. 20, 2012, 5:35 p.m. The output power set by the ISO was 200 MW for Unit 4. Units 1, 2 and 3 were operating as synchronous compensators, and Unit 5 was unavailable. The result of real-time processing indicates Unit 4 was dispatching 202.21 MW, with a turbine discharge of 224.55 m^{3}/sec, determined by hill chart and considering theoretical load losses. In optimization, for the same power, the optimal turbine discharge is 224.39 m^{3}/sec. This difference occurs because interpolations are used to determine the losses, and the tailrace level is a measurement. In the optimization, losses and the tailrace level are calculated using polynomials. Considering the difference of only 0.16 m^{3}/sec, we conclude that the modeling of losses and the tailrace level using polynomial functions and the segmentation of the hill chart are adequate to describe mathematically the plant in the optimization problem presented.

In addition, the system provides results for the optimization levels, which do not depend on current output power of the plant. Table 2 shows the results for one unit. The unit is up to 0.25% below its optimum if its output power is between 214.42 and 230.76 MW. This kind of information is important for monitoring the efficiency of the plant and can contribute for exchanging information with the ISO, when necessary.

## Conclusions

Validation of the system at Ita has shown that detailed modeling of the hydropower production function using the hill chart method gives errors of the same order of magnitude as those obtained with ultrasonic flowmeters, which cost hundreds of thousands of dollars per unit. It was also observed that the proper distribution of power between units can substantially reduce the amount of water used to generate the same amount of energy.

A complete validation of the system will only be possible after some time in operation. We have yet to find how much water can be saved in the long term using the optimization model.

The system also serves to support the decision to modernize and repower units, as a way to improve a plant's technical and economic performance, and to support discussions around the criteria in calculating a plant's guaranteed energy.

## Notes

1. Matos, V.L., and E.C. Finardi, “A Comp-utational Study of a Stochastic Optimization Model for Long Term Hydrothermal Sched-uling,” *International Journal of Electrical Power & Energy Systems*, Vol. 43, No. 1, 2012.

2. Finardi, E.C., E.L. Silva, and C. Sagastizabal, “Solving the Unit Commitment Problem of Hydropower Plants via Lagrangian Relaxation and Sequential Quadratic Pro-gramming,” *Computational & Applied Math-ematics*, Vol. 24, No. 3, 2005, pages 317-341.

3. Silva, E.L., and E.C. Finardi, “Parallel Processing Applied to the Planning of Hydrothermal Systems,” *IEEE Transactions on Parallel and Distributed Systems*, Vol. 14, No. 8, 2003, pages 721-729.

4. Santos, M.L.L., E.L. Silva, E.C. Finardi, and R.E.C. Goncalves, “Practical Aspects in Solving the Medium-term Operation Planning Problem of Hydrothermal Power Systems by Using the Progressive Hedging Method,” *International Journal of Electrical Power & Energy Systems*, Vol. 31, No. 9, 2009, pages 546-552.

5. Finardi, E.C., and M. R. Scuzziato, “Hydro Unit Commitment and Loading Problem for Day-ahead Operation Planning Problem,” *International Journal of Electrical Power & Energy Systems*, Vol. 44, No. 1, 2013.

6. *Hydraulic Turbines, Storage Pumps and Pump-turbines – Model Acceptance Tests*, IEC 60193, International Electrotechnical Commission, Geneva, Switzerland, 1999.

7. Cordova, Marcelo Marcel, et al, “A Support System for Real-Time Performance Evaluation and Production Optimization in a Hydro Power Plant in the Brazilian System,” *Proceedings of HydroVision International 2013*, PennWell Corp., Tulsa, Okla., 2013.

8. “Módulo 8: Programacao Diaria da Operacao Eletroenergetica,” Procedimentos de Rede, Operador Nacional do Sistema Eletrico, 2009.

9. Diniz, A.S.L., and M. E. P. Maceira, “A Four-dimensional Model of Hydro Generation for Short-term Hydrothermal Dispatch Problem Considering Head and Spillage Effects,” *IEEE Transactions on Power Systems*, Vol. 213, No. 3, 2008, pages 1298-1308.

*Marcelo Marcel Cordova, Eng., and Erlon Cristian Finardi, D.Eng., are researchers in the Laboratorio de Planejamento de Sistemas de Energia Eletrica at the Universidade Federal de Santa Catarina. Fernando Antonio Camargo Ribas, Eng., is a specialist in hydraulic turbines in the Department of Hydraulic Generation at Tractebel Energia – GDF Suez. Felipe Azevedo Brown do Coutto, M.Eng., coordinates development and automation at M&D Monitoracao e Diagnose.*